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\newcommand{\Ran}{\operatorname{Ran}} \newcommand{\spn}{\operatorname{span}} \newcommand{\TV}{\mathrm{TV}} \newcommand{\BL}{\mathrm{BL}} \newcommand{\dBL}{d_{\mathrm{BL}}} \begin{document} \frontmatter \npptitlepage{Mobility Edge for\\ Sparse Complex Gaussian Matrices}{Patrick Lopatto} \begin{authorsnote} I have been interested in the problem considered in this manuscript since the completion of \cite{AggarwalBordenaveLopatto2022mobility}. It seemed clear that the transfer operator strategy in \cite{AggarwalBordenaveLopatto2022mobility} was the right way to approach the problem, but also that there were significant model-specific challenges in implementing it in the context of sparse random matrices. For various reasons, my attention became focused on other things, but I always intended to return to this question. Shortly after the release of GPT-5.6 Sol, I became curious about whether the model could complete the details of the argument given an appropriate high-level sketch of the proof, and indeed it could (after exchanging many messages). I had hoped to produce a polished, human-written version of this manuscript, but given the current pace of mathematical progress and recent improvements in LLM writing abilities, it does not feel appropriate to delay the dissemination of these results for the rather long time that would take. The current document, while not perfect, should be accessible to those with the relevant background. Further, given the implications these results have for certain predictions in the physics literature, as described in the introductory material below, I thought it was best to release them now. With the exception of this note and small tweaks to the introduction to add some recent references, the entire document is machine-written. I have taken care to supervise this writing to ensure that the proofs meet a minimum standard of readability and that previous literature is appropriately cited. I welcome suggestions concerning mathematical corrections, omitted citations, and attribution of ideas. P.L.\ was partially supported by NSF grant DMS-2450004. \end{authorsnote} \begin{abstractpage} We consider the sparse Hermitian random matrix supported on the Erd\H{o}s--R\'enyi graph $G(n,p/n)$, whose nonzero entries are independent complex Gaussian variables of variance $1/p$, together with its local limit, a weighted Poisson--Galton--Watson tree. We prove that for every fixed and sufficiently large $p$, the limiting operator on the tree conditioned on survival has exactly one positive mobility edge $E_c(p)$: almost surely, its spectrum is purely absolutely continuous on $(-E_c(p),0)\cup(0,E_c(p))$ and pure point outside $[-E_c(p),E_c(p)]$. Moreover, $E_c(p)/\sqrt p$ converges to $\sqrt\pi/2$ as $p$ tends to infinity. Thus the mobility edge lies far in the tail of the spectrum, rather than near the edge $2$ of the limiting semicircle law, as had been predicted in the physics literature; the spectrum is absolutely continuous in a region whose spectral mass tends to zero as $p\to\infty$, through a resonance mechanism discovered by Aizenman and Warzel. The mobility edge is the unique zero of a free energy associated with a distinguished real solution of the cavity equation, and this free energy crosses zero transversally. We also show that the giant component of the finite graph inherits the dichotomy, in the sense that the local spectral masses in small energy windows are uniformly integrable over vertices below $E_c(p)$ and concentrate on a vanishing fraction of vertices above it. The proofs combine a transfer-operator description of the free energy, the Simon--Wolff criterion applied to the squared moduli of the Gaussian weights, a resonance argument, and an analysis of the cavity equation at large degree. \end{abstractpage} \tableofcontents \mainmatter \chapter{Introduction}\label{sec:intro} \section{Background}\label{sec:intro-background} Disordered quantum systems often exhibit one of two contrasting behaviors. In a \emph{localized} phase, eigenfunctions are concentrated in bounded regions of space and the system is an insulator. In a \emph{delocalized} phase, eigenfunctions are spread out and transport is possible. Anderson proposed in \cite{Anderson1958absence} that sufficiently strong disorder localizes all eigenfunctions of a tight-binding Hamiltonian. At weaker disorder, localized and delocalized states may coexist at different energies, separated by \emph{mobility edges}, a notion introduced by Mott \cite{Mott1967electrons}; see also \cite{EconomouCohen1972existence} and the review \cite{EversMirlin2008anderson}. Localization is by now well understood mathematically in many regimes, for instance through the fractional moment method of Aizenman and Molchanov \cite{AizenmanMolchanov1993localization,Aizenman1994localization}. We refer to \cite{AizenmanWarzel2015random} for an account. Delocalization is much harder to prove, and tree graphs are the main setting in which both phases, and the transition between them, are accessible. \paragraph{Trees.} The Anderson model on the Bethe lattice was studied by Abou-Chacra, Anderson, and Thouless in \cite{AbouChacraAndersonThouless1973}, who wrote self-consistent equations for the local Green function that are exact on the tree. They predicted that the localized phase corresponds to a real solution of these equations, and they located its stability limit by linearizing the equations around this solution. The case of energies near the band edges was treated in \cite{AbouChacraThouless1974}. Kunz and Souillard obtained exact results on the transition between extended and localized states on the Bethe lattice \cite{KunzSouillard1983localization}, and Miller and Derrida studied localization on trees through a weak-disorder expansion \cite{MillerDerrida1994weak}. This linear stability criterion has since been reformulated in several ways: through the free energy of a directed polymer on the tree \cite{DerridaSpohn1988polymers,MonthusGarel2009anderson,KravtsovAltshulerIoffe2018nonergodic,BiroliTarzia2020anomalous}, through supersymmetric field theory \cite{Zirnbauer1986localization,Efetov1987anderson,MirlinFyodorov1991localization}, and through the largest eigenvalue of an integral operator acting on real Green functions \cite{BiroliSemerjianTarzia2010anderson,TarquiniBiroliTarzia2016level,ParisiPascazioPietracaprinaRosScardicchio2020anderson}. In these formulations the criterion is evaluated at the symmetric point of a symmetry of the linearized kernel, which goes back to \cite{AbouChacraAndersonThouless1973}; see \cite{ParisiPascazioPietracaprinaRosScardicchio2020anderson} for a discussion. A pedagogical account of the cavity approach, including its analysis at large connectivity, is given in \cite{PradoBandeiraVecchiarelliTarzia2026anderson}. Biroli, Semerjian, and Tarzia computed the mobility edge and the density of states of the Anderson model on Bethe lattices with the cavity method in \cite{BiroliSemerjianTarzia2010anderson}. More recently, the delocalized phase on trees and random regular graphs has been studied in detail, in particular the question of its ergodicity \cite{BiroliRibeiroTeixeiraTarzia2012difference,DeLucaAltshulerKravtsovScardicchio2014anderson,TikhonovMirlinSkvortsov2016anderson,KravtsovAltshulerIoffe2018nonergodic,BiroliTarzia2018delocalization,BiroliTarzia2020anomalous,TikhonovMirlin2021from,BiroliHartmannTarzia2022critical}. The critical properties of the transition and the localized phase near it were studied in \cite{TarquiniBiroliTarzia2017critical,RizzoTarzia2024localized}. On the mathematical side, Klein proved that the Anderson model on the Bethe lattice has absolutely continuous spectrum at weak disorder \cite{Klein1994absolutely,Klein1998extended}; other proofs and extensions were given in \cite{AizenmanSimsWarzel2006stability,FroeseHaslerSpitzer2007absolutely,KellerLenzWarzel2012absolutely,DroginSmart2025regular,JaksicLastWarzel2026note}. Aizenman and Warzel developed a resonance mechanism for delocalization in \cite{AizenmanWarzel2011extended,AizenmanWarzel2013resonant}. They proved that the spectrum is absolutely continuous wherever a certain free-energy function, denoted $\varphi(1;E)$ in \cite{AizenmanWarzel2013resonant} and describing the decay of fractional moments of the Green function along paths, exceeds the logarithm of the inverse branching number. As a consequence, at weak disorder the absolutely continuous spectrum extends well beyond the band of the hopping operator, into a Lifshitz-tail regime \cite{Lifshitz1964energy} of very small density of states (see \cite{BapstSemerjian2011lifshitz} for Lifshitz tails on the Bethe lattice). For bounded potentials there is no mobility edge of the usual kind at weak disorder \cite{AizenmanWarzel2011absence}; see also the survey \cite{Warzel2013surprises}. Bapst analyzed this free energy at large connectivity in \cite{Bapst2014large} and confirmed the prediction of \cite{AbouChacraAndersonThouless1973} for the critical disorder. Building on these works, Aggarwal and Lopatto proved in \cite{AggarwalLopatto2025mobility} that mobility edges exist for the Anderson model on the Bethe lattice with unbounded potential at large degree, near the locations predicted in \cite{AbouChacraAndersonThouless1973}. Liu and Lopatto transferred this result to random regular graphs in \cite{LiuLopatto2026mobility}. For random trees, much less is known. The spectral measure of the unweighted adjacency operator of the Poisson--Galton--Watson tree, the local limit of the Erd\H{o}s--R\'enyi graph, has a dense set of atoms \cite{Salez2015every,BhamidiEvansSen2012spectra}, and its continuous part was studied in \cite{BordenaveSenVirag2017mean,Salez2020spectral,CosteSalez2021emergence}. Absolutely continuous spectrum for multi-type Galton--Watson trees close to trees of finite cone type was established by Keller \cite{Keller2012absolutely}. Arras and Bordenave proved in \cite{ArrasBordenave2023existence} that, conditionally on survival, the adjacency operator of a Poisson--Galton--Watson tree with large mean degree has a nontrivial absolutely continuous spectral component, which carries most of the spectral mass. \paragraph{Sparse random matrices.} The spectra of sparse random matrices, with a bounded mean number of nonzero entries per row, were first studied in physics in \cite{RodgersBray1988density,RodgersDeDominicis1990density}. Fyodorov and Mirlin showed that such matrices undergo a localization transition, and derived equations for a mobility edge and for the inverse participation ratio in \cite{FyodorovMirlin1991localization,MirlinFyodorov1991universality,FyodorovMirlinSommers1992novel}; numerical studies include \cite{Evangelou1992numerical,EvangelouEconomou1992spectral}. Biroli and Monasson proposed in \cite{BiroliMonasson1999single} that localized states in the tails of the spectra of random graphs are created by rare local defects, such as vertices of abnormally large degree; see also \cite{SemerjianCugliandolo2002sparse}, and \cite{ValigiBaronNeriBiroliCammarota2025eigenvalue} for the analogous picture for non-Hermitian sparse matrices. In the limit of high connectivity, Silva and Metz solved the resolvent equations of sparse random graphs analytically and found that all eigenvectors are extended in this limit \cite{SilvaMetz2022analytic}. The cavity method was developed for sparse matrices in \cite{RogersPerezCastilloKuhnTakeda2008cavity,Kuhn2008spectra,SuscaVivoKuhn2021cavity}, and mobility edges in the tails of the spectra of sparse matrices were computed numerically in \cite{Kuhn2008spectra,MetzNeriBolle2010localization,Slanina2012localization,Kuhn2016disentangling}. Tarzia studied the tails of the spectrum of Erd\H{o}s--R\'enyi graphs in the critical regime in \cite{Tarzia2022fully}. Most relevant for us, Cugliandolo, Schehr, Tarzia, and Venturelli studied in \cite{CugliandoloSchehrTarziaVenturelli2024multifractal} the giant component of Erd\H{o}s--R\'enyi graphs with real Gaussian edge weights, which is the real symmetric analogue of the model considered here. They found a delocalized but weakly multifractal phase at small $p$, separated by a mobility edge from localized states in the tails. The multifractality of extended states in weighted random networks was also studied by Baron and Rogers in \cite{BaronRogers2026local}. The mathematical literature on sparse Erd\H{o}s--R\'enyi graphs is extensive when the mean degree grows with the number of vertices; see for example \cite{ErdosKnowlesYauYin2013spectral,HeKnowlesMarcozzi2019local,BenaychGeorgesBordenaveKnowles2019largest,BenaychGeorgesBordenaveKnowles2020spectral}. In the critical regime where the mean degree is of order $\log n$, Alt, Ducatez, and Knowles proved in a series of works \cite{AltDucatezKnowles2021extremal,AltDucatezKnowles2021delocalization,AltDucatezKnowles2022completely,AltDucatezKnowles2023poisson,AltDucatezKnowles2024localized} that eigenvectors are delocalized in the middle of the spectrum and localized near high-degree vertices at the spectral edges, with a mobility edge between the two regimes. See also the recent work \cite{HenheikKnowles2026Bernoulli}. For bounded mean degree, the global eigenvalue distribution of weighted random graphs was studied in \cite{KhorunzhyShcherbinaVengerovsky2004eigenvalue,BordenaveLelarge2010resolvent}, and the extreme eigenvalues and their eigenvectors in \cite{HiesmayrMcKenzie2023spectral,BucdAlcheKnowles2026semilocalization}. For the model \eqref{eq:intro-model} itself, Afanasiev showed in \cite{Afanasiev2016correlation}, using supersymmetry, that for $p>2$ the second correlation function of characteristic polynomials has a GUE-type limit inside an energy interval and factorizes outside it. \paragraph{Heavy-tailed matrices.} A mobility edge was predicted by Cizeau and Bouchaud for L\'evy matrices, whose entries are in the domain of attraction of an $\alpha$-stable law with $\alpha\in(0,1)$ \cite{CizeauBouchaud1994theory}. Biroli, Bouchaud, and Potters showed that the largest eigenvalues of heavy-tailed matrices are governed by their largest entries \cite{BiroliBouchaudPotters2007top}. Tarquini, Biroli, and Tarzia computed its location in \cite{TarquiniBiroliTarzia2016level}, through the stability of the real solution of the cavity equations, and Biroli and Tarzia used cavity equations to determine the phase diagram of a L\'evy version of the Rosenzweig--Porter ensemble \cite{BiroliTarzia2021levy}. The eigenvalues and eigenvectors of L\'evy matrices were studied rigorously in \cite{BenArousGuionnet2008spectrum,BordenaveGuionnet2013localization,BordenaveGuionnet2017delocalization,AggarwalLopattoYau2021goe,AggarwalLopattoMarcinek2021eigenvector}. Aggarwal, Bordenave, and Lopatto proved in \cite{AggarwalBordenaveLopatto2022mobility} that L\'evy matrices have a mobility edge, in the sense of the imaginary part of the resolvent of their local limit and of eigenvector participation ratios, at the location predicted in \cite{TarquiniBiroliTarzia2016level}, and that it is unique for $\alpha$ close to $0$ or $1$. \paragraph{Recent advances.} Several recent preprints produced by an internal OpenAI large language model announce results on longstanding problems in localization and spectral universality. For the Anderson model with independent uniform site potentials, these include purely absolutely continuous spectrum on a nonempty energy interval at sufficiently weak disorder on $\mathbb Z^d$ for every fixed $d\ge3$ \cite{OpenAI2026AbsolutelyContinuousSpectrum}, and pure-point spectrum throughout the spectrum at every positive disorder strength on $\mathbb Z^2$ \cite{OpenAI2026PurePointSpectrum}. For random regular graphs of every fixed degree $d\ge3$, further preprints announce fixed-energy GOE bulk universality for the adjacency matrix \cite{OpenAI2026GOEUniversalityRRG} and for the Anderson model with sufficiently weak independent uniform diagonal disorder \cite{OpenAI2026AndersonUniversalityRRG}. The fixed-energy GOE statistics result for the adjacency matrix was obtained independently and simultaneously in \cite{HeHuangWang2026Bulk}. \paragraph{This paper.} In this article we establish that the sparse Hermitian matrix with complex Gaussian weights on $G(n,p/n)$ has exactly one positive mobility edge in its local limit, for every fixed sufficiently large $p$, and we determine its location to leading order. Its location, of order $\sqrt p$, is far in the tail of the spectrum, which contradicts the large-$p$ behavior predicted in the physics literature; we discuss this in \Cref{sec:intro-discussion}. We also derive consequences for the giant component of the finite graph. \section{Model and main results}\label{sec:intro-results} \subsection{The model} Fix $p>1$. For each integer $n>p$, let $G_n$ be an Erd\H{o}s--R\'enyi random graph on the vertex set $\{1,\dots,n\}$ in which each edge is present independently with probability $p/n$. Conditional on $G_n$, let $(w_{ij})_{1\le i1$, the graph $G_n$ has with high probability a unique giant component, which contains a positive fraction of the vertices, together with linearly many finite components. We write $\gc$ for the vertex set of the giant component, that is, of the largest connected component of $G_n$ (if there are several, the one containing the smallest vertex; see \Cref{def:tree-model}), $N_n=|\gc|$ for its size, and $H_n^{\gc}$ for the restriction of $H_n$ to $\gc$. Since the finite components produce a positive density of localized eigenvectors for trivial reasons, the natural object of study is the giant component. The local weak limit of $H_n$ is described by a random tree. Let $\mathcal T$ be a Poisson--Galton--Watson tree with mean offspring $p$ and root $o$, and attach to each edge $e$ an independent weight $h_e=w_e/\sqrt p$, where the $w_e$ are standard complex Gaussian variables; we set $h_{vu}=\overline{h_{uv}}$. Let $\mathcal H$ be the weighted adjacency operator of $\mathcal T$, \[ (\mathcal H f)(v)=\sum_{u\sim v}h_{vu}f(u). \] Although the degrees in $\mathcal T$ are unbounded, $\mathcal H$ is almost surely essentially self-adjoint on finitely supported vectors (\Cref{thm:tree-esa}), and we use the same notation for its closure. A uniformly chosen vertex of $G_n$ converges locally to the root of $(\mathcal T,(h_e))$, and a uniformly chosen vertex of the giant component converges locally to the root of the same tree conditioned on the event $\mathcal S$ that it is infinite. We write $\P$ for the law of the weighted tree, $\varrho_p=\P(\mathcal S)$ for its survival probability, and $\P^\infty=\P(\,\cdot\mid\mathcal S)$ for the survival-conditioned law. For an interval $I$, we say that $\mathcal H$ has \emph{purely absolutely continuous spectrum} (respectively \emph{pure point spectrum}) in $I$ if, for every vector $f$, the restriction to $I$ of the spectral measure of $f$ is absolutely continuous with respect to Lebesgue measure (respectively a countable sum of point masses). Intervals of the first kind are regions of delocalization, and intervals of the second kind are regions of localization. An energy separating a region of one kind from a region of the other kind is a \emph{mobility edge}. \subsection{Main results} Set \begin{equation}\label{eq:intro-mustar} \mustar=\E|w|=\frac{\sqrt\pi}{2}, \qquad w\sim\mathcal N_{\C}(0,1). \end{equation} \begin{theorem}\label{thm:main} There exists $p_0<\infty$ such that for every $p\ge p_0$ there exists $E_c(p)>0$ with the following properties. Almost surely with respect to $\P^\infty$, the operator $\mathcal H$ has purely absolutely continuous spectrum in \[ (-E_c(p),0)\cup(0,E_c(p)) \] and pure point spectrum in \[ (-\infty,-E_c(p))\cup(E_c(p),\infty). \] Moreover, \begin{equation}\label{eq:intro-asymptotics} \lim_{p\to\infty}\frac{E_c(p)}{\sqrt p}=\mustar=\frac{\sqrt\pi}2. \end{equation} Finally, $\P^\infty$-almost surely, $\mathcal H$ has no eigenvector with finite support and nonzero eigenvalue, and it has infinitely many linearly independent eigenvectors with finite support and eigenvalue zero. \end{theorem} Thus, for every fixed large $p$, the positive half-line contains exactly one mobility edge. The theorem makes no statement about the spectral type at the two energies $\pm E_c(p)$. The energy zero is exceptional: it is an eigenvalue of infinite multiplicity, created by pairs of leaves attached to a common vertex (localized states of this kind in diluted models go back to Kirkpatrick and Eggarter \cite{KirkpatrickEggarter1972localized}), and \Cref{thm:main} does not describe the spectral measure at zero. The mobility edge lies far in the tail of the spectrum. As $p\to\infty$, the limiting empirical spectral measure of $H_n$ converges to the semicircle law on $[-2,2]$, as a computation with the moment method shows, while $E_c(p)$ is of order $\sqrt p$. Eigenvalues near $E_c(p)$ therefore come from rare local configurations with unusually large edge weights. The constant $\mustar$ arises from a first-moment computation, which we explain in \Cref{sec:outline-largep}: at the energy $E=\xi\sqrt p$, a typical path of length $L$ in the tree contributes a product of about $L$ factors $|w_e|/(\xi p)$ to the resolvent, and there are about $p^L$ such paths. Their total contribution therefore grows or decays like $(\mustar/\xi)^L$. Our second result identifies $E_c(p)$ as the unique zero of a function defined through the cavity equation, which is a recursive distributional equation for the diagonal resolvent entry at the root. Let $\Gamma=\langle\delta_o,(\mathcal H-z)^{-1}\delta_o\rangle$ for $z$ in the upper half-plane. Its law satisfies the cavity equation \[ \Gamma\eqdist-\Bigl(z+\frac1p\sum_{j=1}^N Y_j\Gamma_j\Bigr)^{-1}, \qquad N\sim\Pois(p),\quad Y_j\sim\Exp(1), \] where the variables on the right are independent and the $\Gamma_j$ are distributed as $\Gamma$. For $E>2$, let $g(E)=(E-\sqrt{E^2-4})/2$ and $\theta=g(E)^2$. For a real energy $E$, the variable $B=-g(E)/\Gamma$ formally satisfies \begin{equation}\label{eq:intro-real-rde} B\eqdist1+\theta-\frac\theta p\sum_{j=1}^N\frac{Y_j}{B_j}. \end{equation} For every real solution of this equation we define in \Cref{sec:press} a function $\varphi(s)$, $s\in(0,2)$, as the logarithm of the Perron--Frobenius eigenvalue $\lambda(s)$ of a positive transfer operator on the real projective line. We call it the \emph{free energy}, as in \cite{AizenmanWarzel2013resonant,AggarwalLopatto2025mobility}. For $s\in(0,1)$, $\varphi(s)$ is the exponential growth rate of the expected generation sums of the $s$-th powers of the formal path products. Its limiting value $\varphi(1)=\lim_{s\uparrow1}\varphi(s)$ decides the spectral type. \begin{theorem}\label{thm:critical} Fix $\xi_0\in(0,\mustar)$. There exists $p_1(\xi_0)\ge p_0$ such that for every $p\ge p_1(\xi_0)$ the following holds. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $E\ge\xi_0\sqrt p$, the iterates of the map defined by the right-hand side of \eqref{eq:intro-real-rde}, started from the point mass at one, converge weakly to a solution $\nu_{p,E}$ of \eqref{eq:intro-real-rde}. \item Let $\varphi_E(1)$ be the free energy of $\nu_{p,E}$ at $s=1$. The function $E\mapsto\varphi_E(1)$ is continuous on $[\xi_0\sqrt p,\infty)$. It is positive on $[\xi_0\sqrt p,E_c(p))$, vanishes at $E_c(p)$, and is negative on $(E_c(p),\infty)$, where $E_c(p)$ is the energy of \Cref{thm:main}. It is real-analytic in a neighborhood of $E_c(p)$, and its derivative at $E_c(p)$ is negative. \item For every compact set $\Xi\subset[\xi_0,\infty)$, \[ \lim_{p\to\infty}\sup_{\xi\in\Xi}\Bigl|\exp\varphi_{\xi\sqrt p}(1)-\frac{\mustar}{\xi}\Bigr|=0. \] \end{enumerate} \end{theorem} \Cref{thm:critical} is an instance of the general principle, going back to Abou-Chacra, Anderson, and Thouless \cite{AbouChacraAndersonThouless1973}, that the mobility edge on a tree is located by the linear stability of the real solutions of the cavity equation. Its content is that, at large $p$, the free energy at $s=1$ of the relevant solution changes sign exactly once, and transversally. The asymptotic statement (iii) gives \eqref{eq:intro-asymptotics}. \subsection{The giant component} We now state the consequences for the finite graph. Since the spectral type of a finite matrix is trivial, we describe localization through the distribution of spectral weight over vertices in small energy windows. For an interval $I\subset\R$ and a vertex $v\in\gc$, let \[ \mu_{n,v}(I)=\langle\delta_v,\one_I(H_n^{\gc})\delta_v\rangle, \qquad \bar\mu_n(I)=\frac1{N_n}\Tr\one_I(H_n^{\gc}), \] so that $\bar\mu_n(I)$ is the average of $\mu_{n,v}(I)$ over $v\in\gc$. When $\bar\mu_n(I)>0$, let $P_{I,n}(v)=\mu_{n,v}(I)/\bar\mu_n(I)$ be the normalized local mass, and for $R>0$ define the deficit \[ \mathcal D_{I,n}(R)=1-\frac1{N_n}\sum_{v\in\gc}\min\{P_{I,n}(v),R\}. \] Set $\mathcal D_{I,n}(R)=1$ when $\bar\mu_n(I)=0$. When $\bar\mu_n(I)>0$, the deficit is the excess normalized mass above the truncation level $R$, averaged over the vertices. \begin{definition}\label{def:intro-phases} Let $I_h(E)=(E-h,E+h]$. The giant component is \emph{uniformly integrable delocalized} at $E$ if, for every $\delta>0$, \[ \lim_{R\to\infty}\limsup_{h\downarrow0}\limsup_{n\to\infty} \P\bigl(\mathcal D_{I_h(E),n}(R)>\delta\bigr)=0. \] It is \emph{spectrally condensed} at $E$ if, for every $R<\infty$ and every $\delta>0$, \[ \lim_{h\downarrow0}\liminf_{n\to\infty} \P\bigl(\mathcal D_{I_h(E),n}(R)>1-\delta\bigr)=1. \] \end{definition} In words, the giant component is uniformly integrable delocalized at $E$ if the normalized local masses in small windows around $E$ are uniformly integrable over the vertices, and it is spectrally condensed at $E$ if almost all of the normalized mass is carried by a vanishing fraction of the vertices. In both definitions the limit $n\to\infty$ is taken before the window width $h$ tends to zero. These notions concern averaged spectral projections, and they do not describe individual eigenvectors. \begin{theorem}\label{thm:giant} The following statements hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $p>1$ and every compact set $J\subset\R\setminus\{0\}$, there exist constants $0E_c(p)$. \end{enumerate} \end{theorem} \section{Discussion}\label{sec:intro-discussion} \paragraph{Comparison with the physics literature.} \Cref{thm:main} shows that, for large $p$ and apart from the eigenvalue zero, the localized states of the limiting operator are confined to energies $|E|>E_c(p)\approx\mustar\sqrt p$, far beyond the semicircle bulk $[-2,2]$. This disagrees with the large-$p$ behavior described in the physics literature on this model. For the real symmetric version of \eqref{eq:intro-model} restricted to the giant component, Cugliandolo, Schehr, Tarzia, and Venturelli state that the mobility edge tends to the edge $2$ of the semicircle law as $p\to\infty$ \cite[Section~V]{CugliandoloSchehrTarziaVenturelli2024multifractal}. This statement is an extrapolation: their numerical study covers $p\in[1.6,5]$, and the mobility edge in their phase diagram is drawn only for $p\lesssim2.5$ \cite[Section~V and Fig.~1]{CugliandoloSchehrTarziaVenturelli2024multifractal}. Similarly, Baron and Rogers describe the eigenvalues of weighted Erd\H{o}s--R\'enyi graphs outside the continuous part of the spectrum, which ends at $2+O(p^{-1})$, as produced by well-connected vertices or heavily weighted edges, with eigenvectors exponentially localized around these defects \cite[Section~VIII]{BaronRogers2026local}. This is the picture of Biroli and Monasson \cite{BiroliMonasson1999single}. \Cref{thm:main} shows that this picture is not correct at large $p$ on the limiting tree. At energies $2<|E|1$, or only for $p$ large? For small $p$, the analysis of \cite{CugliandoloSchehrTarziaVenturelli2024multifractal} suggests that the delocalized phase has unusual properties. The second concerns the spectral type at the energies $\pm E_c(p)$ and at zero. The third concerns finite graphs: it would be interesting to prove localization and delocalization of individual eigenvectors of $H_n^{\gc}$ in the two phases, and to determine their local eigenvalue statistics, in analogy with \cite{AggarwalLopattoYau2021goe,AltDucatezKnowles2023poisson}. Finally, it would be natural to study the multifractal properties of the delocalized eigenvectors predicted in \cite{CugliandoloSchehrTarziaVenturelli2024multifractal,BaronRogers2026local}. \section{Notation and conventions}\label{sec:intro-notation} We write $\Hup=\{z\in\C:\Im z>0\}$ for the upper half-plane, and $z=E+\iu\eta$ with $E\in\R$ and $\eta>0$. For a self-adjoint operator $\mathcal K$ and a vector $f$, the spectral measure of $f$ is the measure $\mu$ with $\langle f,(\mathcal K-z)^{-1}f\rangle=\int(\lambda-z)^{-1}\mu(\dd\lambda)$. The letters $c$ and $C$ denote positive constants whose dependence on other parameters is indicated in each statement; within a proof they may change from line to line. We write $\Pois(\lambda)$ and $\Exp(1)$ for the Poisson law with mean $\lambda$ and the exponential law with mean one, and $\eqdist$ for equality in distribution. For a tree vertex $v$ other than the root, $v^-$ is its parent and $|v|$ its generation, and $\mathcal T_v$ is the subtree of descendants of $v$, rooted at $v$. For an edge $e$, we call $Y_e=|w_e|^2=p|h_e|^2$ the squared modulus of the edge weight. The squared moduli have the $\Exp(1)$ law, and are independent of the phases $w_e/|w_e|$, which are uniform on the unit circle. Notation that is used only in one section is introduced in that section. \section{Organization}\label{sec:intro-organization} \Cref{sec:outline} outlines the proofs of our main results. \Cref{part:foundations} develops the basic theory for a fixed degree $p$: the operator on the tree (\Cref{sec:tree}), spine decompositions and estimates for fresh squared moduli (\Cref{sec:palm}), real solutions of the cavity equation (\Cref{sec:rde}), and the transfer operator and the free energy (\Cref{sec:press}). \Cref{part:criteria} shows that the sign of the free energy at $s=1$ determines the spectral type at a fixed degree: \Cref{sec:rank} collects the rank-one perturbation arguments, \Cref{sec:loc} proves that negative free energy implies localization, and \Cref{sec:res,sec:deloc} prove that positive free energy is incompatible with real boundary values. \Cref{part:largep} studies the free energy at large degree: we construct the principal solution of the cavity equation in \Cref{sec:stab}, prove its analyticity and derivative bounds in \Cref{sec:jets}, determine the leading asymptotics of the free energy at $s=1$ in \Cref{sec:asym}, prove the uniqueness of its zero in \Cref{sec:uniq}, and treat small energies in \Cref{sec:inner}. Finally, \Cref{part:proofs} contains the proofs of \Cref{thm:main,thm:critical} in \Cref{sec:main} and of \Cref{thm:giant} in \Cref{sec:giant}. \chapter{Outline of the proof}\label{sec:outline} In this section we explain the main steps of the proofs of \Cref{thm:main,thm:critical}, and indicate where each step is carried out. The discussion is informal, but the objects and identities that appear are the ones used in the proofs. We only consider positive energies, since $\mathcal H$ is unitarily equivalent to $-\mathcal H$ (\Cref{prop:tree-symmetry}). Throughout, $z=E+\iu\eta$ with $E>0$ and $\eta>0$. \section{Cavity variables and the two alternatives}\label{sec:outline-alternatives} Let $\mathcal T$ be the rooted Poisson--Galton--Watson tree with mean degree $p$, with edge weights $h_e=w_e/\sqrt p$, and let $\mathcal H$ be its weighted adjacency operator. Although the degrees of $\mathcal T$ are unbounded, $\mathcal H$ is essentially self-adjoint on finitely supported vectors (\Cref{thm:tree-esa}), and its resolvent $G(z)=(\mathcal H-z)^{-1}$ is canonically defined. For a vertex $v\ne o$, let $\Gamma_v(z)$ denote the diagonal resolvent entry at $v$ of the operator restricted to the subtree of descendants of $v$. The Schur complement formula gives the cavity recursion \begin{equation}\label{eq:outline-cavity} \Gamma_v(z)=-\Bigl(z+\sum_{u:\,u^-=v}|h_{vu}|^2\Gamma_u(z)\Bigr)^{-1}, \end{equation} and, since the subtrees of distinct children are independent copies of $\mathcal T$, the root variable $\Gamma_o=G_{oo}$ satisfies \begin{equation}\label{eq:outline-rde} \Gamma_o\eqdist-\Bigl(z+\frac1p\sum_{j=1}^N Y_j\Gamma_j\Bigr)^{-1}, \qquad N\sim\Pois(p),\quad Y_j\sim\Exp(1), \end{equation} with all variables on the right independent and $\Gamma_j\eqdist\Gamma_o$. Off-diagonal resolvent entries factor along paths: if $o=v_0,v_1,\dots,v_L=v$ is the path from the root to $v$, then \begin{equation}\label{eq:outline-path} G_{ov}(z)=(-1)^L\Gamma_o(z)\prod_{j=1}^L h_{v_{j-1}v_j}\Gamma_{v_j}(z). \end{equation} These facts are proved in \Cref{sec:tree}. At Lebesgue-almost every energy $E$, the boundary values $\Gamma_v(E+\iu0)$ exist, and a zero--one law holds: under the survival-conditioned law $\P^\infty$, either $\Im\Gamma_o(E+\iu0)>0$ almost surely, or $\Im\Gamma_o(E+\iu0)=0$ almost surely (\Cref{prop:rde-boundary}). In the first case, the full diagonal boundary values satisfy $\Im G_{vv}(E+\iu0)>0$ at every vertex of the infinite tree. If this alternative holds for Lebesgue-almost every energy in an interval $I$, a spectral averaging argument shows that the spectral measures are absolutely continuous in $I$ (\Cref{prop:rank-ac}). In the second case, which we call the \emph{no-ac hypothesis} following \cite{AizenmanWarzel2013resonant}, all cavity variables at $E+\iu0$ are finite and real, and the inverse cavity variables $D_v=-1/\Gamma_v(E+\iu0)$, which we call denominators, solve the real cavity equation \begin{equation}\label{eq:outline-real-rde} D\eqdist E-\frac1p\sum_{j=1}^N\frac{Y_j}{D_j}. \end{equation} A law on $\R\setminus\{0\}$ satisfying \eqref{eq:outline-real-rde} is called a \emph{real solution} at energy $E$. The proof decides between the two alternatives by attaching to every real solution a number, the value $\varphi(1)$ at $s=1$ of its free energy, and showing that its sign determines the spectral type. \section{The free energy}\label{sec:outline-pressure} By \eqref{eq:outline-path}, the size of $G_{ov}$ is governed by products of the variables $|h_{v_{j-1}v_j}\Gamma_{v_j}|$ along paths. Under the no-ac hypothesis these products can be generated by a transfer operator. Fix a scaling $g>0$ and set $B=gD$ and $\beta=g^2Y/p$, so that \eqref{eq:outline-real-rde} becomes a recursion for $B$ in which a child with state $x=1/B$ is mapped to the parent state $(B'-\beta x)^{-1}$, where $B'$ collects the contributions of the other children. Each step of this recursion is the projective action of the matrix \begin{equation}\label{eq:outline-matrix} M(B,\beta)=\beta^{-1/2}\begin{pmatrix}0&1\\-\beta&B\end{pmatrix}, \qquad \det M(B,\beta)=1. \end{equation} The Mecke formula for Poisson processes turns sums over vertices of generation $L$ into expectations over a single path, at the cost of a factor $p$ per generation (\Cref{prop:palm-spine}). For $s\in(0,2)$ this leads to the positive transfer operator \[ (\mathcal Q_sF)(x)=p\,\E\bigl[\|M(B,\beta)v_x\|^{-s}F(f_{B,\beta}(x))\bigr], \qquad v_x=\frac{(x,1)}{\sqrt{1+x^2}}, \] acting on continuous functions on the real projective line $\RP$, where $f_{B,\beta}$ is the projective action of $M(B,\beta)$ and $B$ is distributed according to the real solution. We show in \Cref{sec:press} that $\mathcal Q_s^3$ is compact and strongly positive, so that $\mathcal Q_s$ has a simple dominant Perron--Frobenius eigenvalue $\lambda(s)$. The \emph{free energy} is $\varphi(s)=\log\lambda(s)$, and its value $\varphi(1)$ at $s=1$ decides the spectral type. We have \[ \varphi(s)=\log p+\lim_{L\to\infty}\frac1L\log\E\int_{\RP}\|M_L\cdots M_1v\|^{-s}\,m(\dd v), \] where $M_1,M_2,\dots$ are independent copies of $M(B,\beta)$ and $m$ is the uniform measure on $\RP$. For $s<1$, the quantity $\varphi(s)$ is also the exponential growth rate of the expected generation sums of $s$-th powers of the path products in \eqref{eq:outline-path}. The free energy is convex, real-analytic, and independent of the scaling. Its key property is the symmetry \begin{equation}\label{eq:outline-symmetry} \varphi(s)=\varphi(2-s), \end{equation} which follows from the change of variables $v\mapsto Mv/\|Mv\|$ on $\RP$, whose Jacobian is $\|Mv\|^{-2}$, and from the fact that a $2\times2$ matrix of determinant one has the same singular values as its inverse. Then $\varphi'(1)=0$, the point $s=1$ is the minimum of $\varphi$, and $\varphi$ decreases from $\varphi(0+)=\log p$ to $\varphi(1)$ on $(0,1]$. The value $\varphi(1)$ is the analogue, for the present model, of the criterion of Abou-Chacra, Anderson, and Thouless \cite{AbouChacraAndersonThouless1973}, of the free-energy criterion of Aizenman and Warzel \cite{AizenmanWarzel2013resonant}, and of the quantity $\lambda(E,\alpha)$ in the work of Aggarwal, Bordenave, and Lopatto on L\'evy matrices \cite{AggarwalBordenaveLopatto2022mobility}. A symmetry of this kind for the linearized cavity kernel goes back to \cite{AbouChacraAndersonThouless1973}; see \cite{ParisiPascazioPietracaprinaRosScardicchio2020anderson}. As in \cite{AggarwalBordenaveLopatto2022mobility,AggarwalLopatto2025mobility}, and unlike \cite{AizenmanWarzel2013resonant}, we sum over generations rather than normalizing per vertex, and the threshold is zero. \section{Negative free energy implies localization}\label{sec:outline-localization} Suppose that some real solution $\pi$ at energy $E$ has $\varphi_\pi(1)<0$. We do not need to know that $\pi$ is the law of the actual boundary values. By Kolmogorov's extension theorem, we may enlarge the probability space so that every vertex $v$ carries a real variable $D_v$ with law $\pi$, the variables satisfy the recursion \eqref{eq:outline-real-rde} pathwise on the tree, and the tree itself keeps its law (\Cref{lem:rde-rtp}). Set $\Gamma_v=-1/D_v$ and define the formal inverse column \[ F(o)=\Gamma_o,\qquad F(v)=\Gamma_o\prod_{j=1}^{L}\bigl(-h_{v_jv_{j-1}}\Gamma_{v_j}\bigr). \] The recursion gives $(\mathcal H-E)F=\delta_o$ pointwise. By the many-to-one formula, $\E\sum_{|v|=L}|F(v)|^s$ grows at the exponential rate $\varphi_\pi(s)$, and since $\varphi_\pi(1)<0$ we may choose $s<1$ with $\varphi_\pi(s)<0$. Then $F\in\ell^s\subset\ell^2$ almost surely. In particular, $F$ lies in the domain of $\mathcal H$ and $\delta_o=(\mathcal H-E)F$ lies in the range of $\mathcal H-E$. By the spectral theorem, $\int\mu_o(\dd\lambda)/(\lambda-E)^2<\infty$, where $\mu_o$ is the spectral measure at the root. The same holds at every vertex and on every component of the tree with one vertex removed (\Cref{prop:loc-columns}). These reciprocal-square integrals are the input of the Simon--Wolff criterion \cite{SimonWolff1986singular}. Since the model has no random potential, we need another source of rank-one randomness. It is supplied by the complex Gaussian weights. The tree is bipartite, so $\mathcal H^2=A^*A\oplus AA^*$, and \[ A^*A=\sum_{y}U_y\,|v_y\rangle\langle v_y|, \] where $y$ runs over the odd generations, $v_y$ is the normalized vector of edge weights adjacent to $y$, and $U_y$ is its squared norm. Conditional on the tree, the squared row norm $U_y$ has a Gamma density and is independent of the direction $v_y$ and of all other rows (\Cref{prop:tree-bipartite}). Thus each $U_y$ is the coupling constant of a rank-one perturbation with an absolutely continuous law, and the Simon--Wolff argument applies in the square blocks. This yields pure point spectrum on every open set of energies at which a real solution with negative free energy at $s=1$ exists (\Cref{thm:loc-criterion}). \section{Positive free energy implies delocalization}\label{sec:outline-delocalization} Suppose now that the no-ac hypothesis holds at a regular energy $E$, and let $\pi$ be the law of the boundary denominators. We show that $\varphi_\pi(1)\le0$ (\Cref{thm:deloc-criterion}). Together with the spectral averaging argument mentioned above, this proves pure absolute continuity on any interval where, for almost every regular energy, all possible real boundary laws have positive free energy at $s=1$ (\Cref{cor:deloc-ac}). The argument follows the resonance strategy of Aizenman and Warzel \cite{AizenmanWarzel2013resonant}: we look for vertices at large depth $n$ that are strongly coupled to the root, and show that their contributions force $\Im\Gamma_o$ to be large. Variants of this strategy were used in \cite{AggarwalBordenaveLopatto2022mobility,AggarwalLopatto2025mobility}. Suppose, for a contradiction, that $0<\varphi_\pi(1)$. We always have $\varphi_\pi(1)<\log p$ (\Cref{prop:press-properties}). By convexity and symmetry, there exists $s\in(1/2,1)$ such that \[ \gamma=-\varphi_\pi'(s)>0,\qquad F=\varphi_\pi(s)-(1-s)\gamma>0. \] Consider a path of length $n$ from the root, and the event that the product of the transfer matrices along it is of size about $e^{\gamma n}$. After an exponential tilt of the path law, this is a moderate deviation event for a sum of bounded increments, with probability about $e^{-(\log p-\varphi_\pi(s)-s\gamma)n}$. Since there are about $p^n$ paths, the expected number of such paths is about $e^{(\varphi_\pi(s)+s\gamma)n}$. Along such a path, an exact continuant identity shows that $|G_{ox}/G_{xx}|\gtrsim e^{-\gamma n}$, where $x$ is the endpoint. At $x$ we attach a pendant leaf whose squared edge weight $t$ has an exponential density. Since the diagonal entry $G_{xx}$ is a M\"obius function of $t$ with a real pole, the event $|G_{xx}|\ge T$ has probability of order $T^{-1}$. With $T=e^{(\gamma+\delta)n}$, we obtain an expected number \[ \E N_n\ge e^{(F-\delta)n-o(n)} \] of resonant endpoints $x$, each satisfying $|G_{ox}|\ge e^{\delta n}$. A surviving child of $x$, whose imaginary part is at least a fixed quantile $\zeta_\alpha$ of the distribution of $\Im\Gamma$, then gives $\Im\Gamma_o\ge e^{2\delta n-o(n)}\zeta_\alpha$ by the Ward identity. The main difficulty is the second moment. For two resonant endpoints $x$ and $y$ whose paths split at depth $j$, the two pendant tunings are nearly independent unless the Green function between the two branches is large. This interaction is controlled by the off-diagonal entry of the effective $2\times2$ Feshbach matrix at $x$ and $y$, which is the bilinear form of a product of transfer matrices along the path from $x$ to $y$. To average this product we reserve, near the split and near both endpoints, three consecutive factors whose squared moduli are not constrained by any of the events retained in the second moment. Three fresh factors suffice to smooth an arbitrary fixed bilinear form (\Cref{prop:press-three-rows}). The interaction contribution is then bounded by the norm moments of the two arms, which decay at the rate $\varphi_\pi(s)-\log p$ per step. The resulting exponent is an affine function of $j$, and the choice of $s$ makes it negative at $j=0$ and at $j=n$. Thus $\E[N_n(N_n-1)]\le e^{o(n)}(\E N_n)^2$, and the Paley--Zygmund inequality gives $\P(N_n>0)\ge e^{-o(n)}$. To conclude, we need to raise this subexponentially small probability to a fixed level. We use the Poisson structure of the tree: the root has many children, and each of them independently carries a copy of the rare event. An amplification argument, which uses only a fixed number of amplifying children per generation and a reserve of other children that keeps the cavity variables in a fixed compact region, raises the probability to a fixed number $q_*>\alpha$ after $o(n)$ generations, at the cost of a factor $e^{-o(n)}$ in the lower bound for $\Im\Gamma_o$. We obtain $\P^\infty(\Im\Gamma_o\ge e^{2\delta n-o(n)}\zeta_\alpha)\ge q_*>\alpha$, which contradicts the definition of the quantile $\zeta_\alpha$ once $n$ is large. All estimates are first made for the real recursive tree process at a fixed depth and then transferred to a small positive imaginary part $\eta$, chosen after $n$. \section{The free energy at large degree}\label{sec:outline-largep} By the previous two steps, \Cref{thm:main} reduces to determining the sign of the free energy at $s=1$ of the relevant real solutions. This is where the degree $p$ is taken large. \paragraph{The principal solution.} For $E>2$ it is convenient to use the scaling $g(E)=(E-\sqrt{E^2-4})/2$, in which the real cavity equation becomes \[ B\eqdist1+\theta-\frac\theta p\sum_{j=1}^N\frac{Y_j}{B_j},\qquad \theta=g(E)^2. \] At energies $E=\xi\sqrt p$ with $\xi$ of order one, we have $\theta\asymp1/p$, so the sum is an average of about $p$ terms of size $\theta/p$, and one expects $B\approx1$. This is correct except on rare events where some $B_j$ is close to zero. In \Cref{sec:stab} we construct, for every compact window of scaled energies $\xi$ and all large $p$, a solution $\nu_{p,E}$ whose law is concentrated within $p^{-1/4}$ of one, has density at most $p^Ae^{-ap}$ near zero, and has a tail decaying like $p^Ae^{-ap}/L$. We denote by $\mathfrak S$ the class of laws with these three properties. The solution $\nu_{p,E}$, which we call the \emph{principal solution}, is the unique fixed point of the cavity map in $\mathfrak S$, on which the map is contracted by a metric adapted to these three regions. The same construction covers all energies $E\ge M_0\sqrt p$, where the cavity map is a contraction in the same class. We then show that at large $p$ and energies $E=\xi\sqrt p$ in a fixed window, every real solution of the cavity equation is the principal solution (\Cref{prop:stab-selection}). The proof is a one-big-jump bootstrap: it starts from the bound $\sqrt{p+1}$ on the density near zero, which every real solution satisfies, and improves it until the solution lies in the class $\mathfrak S$. In particular, under the no-ac hypothesis the boundary law can only be the principal solution. \paragraph{The location of the mobility edge.} The leading behavior of the free energy at $s=1$ of the principal solution is easy to guess. Since $B\approx1$, we have $|\Gamma_v|\approx g(E)\approx E^{-1}$ along a typical path. Each step of \eqref{eq:outline-path} contributes a factor $|h_e|/E=|w_e|/(\xi p)$, and there are about $p$ choices of the next vertex. The expected sum of the products over the $p^L$ paths of length $L$ is therefore about \[ \Bigl(p\cdot\frac{\E|w|}{\xi p}\Bigr)^L=\Bigl(\frac{\mustar}{\xi}\Bigr)^L, \qquad \mustar=\E|w|=\frac{\sqrt\pi}2. \] This suggests $\exp\varphi_{\xi\sqrt p}(1)\approx\mustar/\xi$, and in \Cref{sec:asym} we prove that this is correct uniformly on compact sets of $\xi$ (\Cref{thm:asym-leading}). The lower bound follows from the behavior of the transfer operator on states near one. The upper bound is harder, because the rare cancellations $B\approx0$ produce large states, which may feed back into the core region of states near one. We decompose the state space into six regions and show that the only order-one transitions between the five other regions form an acyclic graph, so that every return to the core region passes through a transition of small weight. A finite-dimensional diagonal balancing then bounds the growth rate. \paragraph{Uniqueness of the crossing.} The leading asymptotics show that the free energy at $s=1$ changes sign near $\mustar\sqrt p$, but by themselves they do not exclude several sign changes in a window of width $o(\sqrt p)$. In \Cref{sec:jets} we show that the principal law is a real-analytic function of the energy, with polynomially bounded derivatives in suitable norms, and that its first two derivatives are small when tested against smooth functions supported near one. In \Cref{sec:uniq} we work with the $L^2$ version of the critical transfer operator, on a fixed space on which the energy enters only through the law of $B$ and through exponential densities. We then decompose its square exactly as \[ \mathsf A_\xi^2=G_\xi+h_{\mathrm c}\otimes\ell_{\mathrm c}+h_{\mathrm t}\otimes\ell_{\mathrm t}, \] where $\mathsf A_\xi$ is the critical operator on this space, the first rank-one term is a positive operator describing the region where $B$ is close to $1$, the second is its image under an involution $J$ that exchanges this region with the region where $B$ is close to $0$, and $G_\xi$ is a positive operator with small spectral radius. The Perron--Frobenius eigenvalue of $\mathsf A_\xi^2$ is then determined by the Perron--Frobenius eigenvalue of a $2\times2$ positive renewal matrix. Its diagonal entries are equal, by the symmetry under $J$, and decrease strictly in $\xi$. Its off-diagonal entries are positive, and one of them is exponentially small. A positivity argument, based on Glaeser's inequality, controls the derivative of this Perron--Frobenius eigenvalue without dividing by the possibly tiny gap between the two components. This shows that, for scaled energies in a fixed neighborhood of $\mustar$, the free energy at $s=1$ has exactly one zero, at which it crosses zero transversally and downward. Outside this neighborhood, its sign is given by the leading asymptotics and, at large energies, by the large-energy estimate. Together, these facts give exactly one zero on the half-line, and the crossing is transverse (\Cref{thm:uniq-window,cor:uniq-halfline}). \paragraph{Small energies.} For energies below a small multiple of $\sqrt p$, the real cavity equation need not have a principal solution, and we do not try to identify the real boundary law. Instead we show in \Cref{sec:inner} that \emph{every} real solution at an energy $00$ (\Cref{prop:inner-pressure}). The proof bounds the critical eigenvalue from below by $p\E\|M\|^{-1}$ for two scalings and shows, using only the recursion, that one of the two lower bounds exceeds one. Since the delocalization criterion applies to arbitrary real boundary laws, this gives absolute continuity at all small positive energies for the same fixed $p$. \section{Assembly and the finite graph}\label{sec:outline-assembly} The proof of \Cref{thm:main} in \Cref{sec:main} combines these results. On $(0,\xi_{\mathrm{in}}\sqrt p]$, every possible real boundary law has positive free energy at $s=1$, and the delocalization criterion shows that the spectrum is absolutely continuous. At larger energies of order $\sqrt p$, a real boundary law must be the principal solution, whose free energy at $s=1$ is positive below $E_c(p)$ and negative above it. Above $E_c(p)$ the principal solution is a real solution with negative free energy at $s=1$, which gives pure point spectrum, including at all large energies. The two absolutely continuous regions overlap, which leaves $E_c(p)$ as the only transition. Finally, in \Cref{sec:giant} we transfer the conclusions to the giant component of the finite graph. A uniformly chosen vertex of the giant component converges locally to the survival-conditioned tree, jointly for finitely many vertices, and the local spectral measures converge accordingly. The density of states of the limiting operator is bounded above and below on compact sets away from zero. On the absolutely continuous side, the normalized local spectral masses are uniformly integrable over vertices, by Lebesgue differentiation and Scheff\'e's lemma; on the pure point side, they concentrate on a vanishing fraction of vertices. This proves \Cref{thm:giant}. \part{The limiting tree and the cavity equation}\label{part:foundations} \chapter{The weighted tree and its operator}\label{sec:tree} In this section, we define the finite model and the marked Poisson--Galton--Watson tree, construct the self-adjoint operator of the tree and of the forests obtained from it by finitely many deletions, and collect the algebraic identities for its resolvent that are used throughout the paper. \Cref{sec:tree-models} defines the models. \Cref{sec:tree-depletions} defines finite removals and shows that marks which are not inspected by a selection remain independent standard Gaussians. \Cref{sec:tree-esa} proves essential self-adjointness of all these operators on finitely supported vectors, identifies their domains, and proves convergence of Dirichlet truncations. \Cref{sec:tree-resolvents} proves the Schur complement formula (the cavity recursion), the path product formula, and the Ward identity. \Cref{sec:tree-symmetry} records a gauge transformation and the parity symmetry. \Cref{sec:tree-bipartite} develops the bipartite factorization of the square of the operator and its decomposition into rank-one terms indexed by rows, which are the basis of the rank-one arguments in \Cref{sec:rank}. Finally, \Cref{sec:tree-compact} shows that on the survival event the operator has no finitely supported eigenvector with nonzero eigenvalue, while finitely supported eigenvectors with eigenvalue zero occur infinitely often. \paragraph{Conventions.} For a countable set $S$, we write $\ell^2(S)$ for the Hilbert space of square-summable functions $f\colon S\to\C$, with the inner product \[ \langle f,g\rangle=\sum_{s\in S}\overline{f(s)}\,g(s), \] which is antilinear in the first argument. We write $\delta_s$ for the coordinate vector at $s\in S$ and $c_c(S)$ for the space of finitely supported functions on $S$. For $S'\subset S$, we identify $\ell^2(S')$ with the subspace of $\ell^2(S)$ of functions vanishing off $S'$, and we write $P_{S'}$ for the orthogonal projection onto it. For a self-adjoint operator $T$ and a vector $f$, we write $\mu^T_f$ for the spectral measure of $f$, defined by $\langle f,\phi(T)f\rangle=\int\phi\dd\mu^T_f$ for bounded Borel functions $\phi$. For a Borel set $I\subset\R$, we say that $T$ has \emph{pure point spectrum in $I$} if $\Ran\one_I(T)$ is contained in the closed span of the eigenvectors of $T$, and that $T$ is \emph{purely absolutely continuous in $I$} if the restriction of $\mu^T_f$ to $I$ is absolutely continuous with respect to Lebesgue measure for every vector $f$. Since the closed span of the eigenvectors consists of the vectors whose spectral measure is purely atomic \cite[Section~VII.2]{ReedSimon1980methods}, the operator $T$ has pure point spectrum in $I$ if and only if the restriction of $\mu^T_f$ to $I$ is purely atomic for every $f$. Spectral parameters are written $z=E+\iu\eta$ with $E,\eta\in\R$, and $\Hup=\{z\in\C:\Im z>0\}$. \section{The finite model and the marked tree}\label{sec:tree-models} A standard complex Gaussian variable is a random variable $w$ with density $\pi^{-1}e^{-|\zeta|^2}$ with respect to Lebesgue measure on $\C$. Writing $\zeta=re^{\iu\vartheta}$, Lebesgue measure on $\C$ is $r\dd r\dd\vartheta$, and in the variables $t=r^2$ and $\vartheta\in[0,2\pi)$ the law of $w$ is $e^{-t}\dd t\otimes\dd\vartheta/(2\pi)$. This implies that $Y=|w|^2$ has the law $\Exp(1)$, the phase $w/|w|$ is uniform on the unit circle, and the two are independent. Also $\overline w$ and $-w$ are standard complex Gaussian variables. \begin{definition}\label{def:tree-model} Let $p>0$ and let $n\ge1$ be an integer with $n>p$. Let $G_n$ be the Erd\H{o}s--R\'enyi graph on $[n]=\{1,\dots,n\}$, in which each pair $\{i,j\}$ is an edge independently with probability $p/n$. Let $(w_{ij})_{1\le i0$. Let $(\Omega,\mathcal F,\P)$ be a probability space carrying three independent families: random variables $(N_u)_{u\in\mathbb U}$, independent with the law $\Pois(p)$; random variables $(w_u)_{u\in\mathbb U_*}$, independent standard complex Gaussians; and an auxiliary random element $\mathfrak a$ of a standard Borel space. \begin{enumerate}[label=\textup{(\roman*)}] \item The \emph{genealogy} is the rooted tree $\mathcal T$ with root $o$, vertex set \[ V=\bigl\{j_1j_2\cdots j_k\in\mathbb U:\ k\ge0,\ j_i\le N_{j_1\cdots j_{i-1}} \text{ for }1\le i\le k\bigr\}, \] and edges $\{u^-,u\}$ for $u\in V\setminus\{o\}$. It is a $\PGW(p)$ tree: the children of $v\in V$ are $v1,\dots,vN_v$. The \emph{generation} of $v$ is $|v|$, the vertex $v^-$ is the \emph{parent} of $v\neq o$, and we write $u\sim v$ when $u$ and $v$ are adjacent. The degree of $v$ is $\deg(v)=N_v+\one_{\{v\ne o\}}$. We write $N=N_o$ for the number of children of the root. \item The edge $\{u^-,u\}$ is identified with the word $u$, and $w_u$ is its \emph{mark}; we also write $w_e$ for the mark of an edge $e$. The \emph{edge weights} are \[ h_{u^-u}=\frac{w_u}{\sqrt p}, \qquad h_{uu^-}=\overline{h_{u^-u}}, \qquad u\in V\setminus\{o\}, \] and $h_{uv}=0$ when $u$ and $v$ are not adjacent. The \emph{squared modulus} of the mark of the edge $u$ is $Y_u=|w_u|^2=p|h_{u^-u}|^2$, which has the law $\Exp(1)$ and is independent of the phase $w_u/|w_u|$. The pair $(\mathcal T,(h_{uv})_{u,v\in V})$ is the \emph{marked tree}. \item For $v\in V$, the \emph{forward subtree} $\mathcal T_v$ is the subtree of $\mathcal T$ induced on $V(\mathcal T_v)=\{vu:u\in\mathbb U\}\cap V$, rooted at $v$. In particular $\mathcal T_o=\mathcal T$. \item The \emph{survival event} is $\mathcal S=\{|V|=\infty\}$, and $\varrho_p=\P(\mathcal S)$. When $p>1$, we have $\varrho_p\in(0,1)$, and $\varrho_p$ is the unique solution in $(0,1]$ of $1-\varrho=e^{-p\varrho}$. For $p>1$ we set $\P^\infty=\P(\,\cdot\mid\mathcal S)$ and write $\E^\infty$ for the corresponding expectation. \end{enumerate} We write $\mathcal G_0=\sigma\bigl((N_u)_{u\in\mathbb U},\mathfrak a\bigr)$ for the $\sigma$-algebra generated by the genealogy variables and the auxiliary randomness. \end{definition} The marks $w_u$ with $u\notin V$ do not enter the marked tree; including them makes the edge marks of $\mathcal T$ a subfamily of an independent family indexed by the fixed set $\mathbb U_*$. Conditionally on $\mathcal G_0$, the marks of the edges of $\mathcal T$ are independent standard complex Gaussians, and the same holds under $\P^\infty$, since $\mathcal S\in\mathcal G_0$. The value of $\varrho_p$ is the classical extinction computation for Galton--Watson trees \cite[Chapter~3]{vanderHofstad2017random}. The auxiliary element $\mathfrak a$ is used in later sections for selections by independent ranks; in \Cref{sec:palm}, it is realized as a family $(\zeta_v)_{v\in\mathbb U}$ of independent auxiliary marks indexed by the words. The families indexed by disjoint subsets of $\mathbb U$ are independent. We record the consequence that is used to derive distributional identities. For $j\in\N$, let $\theta_j$ be the \emph{shift} which maps the families above to $\bigl((N_{ju})_{u\in\mathbb U}, (w_{ju})_{u\in\mathbb U_*}\bigr)$. The genealogy and marks built from $\theta_j$ by \Cref{def:tree-pgw} form a marked tree $\theta_j\mathcal T$, and if $j\le N_o$, then $\theta_j\mathcal T$ is the forward subtree $\mathcal T_j$ relabelled by $ju\mapsto u$. The shifted families use the variables indexed by words beginning with $j$, other than $w_j$, and we conclude that \begin{equation}\label{eq:tree-branching} \theta_1\mathcal T,\theta_2\mathcal T,\dots\text{ are independent copies of }\mathcal T, \text{ independent of }\bigl(N_o,(w_j)_{j\ge1}\bigr). \end{equation} \section{Finite removals}\label{sec:tree-depletions} Many arguments in this paper delete a vertex, an edge, or a few selected leaves from the tree and study the resulting operator. We now define these operations and the class of selections under which the retained marks keep their law. \begin{definition}\label{def:tree-depletion} \leavevmode \begin{enumerate}[label=\textup{(\roman*)}] \item A \emph{finite removal} of $\mathcal T$ is given by a pair $(F,D)$, called its \emph{removed set}, where $F\subset V$ is a finite set of vertices and $D$ is a finite set of edges of $\mathcal T$; we also write $(F,D)$ for the finite removal. The \emph{deleted edges} are the edges in $D$ and the edges with an endpoint in $F$; all other edges of $\mathcal T$ are \emph{retained}. The \emph{reduced forest} $\mathcal T^{F,D}$ is the graph with vertex set $V\setminus F$ whose edges are the retained edges. A \emph{component} of $(F,D)$ is the vertex set of a connected component of $\mathcal T^{F,D}$. \item An \emph{inspected set} is a random finite set $Q\subset\mathbb U_*$ such that $\{Q=q\}\in\mathcal G_0^q$ for every finite $q\subset\mathbb U_*$, where $\mathcal G_0^q=\mathcal G_0\vee\sigma(w_u:u\in q)$. Its $\sigma$-algebra is \[ \mathcal G_Q=\bigl\{A\in\mathcal F:\ A\cap\{Q=q\}\in\mathcal G_0^q \text{ for every finite }q\subset\mathbb U_*\bigr\}. \] \item A random finite removal $(F,D)$ is \emph{selected without inspecting the retained marks} if there exists an inspected set $Q$ such that $\{(F,D)=(f,d)\}\in\mathcal G_Q$ for all finite sets $f,d$, and such that almost surely no edge with label in $Q$ is retained. If this holds with $Q=\emptyset$, that is, if $(F,D)$ is a function of the genealogy and the auxiliary randomness, we say that $(F,D)$ is \emph{selected from the genealogy}. \end{enumerate} \end{definition} An inspected set is the set of labels of the marks that a selection procedure looks at; the procedure may decide which marks to look at using what it has seen so far, and $\mathcal G_Q$ is the information available at the end. The finite removals used later are the following. \begin{enumerate}[label=(\alph*)] \item \emph{Deleting a vertex $y$}: $(F,D)=(\{y\},\emptyset)$. Its components are the forward subtrees $V(\mathcal T_{yj})$, $1\le j\le N_y$, and, if $y\ne o$, the set $V\setminus V(\mathcal T_y)$. \item \emph{Deleting an edge $\{v^-,v\}$}: $(F,D)=(\emptyset,\{v\})$. Its components are $V(\mathcal T_v)$ and $V\setminus V(\mathcal T_v)$. In particular, every forward subtree is a component of a finite removal. \item \emph{Deleting selected pendant leaves}: $F$ is a finite set of leaf children (children $\ell$ with $N_\ell=0$) chosen as a function of the genealogy and of auxiliary ranks carried by $\mathfrak a$, and $D=\emptyset$. This finite removal is selected from the genealogy. \item \emph{Dirichlet balls}: deleting the finitely many edges between generations $R$ and $R+1$ leaves the ball $\{v\in V:|v|\le R\}$ as the component of $o$. \end{enumerate} \begin{lemma}\label{lem:tree-depletion-marks} Let $Q$ be an inspected set. For every $A\in\mathcal G_Q$, every finite $q\subset\mathbb U_*$, and every bounded measurable function $\Psi$ of the family $(w_u)_{u\in\mathbb U_*\setminus q}$, we have \begin{equation}\label{eq:tree-fresh} \E\Bigl[\one_{A\cap\{Q=q\}}\,\Psi\bigl((w_u)_{u\notin q}\bigr)\Bigr] =\P\bigl(A\cap\{Q=q\}\bigr)\,\E\Bigl[\Psi\bigl((w_u)_{u\notin q}\bigr)\Bigr]. \end{equation} The same identity holds with $\P,\E$ replaced by $\P^\infty,\E^\infty$ when $p>1$. If $(F,D)$ is selected without inspecting the retained marks, then conditionally on $\mathcal G_Q$ the marks of the retained edges are independent standard complex Gaussian variables, under $\P$ and under $\P^\infty$. \end{lemma} \begin{proof} Fix $A$ and $q$. By the definition of $\mathcal G_Q$, the event $A\cap\{Q=q\}$ belongs to $\mathcal G_0^q=\sigma\bigl((N_u)_{u\in\mathbb U},\mathfrak a,(w_u)_{u\in q}\bigr)$. Since the three families in \Cref{def:tree-pgw} are independent and the marks are independent of each other, $\mathcal G_0^q$ is independent of $(w_u)_{u\notin q}$, and \eqref{eq:tree-fresh} follows. Since $\mathcal S\in\mathcal G_0$, the event $A\cap\mathcal S$ belongs to $\mathcal G_Q$; applying \eqref{eq:tree-fresh} to it and dividing by $\varrho_p$ gives the identity under $\P^\infty$, since $\E[\Psi]=\E^\infty[\Psi]$ by the independence of $\mathcal S\in\mathcal G_0$ and the marks. For the last assertion, let $(w'_u)_{u\in\mathbb U_*}$ be an independent copy of the marks, defined on a product extension of the probability space, and set $\widetilde w_u=w_u$ for $u\notin Q$ and $\widetilde w_u=w'_u$ for $u\in Q$. For fixed $q$, the family $\bigl((w_u)_{u\notin q},(w'_u)_{u\in q}\bigr)$ has the law of $(w_u)_{u\ne o}$ and is independent of $\mathcal G_0^q$, by the argument above and since $w'$ is independent of everything else. For $A\in\mathcal G_Q$ and every bounded measurable function $\Psi$ of the whole family, we then have \begin{align*} \E\bigl[\one_A\Psi(\widetilde w)\bigr] &=\sum_q\E\Bigl[\one_{A\cap\{Q=q\}}\Psi\bigl((w_u)_{u\notin q},(w'_u)_{u\in q}\bigr)\Bigr]\\ &=\sum_q\P\bigl(A\cap\{Q=q\}\bigr)\E\bigl[\Psi(w)\bigr]=\P(A)\,\E\bigl[\Psi(w)\bigr], \end{align*} where the sums run over the countably many finite sets $q$. This shows that $\widetilde w$ is independent of $\mathcal G_Q$ and is a family of independent standard complex Gaussians. The set of retained edges is a function of $\mathcal T$ and $(F,D)$, and in particular it is $\mathcal G_Q$-measurable. Since it is almost surely disjoint from $Q$, the retained marks coincide with the corresponding entries of $\widetilde w$. The conditional law, given $\mathcal G_Q$, of the entries of an independent i.i.d.\ family indexed by a $\mathcal G_Q$-measurable random set is the product law over that set \cite[Theorem~6.4]{Kallenberg2002foundations}. This proves the claim under $\P$, and the same argument applies under $\P^\infty$. \end{proof} For a finite removal selected from the genealogy, conditionally on $\mathcal G_0$ all marks, deleted or retained, are independent standard complex Gaussians. For the pendant leaves in (c), this means that the squared moduli of the marks of the deleted leaf edges are independent $\Exp(1)$ variables, independent of the genealogy and of all retained marks. We next record some elementary facts about the shape of the components. For a component $C$ of a finite removal and adjacent vertices $u,w\in C$, the \emph{branch} $C_{u\to w}$ is the component containing $u$ of the graph obtained from $\mathcal T^{F,D}$ by also deleting the edge $\{u,w\}$. \begin{lemma}\label{lem:tree-branch} Let $(F,D)$ be a finite removal of $\mathcal T$. \begin{enumerate}[label=\textup{(\roman*)}] \item The reduced forest $\mathcal T^{F,D}$ has finitely many components. Every component $C$ is the vertex set of an induced subtree of $\mathcal T$: every edge of $\mathcal T$ with both endpoints in $C$ is retained. \item Let $C$ be a component and let $u,w\in C$ be adjacent. Then $C_{u\to w}$ is a component of the finite removal $(F,D\cup\{\{u,w\}\})$, and it consists of the vertices $x\in C$ such that the path in $\mathcal T$ from $x$ to $w$ contains $u$. The neighbors of $u$ in $C$ lie in distinct components of $(F\cup\{u\},D)$, and the component containing a neighbor $x$ is $C_{x\to u}$. \item Let $C$ be as in \textup{(ii)}, let $u,w\in C$ be adjacent, and let $C'$ be a component of a finite removal $(F',D')$ with $F\subset F'$ and $D\subset D'$. If $C_{u\to w}\subset C'$ and $w\in C'$, then $C'_{u\to w}=C_{u\to w}$. \end{enumerate} \end{lemma} \begin{proof} Every edge of a tree disconnects it, and every edge of a forest disconnects the component that contains it. By induction on the number of deleted edges, deleting them from $\mathcal T$ produces a forest with exactly one more component than deleted edges. The vertices of $F$ are isolated in this forest, and its other components are the components of $(F,D)$; this proves the first claim of (i). If an edge $\{u,v\}$ of $\mathcal T$ with $u,v\in C$ were deleted, then the path from $u$ to $v$ in the connected graph $\mathcal T^{F,D}$ restricted to $C$ would avoid this edge, and together with the edge it would form a cycle in $\mathcal T$. This proves (i). For (ii), the first assertion holds because adding $\{u,w\}$ to $D$ changes only the component $C$, which splits into two. Since $C$ is a subtree, paths in $C$ are paths in $\mathcal T$. A vertex $x\in C$ is connected to $u$ in $C$ without using the edge $\{u,w\}$ exactly when the path from $x$ to $u$ does not contain $w$, and this holds exactly when the path from $x$ to $w$ is the path from $x$ to $u$ followed by the edge $\{u,w\}$. This proves the description of $C_{u\to w}$. If $x$ and $x'$ are distinct neighbors of $u$ in $C$, then they lie in distinct components after deleting $u$, since the unique path between them is $x,u,x'$. The component of $x$ after deleting $u$ consists of the vertices of $C$ connected to $x$ by a path avoiding $u$, and this is $C_{x\to u}$ by the description just proved. For (iii), we have $C'\subset C$, because every edge retained by $(F',D')$ is retained by $(F,D)$. Since a path in $C'$ which avoids the edge $\{u,w\}$ is such a path in $C$, we have $C'_{u\to w}\subset C_{u\to w}$. Conversely, $C_{u\to w}$ induces a connected subtree of $\mathcal T$ by (i) and (ii), it is contained in $C'$, and $C'$ is an induced subtree by (i). Then every vertex of $C_{u\to w}$ is joined to $u$ by a path in $C'$ avoiding $\{u,w\}$, which gives $C_{u\to w}\subset C'_{u\to w}$. \end{proof} \section{Essential self-adjointness}\label{sec:tree-esa} The weighted adjacency operator of $\mathcal T$ need not be bounded. In this section, we show that it is essentially self-adjoint on finitely supported vectors, simultaneously for all components of all finite removals. The argument follows \cite[Appendix~A]{BordenaveCaputoChafai2011spectrum}: by \cite[Lemma~A.3]{BordenaveCaputoChafai2011spectrum}, an exhaustion of a weighted tree by finite connected sets whose boundary weights are uniformly bounded implies essential self-adjointness, and such an exhaustion of the Poisson weighted infinite tree is constructed in \cite[Proposition~A.2]{BordenaveCaputoChafai2011spectrum} by a subcritical exploration; see also \cite{BordenaveLelargeSalez2011rank} for the unweighted case. We include the proof for our model for completeness. \subsection{Hermitian networks} A \emph{locally finite Hermitian network} is a countable set $\mathcal V$ with weights $k\colon\mathcal V\times\mathcal V\to\C$ such that $k_{vu}=\overline{k_{uv}}$ and $k_{vv}=0$ for all $u,v$, and such that $\{u:k_{vu}\ne0\}$ is finite for every $v$. Its \emph{formal operator} acts on all functions $f\colon\mathcal V\to\C$ by \[ (T_{\mathrm{form}}f)(v)=\sum_{u\in\mathcal V}k_{vu}f(u), \] which is a finite sum. The \emph{minimal operator} $T_0$ is the restriction of $T_{\mathrm{form}}$ to $c_c(\mathcal V)$, viewed as an operator in $\ell^2(\mathcal V)$, and the \emph{maximal domain} is $\{f\in\ell^2(\mathcal V):T_{\mathrm{form}}f\in\ell^2(\mathcal V)\}$. For a finite set $S\subset\mathcal V$, the \emph{boundary operator} $B_S$ is defined by \[ (B_Sf)(v)=\one_S(v)\sum_{u\notin S}k_{vu}f(u), \qquad f\in\ell^2(\mathcal V). \] Since only the finitely many $u\notin S$ with a neighbor in $S$ enter this sum, $B_S$ is a bounded operator of finite rank. On the marked tree, $(V,(h_{uv})_{u,v\in V})$ is a locally finite Hermitian network. If $C$ is a component of a finite removal, then $(C,(h_{uv})_{u,v\in C})$ is a locally finite Hermitian network as well, and by \Cref{lem:tree-branch}(i) its weights are those of the retained edges inside $C$. We write $\mathcal H_{C,\mathrm{form}}$ and $\mathcal H_{C,0}$ for its formal and minimal operators, and $\mathcal H_{\mathrm{form}}$ and $\mathcal H_0$ when $C=V$. \begin{lemma}\label{lem:tree-adjoint} Let $(\mathcal V,k)$ be a locally finite Hermitian network. Then $T_0$ is symmetric, and $T_0^*$ is the restriction of $T_{\mathrm{form}}$ to the maximal domain. If $T_0$ is essentially self-adjoint, then its closure $T$ satisfies \[ \Dom(T)=\{f\in\ell^2(\mathcal V):T_{\mathrm{form}}f\in\ell^2(\mathcal V)\}, \qquad Tf=T_{\mathrm{form}}f\quad\text{for }f\in\Dom(T). \] \end{lemma} \begin{proof} Let $g\in c_c(\mathcal V)$ and $f\in\ell^2(\mathcal V)$. Only the finitely many pairs $(u,v)$ with $u\in\supp g$ and $k_{uv}\ne0$ contribute to the following sums. Exchanging these finite sums and using $\overline{k_{vu}}=k_{uv}$, we have \begin{equation}\label{eq:tree-green} \langle T_0g,f\rangle =\sum_v\overline{\sum_uk_{vu}g(u)}\,f(v) =\sum_u\overline{g(u)}\sum_vk_{uv}f(v) =\langle g,T_{\mathrm{form}}f\rangle. \end{equation} Taking $f\in c_c(\mathcal V)$, we see that $T_0$ is symmetric. If $f$ is in the maximal domain, then \eqref{eq:tree-green} implies that $f\in\Dom(T_0^*)$ and $T_0^*f=T_{\mathrm{form}}f$. Conversely, if $f\in\Dom(T_0^*)$, then applying \eqref{eq:tree-green} with $g=\delta_v$, we have $(T_0^*f)(v)=(T_{\mathrm{form}}f)(v)$ for every $v$, that is, $T_{\mathrm{form}}f=T_0^*f\in\ell^2$. If $T_0$ is essentially self-adjoint, then its closure $T$ is self-adjoint and $T=T^*=T_0^*$, since an operator and its closure have the same adjoint. \end{proof} \begin{lemma}\label{lem:tree-flow} Let $(\mathcal V,k)$ be a locally finite Hermitian network. Suppose that there exist finite sets $S_1\subset S_2\subset\cdots$ with $\bigcup_nS_n=\mathcal V$ and $M=\sup_n\|B_{S_n}\|<\infty$. Then $T_0$ is essentially self-adjoint on $c_c(\mathcal V)$. \end{lemma} \begin{proof} By \cite[Theorem~VIII.3 and its corollary]{ReedSimon1980methods}, the symmetric operator $T_0$ is essentially self-adjoint if $\ker(T_0^*-z)=\{0\}$ for $z=\iu$ and $z=-\iu$. Fix $z\in\C\setminus\R$ and $\psi\in\ker(T_0^*-z)$. By \Cref{lem:tree-adjoint}, we have $T_{\mathrm{form}}\psi=z\psi$ pointwise. Let $S\subset\mathcal V$ be finite. Multiplying the equation at $v$ by $\overline{\psi(v)}$ and summing over $v\in S$, we obtain \[ z\|P_S\psi\|^2 =\sum_{v\in S}\sum_{u\in S}\overline{\psi(v)}k_{vu}\psi(u) +\sum_{v\in S}\sum_{u\notin S}\overline{\psi(v)}k_{vu}\psi(u) =\sum_{u,v\in S}\overline{\psi(v)}k_{vu}\psi(u)+\langle P_S\psi,B_S\psi\rangle. \] The first sum on the right is real, because the matrix $(k_{vu})_{u,v\in S}$ is Hermitian. Taking imaginary parts, and using that $B_S\psi=B_SP_{\mathcal V\setminus S}\psi$, we find \begin{equation}\label{eq:tree-flow} |\Im z|\,\|P_S\psi\|^2 \le\bigl|\langle P_S\psi,B_SP_{\mathcal V\setminus S}\psi\rangle\bigr| \le\|B_S\|\,\|P_S\psi\|\,\|P_{\mathcal V\setminus S}\psi\|. \end{equation} We apply \eqref{eq:tree-flow} with $S=S_n$. As $n\to\infty$, we have $\|P_{S_n}\psi\|\to\|\psi\|$ and $\|P_{\mathcal V\setminus S_n}\psi\|\to0$, since $S_n\uparrow\mathcal V$ and $\psi\in\ell^2(\mathcal V)$. The right side of \eqref{eq:tree-flow} is at most $M\|\psi\|\|P_{\mathcal V\setminus S_n}\psi\|$, which tends to zero. We conclude that $|\Im z|\|\psi\|^2=0$ and $\psi=0$. \end{proof} \subsection{A bounded-boundary exhaustion of the tree} For $u\in\mathbb U$, set \[ R_u^2=\frac1p\sum_{j=1}^{N_u}|w_{uj}|^2 . \] For $u\in V$, this is $\sum_{j\le N_u}|h_{u,uj}|^2$, the sum of the squared weights of the edges from $u$ to its children. Since $R_o<\infty$ almost surely and $N_o\one_{\{R_o>M\}}\le N_o$, which is integrable, by dominated convergence we have $\E[N_o\one_{\{R_o>M\}}]\to0$ as $M\to\infty$. We fix a deterministic $M=M(p)<\infty$ such that \begin{equation}\label{eq:tree-subcritical} m_M=\E\bigl[N_o\one_{\{R_o>M\}}\bigr]<1. \end{equation} We call $u\in\mathbb U$ \emph{good} if $R_u\le M$ and \emph{bad} otherwise. For $v\in V$, let $\lambda(v)$ be the number of good vertices among the strict ancestors of $v$, and set \[ S_n=\{v\in V:\lambda(v)\le n\},\qquad n\ge0. \] \begin{lemma}\label{lem:tree-exhaustion} Let $p>0$. There exists an event $\Omega_{\mathrm{sa}}$ with $\P(\Omega_{\mathrm{sa}})=1$ on which the following holds. Every $S_n$ is finite, $o\in S_n\subset S_{n+1}$, every ancestor of a vertex of $S_n$ belongs to $S_n$, and $\bigcup_nS_n=V$. Further, for every $n$, every vertex outside $S_n$ has at most one neighbor in $S_n$, and \begin{equation}\label{eq:tree-boundary-sum} \sum_{u\notin S_n,\,u\sim v}|h_{vu}|^2\le M^2\qquad\text{for every }v\in S_n. \end{equation} In particular, the boundary operators of the network $(V,h)$ satisfy $\|B_{S_n}\|\le M$. \end{lemma} \begin{proof} The proof shows that the vertices reachable from a given vertex through bad vertices form a subcritical Galton--Watson tree, and that the sets $S_n$ are finite unions of such trees. \proofstep{1}{The bad clusters are subcritical Galton--Watson trees.} For $u\in\mathbb U$, let $\chi_u=N_u\one_{\{R_u>M\}}$. This is a function of $N_u$ and of $(w_{uj})_{j\in\N}$. Since for distinct $u$ the variables $N_u$ are distinct and the families $(w_{uj})_{j\in\N}$ are indexed by disjoint subsets of $\mathbb U$, the family $(\chi_u)_{u\in\mathbb U}$ is independent and identically distributed, with mean $m_M<1$. For $x\in\mathbb U$, define the \emph{bad cluster} \[ K_x=\bigl\{xj_1\cdots j_k:\ k\ge0,\ j_i\le\chi_{xj_1\cdots j_{i-1}}\text{ for }1\le i\le k\bigr\}. \] Then $K_x$ is a Galton--Watson tree whose offspring numbers $(\chi_{xu})_{u\in\mathbb U}$ are independent with mean $m_M$. Let $Z^x_k$ be the number of words of length $|x|+k$ in $K_x$. The event $\{u\in K_x\}$ is determined by the variables $\chi$ at the strict ancestors of $u$, which are independent of $\chi_u$. Summing over the countably many words $u$ of length $|x|+k$, we obtain \[ \E Z^x_{k+1}=\sum_{|u|=|x|+k}\E\bigl[\one_{\{u\in K_x\}}\chi_u\bigr] =m_M\,\E Z^x_k, \] so $\E Z^x_k=m_M^k$ and $\E|K_x|=(1-m_M)^{-1}<\infty$. Let $\Omega_{\mathrm{sa}}$ be the event that $K_x$ is finite for every $x\in\mathbb U$; as a countable intersection of events of probability one, it satisfies $\P(\Omega_{\mathrm{sa}})=1$. Since $j\le\chi_u$ holds exactly when $u$ is bad and $j\le N_u$, we have, for $x\in V$, \begin{equation}\label{eq:tree-bad-cluster} K_x=\bigl\{y\in V(\mathcal T_x):\ \text{every vertex }u\ne y\text{ on the path from }x\text{ to }y \text{ is bad}\bigr\}. \end{equation} \proofstep{2}{The sets $S_n$ are finite and exhaust $V$.} For $v\in V$ and $j\le N_v$, we have $\lambda(vj)=\lambda(v)+\one_{\{v\text{ good}\}}$. This implies that $\lambda$ is nondecreasing along every path away from the root, and $S_n$ contains every ancestor of its vertices, and $o\in S_n$ since $\lambda(o)=0$. Also $\lambda(v)\le|v|$, which gives $\bigcup_nS_n=V$. We show by induction that $S_n$ is finite on $\Omega_{\mathrm{sa}}$. By \eqref{eq:tree-bad-cluster}, we have $S_0=K_o$. Let $n\ge0$ and let $v\in V$ satisfy $\lambda(v)=n+1$. Let $g$ be the strict ancestor of $v$ of largest generation which is good, and let $t$ be the child of $g$ on the path from $g$ to $v$ (possibly $t=v$). Since the good strict ancestors of $v$ are $g$ and the good strict ancestors of $g$, we have $\lambda(g)=n$. All vertices on the path from $t$ to $v$, other than $v$, are bad, and \eqref{eq:tree-bad-cluster} implies that $v\in K_t$. We conclude that \[ \{v\in V:\lambda(v)=n+1\}\subset\bigcup_{g\in S_n}\ \bigcup_{j=1}^{N_g}K_{gj}, \] which is a finite union of finite sets on $\Omega_{\mathrm{sa}}$ when $S_n$ is finite. \proofstep{3}{The boundary of $S_n$.} Let $v\in S_n$ and let $u\sim v$ with $u\notin S_n$. Since the parent of $v$ belongs to $S_n$, the vertex $u$ is a child of $v$, and $\lambda(u)=\lambda(v)+\one_{\{v\text{ good}\}}>n$ forces $v$ to be good. It follows that the left side of \eqref{eq:tree-boundary-sum} vanishes if $v$ is bad, and is at most $R_v^2\le M^2$ if $v$ is good. If $u\notin S_n$, then the children of $u$ satisfy $\lambda\ge\lambda(u)>n$, and the only possible neighbor of $u$ in $S_n$ is its parent. The boundary operator $B=B_{S_n}$ has matrix entries $B_{vu}=h_{vu}$ for $v\in S_n$ and $u\notin S_n$, and all its other entries vanish. Since each column of $B$ has at most one nonzero entry, the matrix $BB^*$, which acts on $\ell^2(S_n)$ and has entries $\sum_{u\notin S_n}h_{vu}\overline{h_{v'u}}$, is diagonal. Its diagonal entries are the left side of \eqref{eq:tree-boundary-sum}, which gives $\|B\|^2=\|BB^*\|\le M^2$. \end{proof} \subsection{Dirichlet truncations} \begin{lemma}\label{lem:tree-core} Let $T$ and $T_1,T_2,\dots$ be self-adjoint operators on a Hilbert space $\mathfrak h$. Let $\mathcal D$ be a core for $T$ with $\mathcal D\subset\Dom(T_n)$ for every $n$, and suppose that $T_n\varphi\to T\varphi$ for every $\varphi\in\mathcal D$. Then $(T_n-z)^{-1}f\to(T-z)^{-1}f$ for every $f\in\mathfrak h$ and $z\in\C\setminus\R$. Further, for all $f,g\in\mathfrak h$, we have $\langle f,(T_n-z)^{-1}g\rangle\to\langle f,(T-z)^{-1}g\rangle$ uniformly for $z$ in compact subsets of $\C\setminus\R$. \end{lemma} \begin{proof} The first assertion is a variant of \cite[Theorem~VIII.25(a)]{ReedSimon1980methods}, where $\mathcal D$ is assumed to be a common core for $T$ and all $T_n$; the following argument does not use this assumption. Fix $z\in\C\setminus\R$. For $\varphi\in\mathcal D$ and $f=(T-z)\varphi$, we have \[ (T_n-z)^{-1}f-(T-z)^{-1}f=(T_n-z)^{-1}\bigl((T-z)\varphi-(T_n-z)\varphi\bigr) =(T_n-z)^{-1}(T-T_n)\varphi, \] whose norm is at most $|\Im z|^{-1}\|(T-T_n)\varphi\|\to0$. The set $(T-z)\mathcal D$ is dense in $\mathfrak h$: every $f\in\mathfrak h$ equals $(T-z)\psi$ for some $\psi\in\Dom(T)$, and if $\varphi_k\in\mathcal D$ converge to $\psi$ in the graph norm of $T$, then $(T-z)\varphi_k\to f$. Since all resolvents have norm at most $|\Im z|^{-1}$, the convergence extends from this dense set to all $f$. For the second assertion, let $K\subset\C\setminus\R$ be compact and $d=\dist(K,\R)>0$. For every self-adjoint $S$ and $z,z'\in K$, the resolvent identity $(S-z)^{-1}-(S-z')^{-1}=(z-z')(S-z)^{-1}(S-z')^{-1}$ gives \[ \bigl|\langle f,(S-z)^{-1}g\rangle-\langle f,(S-z')^{-1}g\rangle\bigr| \le\frac{|z-z'|\,\|f\|\,\|g\|}{d^2}. \] Then the functions $z\mapsto\langle f,(T_n-z)^{-1}g\rangle$ and $z\mapsto\langle f,(T-z)^{-1}g\rangle$ are Lipschitz on $K$ with a common constant. Given $\eps>0$, we cover $K$ by finitely many discs of radius $\eps d^2/(3\|f\|\|g\|+1)$ centered in $K$, and choose $n_0$ such that the pointwise convergence holds within $\eps/3$ at the finitely many centers for $n\ge n_0$. Then the difference is at most $\eps$ on $K$ for $n\ge n_0$. \end{proof} For a component $C$ of a finite removal and a finite set $\Lambda\subset C$, the \emph{Dirichlet truncation} $\mathcal H_C^{[\Lambda]}=P_\Lambda\mathcal H_{C,0}P_\Lambda$ is the bounded self-adjoint operator on $\ell^2(C)$ which acts on $\ell^2(\Lambda)$ by the finite matrix $(h_{uv})_{u,v\in\Lambda}$ and vanishes on $\ell^2(C\setminus\Lambda)$. When $C=V$, we write $\mathcal H^{[\Lambda]}$. \subsection{The self-adjoint operators} For a finite removal $(F,D)$, define the bounded self-adjoint operator of finite rank \[ W_{F,D}=\sum_{\{u,v\}\text{ deleted}} \bigl(h_{uv}|\delta_u\rangle\langle\delta_v|+h_{vu}|\delta_v\rangle\langle\delta_u|\bigr) \] on $\ell^2(V)$, where the sum runs over the deleted edges of $(F,D)$, and $|f\rangle\langle g|$ denotes the operator $\varphi\mapsto\langle g,\varphi\rangle f$. \begin{theorem}\label{thm:tree-esa} Let $p>0$, and let $\Omega_{\mathrm{sa}}$ be the event of \Cref{lem:tree-exhaustion}; we have $\P(\Omega_{\mathrm{sa}})=1$, and $\P^\infty(\Omega_{\mathrm{sa}})=1$ when $p>1$. On $\Omega_{\mathrm{sa}}$, the following statements hold simultaneously for every finite removal $(F,D)$ of $\mathcal T$ and every component $C$ of $(F,D)$. \begin{enumerate}[label=\textup{(\roman*)}] \item The operator $\mathcal H_{C,0}$ is essentially self-adjoint on $c_c(C)$. We write $\mathcal H_C$ for its closure, $\mathcal H=\mathcal H_V$ for the operator of the full tree, and $\mathcal H_v=\mathcal H_{V(\mathcal T_v)}$ for the operator of the forward subtree at $v\in V$. \item The domain of $\mathcal H_C$ is the maximal domain: \[ \Dom(\mathcal H_C)=\bigl\{f\in\ell^2(C):\mathcal H_{C,\mathrm{form}}f\in\ell^2(C)\bigr\}, \] and $\mathcal H_Cf=\mathcal H_{C,\mathrm{form}}f$ for $f\in\Dom(\mathcal H_C)$. \item Let $\Lambda_1\subset\Lambda_2\subset\cdots$ be finite sets with $\bigcup_n\Lambda_n=C$. Then $(\mathcal H_C^{[\Lambda_n]}-z)^{-1}f\to(\mathcal H_C-z)^{-1}f$ for every $f\in\ell^2(C)$ and $z\in\C\setminus\R$, and for all $u,v\in C$, \[ \bigl\langle\delta_u,(\mathcal H_C^{[\Lambda_n]}-z)^{-1}\delta_v\bigr\rangle \to\bigl\langle\delta_u,(\mathcal H_C-z)^{-1}\delta_v\bigr\rangle \] uniformly for $z$ in compact subsets of $\C\setminus\R$. This applies to the balls $\Lambda_n=\{v\in C:\dist_C(v,v_0)\le n\}$ around any $v_0\in C$, and for $C=V$ to the balls $\{v\in V:|v|\le n\}$. \item Let $C_1,\dots,C_m$ be the components of $(F,D)$. Then \[ \mathcal H-W_{F,D}=\mathcal H_{C_1}\oplus\cdots\oplus\mathcal H_{C_m}\oplus0_F \] with respect to $\ell^2(V)=\ell^2(C_1)\oplus\cdots\oplus\ell^2(C_m)\oplus\ell^2(F)$, where $0_F$ is the zero operator on $\ell^2(F)$. In particular $\Dom(\mathcal H)=\Dom(\mathcal H_{C_1})\oplus\cdots\oplus\Dom(\mathcal H_{C_m})\oplus\ell^2(F)$. \end{enumerate} \end{theorem} We write $\mathcal H^{F,D}=\mathcal H_{C_1}\oplus\cdots\oplus\mathcal H_{C_m}$ for the \emph{reduced operator} on $\ell^2(V\setminus F)$, and $\mathcal H^{(y)}=\mathcal H^{\{y\},\emptyset}$ for the operator with the vertex $y$ deleted. \begin{proof} We fix a realization in $\Omega_{\mathrm{sa}}$; all statements below are deterministic consequences of the conclusions of \Cref{lem:tree-exhaustion}, which do not depend on $(F,D)$. \proofstep{1}{The full tree.} By \Cref{lem:tree-exhaustion}, the network $(V,h)$ satisfies the hypotheses of \Cref{lem:tree-flow}, which implies that $\mathcal H_0$ is essentially self-adjoint. \Cref{lem:tree-adjoint} identifies the domain of its closure $\mathcal H$ with the maximal domain. \proofstep{2}{Finite removals.} Let $(F,D)$ be a finite removal and $W=W_{F,D}$. Since $W$ is bounded and self-adjoint, the Kato--Rellich theorem \cite[Theorem~X.12]{ReedSimon1975methods} (with relative bound zero) shows that $\mathcal H-W$ is self-adjoint on $\Dom(\mathcal H)$ and essentially self-adjoint on every core of $\mathcal H$, in particular on $c_c(V)$. On $c_c(V)$, the operator $\mathcal H_0-W$ is the weighted adjacency operator of the graph on $V$ with the retained edges. By \Cref{lem:tree-branch}(i), this graph has the components $C_1,\dots,C_m$ and the isolated vertices of $F$. Hence \begin{equation}\label{eq:tree-sum-minimal} (\mathcal H_0-W)|_{c_c(V)}=\mathcal H_{C_1,0}\oplus\cdots\oplus\mathcal H_{C_m,0}\oplus0 \end{equation} with respect to $c_c(V)=c_c(C_1)\oplus\cdots\oplus c_c(C_m)\oplus c_c(F)$. The adjoint of a finite orthogonal sum of densely defined operators is the orthogonal sum of their adjoints. Indeed, a vector $(\varphi_j)_j$ is in the domain of the adjoint of $\bigoplus_jT_j$ exactly when $(f_j)_j\mapsto\sum_j\langle\varphi_j,T_jf_j\rangle$ is bounded, and taking $f$ supported in a single summand shows that this holds exactly when $\varphi_j\in\Dom(T_j^*)$ for every $j$. For $z=\pm\iu$, it follows that the kernel of $T^*-z$, where $T$ is the operator in \eqref{eq:tree-sum-minimal}, is the orthogonal sum of the kernels of $\mathcal H_{C_j,0}^*-z$ and of the kernel of $-z$ on $\ell^2(F)$, which is trivial. Since $T$ is essentially self-adjoint, the kernel of $T^*-z$ is trivial, and so are the kernels of $\mathcal H_{C_j,0}^*-z$. Then \cite[Theorem~VIII.3 and its corollary]{ReedSimon1980methods} shows that every $\mathcal H_{C_j,0}$ is essentially self-adjoint. The closure of a finite orthogonal sum is the orthogonal sum of the closures, since the graph of the sum is the orthogonal sum of the graphs. Taking closures in \eqref{eq:tree-sum-minimal} proves (iv), and (i) holds for every component. Statement (ii) follows from \Cref{lem:tree-adjoint} applied to the network $(C,h)$. \proofstep{3}{Dirichlet truncations.} Let $\Lambda_n$ be as in (iii) and $f\in c_c(C)$. The vector $\mathcal H_{C,0}f$ is supported on the union of $\supp f$ and its neighbors in $C$, which is finite. When $\Lambda_n$ contains this union, we have $\mathcal H_C^{[\Lambda_n]}f=P_{\Lambda_n}\mathcal H_{C,0}f=\mathcal H_{C,0}f$. Since $c_c(C)$ is a core for $\mathcal H_C$, \Cref{lem:tree-core} implies (iii). Balls in the locally finite graph $C$ are finite and exhaust $C$. \end{proof} \begin{remark}\label{rem:tree-maximal} Statement (ii) is used in the following form. Let $C$ be as in \Cref{thm:tree-esa}, let $E\in\R$, and let $g\in\ell^2(C)$. If $F\in\ell^2(C)$ satisfies $(\mathcal H_{C,\mathrm{form}}F)(v)-EF(v)=g(v)$ for every $v\in C$, then $\mathcal H_{C,\mathrm{form}}F=EF+g\in\ell^2(C)$, and (ii) implies that $F\in\Dom(\mathcal H_C)$ and $(\mathcal H_C-E)F=g$. In particular, a vector $F\in\ell^2(V)$ satisfying $(\mathcal H_{\mathrm{form}}-E)F=\delta_o$ pointwise lies in $\Dom(\mathcal H)$ and satisfies $(\mathcal H-E)F=\delta_o$; in particular, $\delta_o\in\Ran(\mathcal H-E)$. \end{remark} \begin{remark}\label{rem:tree-direct-exhaustion} The exhaustion of \Cref{lem:tree-exhaustion} also applies directly to a component $C$ of a finite removal: the sets $S_n\cap C$ exhaust $C$, each vertex of $C\setminus S_n$ has at most one neighbor in $S_n\cap C$, and the boundary sums in $C$ are bounded by the left side of \eqref{eq:tree-boundary-sum}. Then \Cref{lem:tree-flow} yields (i) without the perturbation argument. Statement (iv), which identifies the domains, is used repeatedly below. \end{remark} The following consequence of (iv) decouples a component along finitely many further deleted edges. \begin{lemma}\label{lem:tree-decouple} On $\Omega_{\mathrm{sa}}$, let $C$ be a component of a finite removal $(F,D)$, and let $D'$ be a finite set of edges of $\mathcal T$ with both endpoints in $C$. Let $C'_1,\dots,C'_k$ be the components of $(F,D\cup D')$ contained in $C$. Then \[ \mathcal H_C=\bigl(\mathcal H_{C'_1}\oplus\cdots\oplus\mathcal H_{C'_k}\bigr)+W_{D'}, \qquad \Dom(\mathcal H_C)=\Dom(\mathcal H_{C'_1})\oplus\cdots\oplus\Dom(\mathcal H_{C'_k}), \] where $W_{D'}=\sum_{\{u,v\}\in D'}(h_{uv}|\delta_u\rangle\langle\delta_v|+ h_{vu}|\delta_v\rangle\langle\delta_u|)$ acts on $\ell^2(C)$. In particular, if $y\in C$ has the neighbors $x_1,\dots,x_d$ in $C$, then with respect to $\ell^2(C)=\C\delta_y\oplus\ell^2(C_{x_1\to y})\oplus\cdots\oplus\ell^2(C_{x_d\to y})$, \begin{equation}\label{eq:tree-star} \mathcal H_C=\bigl(0\oplus\mathcal H_{C_{x_1\to y}}\oplus\cdots\oplus\mathcal H_{C_{x_d\to y}}\bigr) +\sum_{i=1}^d\bigl(h_{yx_i}|\delta_y\rangle\langle\delta_{x_i}| +h_{x_iy}|\delta_{x_i}\rangle\langle\delta_y|\bigr). \end{equation} \end{lemma} \begin{proof} Since the edges of $D'$ are retained by $(F,D)$, we have $W_{F,D\cup D'}=W_{F,D}+W_{D'}$. The components of $(F,D\cup D')$ are the components of $(F,D)$ other than $C$, together with $C'_1,\dots,C'_k$. Applying \Cref{thm:tree-esa}(iv) to both finite removals, we find that $\mathcal H-W_{F,D}-W_{D'}$ equals the orthogonal sum of the operators $\mathcal H_{C''}$ over the components $C''\neq C$ of $(F,D)$, of $\mathcal H_C-W_{D'}$, and of $0_F$, and that it also equals the same sum with $\mathcal H_C-W_{D'}$ replaced by $\mathcal H_{C'_1}\oplus\cdots\oplus\mathcal H_{C'_k}$. Both decompositions are orthogonal with respect to the same splitting of $\ell^2(V)$, and their parts acting in $\ell^2(C)$ agree, including their domains. Since $W_{D'}$ is bounded, $\mathcal H_C-W_{D'}$ has the domain $\Dom(\mathcal H_C)$. For \eqref{eq:tree-star}, take $D'$ to be the set of edges of $C$ at $y$. By \Cref{lem:tree-branch}(ii), the components of $(F,D\cup D')$ inside $C$ are $\{y\}$ and the branches $C_{x_i\to y}$, and the operator of the isolated vertex $y$ is zero. \end{proof} \section{Resolvents and the cavity recursion}\label{sec:tree-resolvents} \subsection{Definitions} We begin with the elementary properties of resolvent entries of a self-adjoint operator. \begin{lemma}\label{lem:tree-herglotz} Let $T$ be a self-adjoint operator on a Hilbert space $\mathfrak h$, let $f,g\in\mathfrak h$, and let $z=E+\iu\eta\in\C\setminus\R$. \begin{enumerate}[label=\textup{(\roman*)}] \item The function $z\mapsto\langle f,(T-z)^{-1}g\rangle$ is analytic on $\C\setminus\R$, bounded by $\|f\|\|g\|/|\eta|$, and satisfies $\langle f,(T-z)^{-1}g\rangle=\overline{\langle g,(T-\bar z)^{-1}f\rangle}$. \item The function $m(z)=\langle f,(T-z)^{-1}f\rangle$ satisfies \[ m(z)=\int_\R\frac{\mu^T_f(\dd\lambda)}{\lambda-z}, \qquad \Im m(z)=\eta\,\bigl\|(T-z)^{-1}f\bigr\|^2=\eta\int_\R\frac{\mu^T_f(\dd\lambda)}{|\lambda-z|^2}, \] and $m(\bar z)=\overline{m(z)}$. In particular, if $f\ne0$ and $\eta>0$, then $\Im m(z)>0$. \end{enumerate} \end{lemma} \begin{proof} The resolvent is an analytic operator-valued function on $\C\setminus\R$ with norm at most $|\eta|^{-1}$, and $((T-z)^{-1})^*=(T-\bar z)^{-1}$; this gives (i). The integral formula is the spectral theorem. By the resolvent identity, $(T-z)^{-1}-(T-\bar z)^{-1}=(z-\bar z)(T-\bar z)^{-1}(T-z)^{-1}$, and we have \begin{align*} 2\iu\Im m(z)=m(z)-\overline{m(z)} &=\bigl\langle f,\bigl((T-z)^{-1}-(T-\bar z)^{-1}\bigr)f\bigr\rangle\\ &=2\iu\eta\,\bigl\langle(T-z)^{-1}f,(T-z)^{-1}f\bigr\rangle, \end{align*} where we used (i) in the form $\overline{m(z)}=m(\bar z)$. Since the resolvent is injective, $\Im m(z)>0$ when $f\ne0$ and $\eta>0$. \end{proof} \begin{definition}\label{def:tree-resolvent} On $\Omega_{\mathrm{sa}}$, let $(F,D)$ be a finite removal, let $C$ be one of its components, and let $z\in\C\setminus\R$. \begin{enumerate}[label=\textup{(\roman*)}] \item The \emph{resolvent entries} are \[ G_{uv}(z)=\bigl\langle\delta_u,(\mathcal H-z)^{-1}\delta_v\bigr\rangle, \qquad G^C_{uv}(z)=\bigl\langle\delta_u,(\mathcal H_C-z)^{-1}\delta_v\bigr\rangle, \] \[ G^{F,D}_{uv}(z)=\bigl\langle\delta_u,(\mathcal H^{F,D}-z)^{-1}\delta_v\bigr\rangle, \] for $u,v\in V$, $u,v\in C$, and $u,v\in V\setminus F$, respectively. We write $G^{(y)}$ for $G^{\{y\},\emptyset}$. Since $\mathcal H^{F,D}$ is the orthogonal sum of the operators of its components, $G^{F,D}_{uv}=G^C_{uv}$ if $u,v\in C$, and $G^{F,D}_{uv}=0$ if $u$ and $v$ lie in different components. \item For adjacent $u,w\in C$, the \emph{cavity resolvent} is the diagonal entry at $u$ of the operator of the branch $C_{u\to w}$: \[ \Gamma^C_{u\to w}(z)=G^{C_{u\to w}}_{uu}(z). \] For $C=V$ we write $\Gamma_{u\to w}$. The \emph{forward cavity} at $v\in V\setminus\{o\}$ is $\Gamma_v=\Gamma_{v\to v^-}$; since $V_{v\to v^-}=V(\mathcal T_v)$, it equals $\langle\delta_v,(\mathcal H_v-z)^{-1}\delta_v\rangle$. At the root we set $\Gamma_o=G_{oo}$, the diagonal entry at the root of the operator of its forward subtree $\mathcal T_o=\mathcal T$. \item The \emph{spectral measure} of $v\in V$ is $\mu_v=\mu^{\mathcal H}_{\delta_v}$, and $G_{vv}(z)=\int\mu_v(\dd\lambda)/(\lambda-z)$. \end{enumerate} \end{definition} By \Cref{lem:tree-herglotz}, all these entries are analytic in $z\in\C\setminus\R$ and bounded by $1/|\Im z|$, and the diagonal entries and cavity resolvents map $\Hup$ to $\Hup$. They are also measurable functions of the marked tree. Indeed, for a fixed finite removal and fixed words $u,v$, \Cref{thm:tree-esa}(iii) with $\Lambda_n=\{x\in C:|x|\le n\}$ expresses $G^C_{uv}(z)$ as the limit of entries of the inverses of finite matrices, which are measurable functions of $(N_x)_{|x|0$. On $\Omega_{\mathrm{sa}}$, the following statements hold for every $z\in\C\setminus\R$; statements \textup{(i)} and \textup{(ii)} hold for every finite removal $(F,D)$ and every component $C$ of $(F,D)$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $u\in C$, \begin{equation}\label{eq:tree-schur-diag} G^C_{uu}(z)=-\Bigl(z+\sum_{x\in C,\,x\sim u}|h_{ux}|^2\,\Gamma^C_{x\to u}(z)\Bigr)^{-1}. \end{equation} \item For all adjacent $u,w\in C$, \begin{equation}\label{eq:tree-schur-cavity} \Gamma^C_{u\to w}(z)=-\Bigl(z+\sum_{x\in C,\,x\sim u,\,x\ne w}|h_{ux}|^2\,\Gamma^C_{x\to u}(z)\Bigr)^{-1}. \end{equation} \item Let $N=N_o$, $Y_j=|w_j|^2$, and $\Gamma_j(z)=\Phi_z(\theta_j\mathcal T)$ for $j\ge1$. Then \begin{equation}\label{eq:tree-cavity-law} \Gamma_o(z)=-\Bigl(z+\frac1p\sum_{j=1}^NY_j\,\Gamma_j(z)\Bigr)^{-1}, \end{equation} where $N$ has the law $\Pois(p)$, the $Y_j$ are independent with the law $\Exp(1)$, the random analytic functions $\Gamma_j$ on $\C\setminus\R$ are independent copies of $\Gamma_o$, and the three families $N$, $(Y_j)_{j\ge1}$, $(\Gamma_j)_{j\ge1}$ are independent. In particular, $\Gamma_o(z)\eqdist-\bigl(z+p^{-1}\sum_{j=1}^NY_j\Gamma_j(z)\bigr)^{-1}$. For $j\le N$, the function $\Gamma_j$ is the forward cavity at the child $j$ of the root, and $\Gamma_o(z)$ is the limit of the root entries of the resolvents of the Dirichlet truncations to the balls $\{v\in V:|v|\le n\}$. \end{enumerate} In (i) and (ii), the imaginary part of the expression in parentheses has the sign of $\Im z$ and absolute value at least $|\Im z|$. For the full tree, these formulas read \begin{align} \Gamma_v(z)&=-\Bigl(z+\sum_{j=1}^{N_v}|h_{v,vj}|^2\,\Gamma_{vj}(z)\Bigr)^{-1}, \qquad v\in V,\label{eq:tree-forward}\\ G_{vv}(z)&=-\Bigl(z+|h_{vv^-}|^2\,\Gamma_{v^-\to v}(z) +\sum_{j=1}^{N_v}|h_{v,vj}|^2\,\Gamma_{vj}(z)\Bigr)^{-1}, \qquad v\in V\setminus\{o\},\label{eq:tree-full-diag}\\ \Gamma_{u\to v}(z)&=-\Bigl(z+\one_{\{u\ne o\}}|h_{uu^-}|^2\,\Gamma_{u^-\to u}(z) +\sum_{j\le N_u,\,uj\ne v}|h_{u,uj}|^2\,\Gamma_{uj}(z)\Bigr)^{-1},\label{eq:tree-rootward} \end{align} where in \eqref{eq:tree-rootward}, $u\in V$ and $v$ is a child of $u$. \end{proposition} The cavity $\Gamma_{u\to v}$ in \eqref{eq:tree-rootward} is a \emph{rootward} cavity: it is the diagonal entry at $u$ of the operator on $V\setminus V(\mathcal T_v)$. Formula \eqref{eq:tree-rootward} expresses it through the rootward cavity one generation closer to the root and through forward cavities; along a path from the root, the rootward cavities can be computed recursively from the root downwards. Note that \eqref{eq:tree-forward} with $v=o$ is the formula for $\Gamma_o=G_{oo}$. \begin{proof} Fix $C$, $u\in C$ and $z$, and let $x_1,\dots,x_d$ be the neighbors of $u$ in $C$. By \Cref{lem:tree-decouple}, with respect to $\ell^2(C)=\C\delta_u\oplus\mathfrak k$, where $\mathfrak k=\bigoplus_{i}\ell^2(C_{x_i\to u})$, we have $\Dom(\mathcal H_C)=\C\delta_u\oplus\Dom(\mathcal D)$ with $\mathcal D=\bigoplus_i\mathcal H_{C_{x_i\to u}}$, and \[ \mathcal H_C(\alpha\delta_u+g)=\langle a,g\rangle\,\delta_u+\alpha a+\mathcal Dg, \qquad \alpha\in\C,\ g\in\Dom(\mathcal D), \] where $a=\sum_ih_{x_iu}\delta_{x_i}\in\mathfrak k$. Here we used that $\langle a,g\rangle=\sum_i\overline{h_{x_iu}}g(x_i)=\sum_ih_{ux_i}g(x_i)$. The operator $\mathcal D$ is self-adjoint on $\mathfrak k$, and since it is block diagonal, \[ \bigl\langle a,(\mathcal D-z)^{-1}a\bigr\rangle =\sum_{i=1}^d|h_{ux_i}|^2\bigl\langle\delta_{x_i},(\mathcal H_{C_{x_i\to u}}-z)^{-1} \delta_{x_i}\bigr\rangle =\sum_{i=1}^d|h_{ux_i}|^2\,\Gamma^C_{x_i\to u}(z). \] By \Cref{lem:tree-herglotz}(ii), the imaginary part of this quantity has the sign of $\Im z$ (or vanishes), and $s=-z-\langle a,(\mathcal D-z)^{-1}a\rangle$ satisfies $|\Im s|\ge|\Im z|>0$. Set $\alpha=s^{-1}$ and $g=-\alpha(\mathcal D-z)^{-1}a\in\Dom(\mathcal D)$. Then $f=\alpha\delta_u+g\in\Dom(\mathcal H_C)$, and \[ (\mathcal H_C-z)f=\bigl(\langle a,g\rangle-z\alpha\bigr)\delta_u+\alpha a+(\mathcal D-z)g =\alpha\bigl(-\langle a,(\mathcal D-z)^{-1}a\rangle-z\bigr)\delta_u=\delta_u. \] This shows that $f=(\mathcal H_C-z)^{-1}\delta_u$, and $G^C_{uu}(z)=\langle\delta_u,f\rangle=\alpha=s^{-1}$, which is \eqref{eq:tree-schur-diag}. Reading off the other coordinates of $f$, we also obtain, for every $i$ and every $v\in C_{x_i\to u}$, \begin{equation}\label{eq:tree-column} G^C_{vu}(z)=-G^C_{uu}(z)\,h_{x_iu}\,G^{C_{x_i\to u}}_{vx_i}(z). \end{equation} For (ii), let $C'=C_{u\to w}$, which is a component of the finite removal $(F,D\cup\{\{u,w\}\})$ by \Cref{lem:tree-branch}(ii). The neighbors of $u$ in $C'$ are its neighbors in $C$ other than $w$. For such a neighbor $x$, every vertex of $C_{x\to u}$ is joined to $w$ by a path through $x$ and $u$. This gives $C_{x\to u}\subset C'$, and \Cref{lem:tree-branch}(iii) implies that $C'_{x\to u}=C_{x\to u}$. Applying \eqref{eq:tree-schur-diag} to $C'$ at $u$, and using $G^{C'}_{uu}=\Gamma^C_{u\to w}$ and $\Gamma^{C'}_{x\to u}=\Gamma^C_{x\to u}$, we obtain \eqref{eq:tree-schur-cavity}. For the full tree, the branch $V_{vj\to v}$ is $V(\mathcal T_{vj})$ and $\Gamma_{vj\to v}=\Gamma_{vj}$, and the neighbors of $v$ are its children and, if $v\ne o$, its parent. With these identifications, \eqref{eq:tree-forward} and \eqref{eq:tree-full-diag} are \eqref{eq:tree-schur-cavity} with $(u,w)=(v,v^-)$ (or \eqref{eq:tree-schur-diag} at $u=o$) and \eqref{eq:tree-schur-diag} at $u=v$. Formula \eqref{eq:tree-rootward} is \eqref{eq:tree-schur-cavity} with $(u,w)=(u,v)$, since the neighbors of $u$ other than $v$ are its parent (if $u\ne o$) and its other children. For (iii), on $\Omega_{\mathrm{sa}}$, \eqref{eq:tree-forward} at $v=o$, the identity $|h_{oj}|^2=Y_j/p$, and \eqref{eq:tree-shift-cavity} give \eqref{eq:tree-cavity-law}; the functions $\Gamma_j$ with $j>N$ do not enter it. By \eqref{eq:tree-branching}, the marked trees $\theta_j\mathcal T$ are independent copies of $\mathcal T$, independent of $N_o$ and $(w_j)_{j\ge1}$. The functions $\Gamma_j=\Phi(\theta_j\mathcal T)$ are independent copies of $\Phi(\mathcal T)$, which equals $\Gamma_o$ almost surely, and they are independent of $N$ and $(Y_j)_{j\ge1}$. The variables $N_o$ and $(w_j)_{j\ge1}$ are independent by construction, and $Y_j=|w_j|^2$ has the law $\Exp(1)$. The last statement is \Cref{thm:tree-esa}(iii). \end{proof} \begin{remark}\label{rem:tree-reroot} Let $x\in V$ and let $o=v_0,v_1,\dots,v_k=x$ be the path from the root to $x$. When $\mathcal T$ is rerooted at $x$, the parent of a vertex $v\neq x$ is the next vertex on the path from $v$ to $x$. If $v$ is not on the path, this is $v^-$, since the path from $v$ to $x$ first climbs to the last common ancestor of $v$ and $x$. The forward cavities of the rerooted tree are $\Gamma_v$ for $v$ off the path, and $\Gamma_{v_j\to v_{j+1}}$ for $0\le j0$, then $\Im G^C_{uu}(z)>0$ and $\Im\Gamma^C_{u\to w}(z)>0$ for all $u$ and all neighbors $w$ of $u$ in $C$. \item For all adjacent $u,w\in C$, \begin{align*} \Im\Gamma^C_{u\to w}(z)&=|\Gamma^C_{u\to w}(z)|^2\Bigl(\eta +\sum_{x\in C,\,x\sim u,\,x\ne w}|h_{ux}|^2\,\Im\Gamma^C_{x\to u}(z)\Bigr),\\ \Im G^C_{uu}(z)&=|G^C_{uu}(z)|^2\Bigl(\eta +\sum_{x\in C,\,x\sim u}|h_{ux}|^2\,\Im\Gamma^C_{x\to u}(z)\Bigr). \end{align*} \item Let $u,x\in C$, and let $y$ be a neighbor of $x$ in $C$ which does not lie on the path from $u$ to $x$. If $\eta>0$, then \[ \Im G^C_{uu}(z)\ge|G^C_{ux}(z)|^2\,|h_{xy}|^2\,\Im\Gamma^C_{y\to x}(z). \] In particular, $\Im G_{oo}(z)\ge|G_{ox}(z)|^2|h_{xy}|^2\Im\Gamma_y(z)$ for every $x\in V$ and every child $y$ of $x$. \end{enumerate} \end{lemma} \begin{proof} For (i), we have $\sum_v|G^C_{vu}(z)|^2=\|(\mathcal H_C-z)^{-1}\delta_u\|^2$, which equals $\Im G^C_{uu}(z)/\eta$ by \Cref{lem:tree-herglotz}(ii). By \Cref{lem:tree-herglotz}(i), we have $|G^C_{uv}(z)|=|G^C_{vu}(\bar z)|$, and the first sum equals $\Im G^C_{uu}(\bar z)/(-\eta) =\Im G^C_{uu}(z)/\eta$. The positivity statements follow from \Cref{lem:tree-herglotz}(ii), since $\Gamma^C_{u\to w}$ is a diagonal entry of the operator of $C_{u\to w}$. For (ii), write \eqref{eq:tree-schur-cavity} as $\Gamma=-1/s$. Then $\Im\Gamma=\Im s/|s|^2=|\Gamma|^2\Im s$, and $\Im s$ is the expression in parentheses. The same computation applies to \eqref{eq:tree-schur-diag}. For (iii), let $B=C_{y\to x}$. Let $u=v_0,\dots,v_k=x$ be the path from $u$ to $x$ (with $k=0$ if $u=x$). No vertex of this path lies in $B$: otherwise, by \Cref{lem:tree-branch}(ii), its path to $x$, which is part of the path from $u$ to $x$, would contain $y$. Let $v\in B$ and let $y=v_{k+1},\dots,v_L=v$ be the path from $y$ to $v$, which lies in $B$. Then $v_0,\dots,v_L$ is the path from $u$ to $v$. Applying \eqref{eq:tree-path-row} in $C$ to the paths from $u$ to $x$ and from $u$ to $v$, and in $B$ to the path from $y$ to $v$, and using $G^B_{yy}=\Gamma^C_{y\to x}$ and $\Gamma^B_{v_j\to v_{j-1}}=\Gamma^C_{v_j\to v_{j-1}}$ for $j\ge k+2$ (by \Cref{lem:tree-branch}(iii), as in the proof of \Cref{prop:tree-path}), we obtain \[ G^C_{uv}(z)=-G^C_{ux}(z)\,h_{xy}\,G^B_{yv}(z),\qquad v\in B. \] Using (i) in $C$ and in $B$, we find \[ \frac{\Im G^C_{uu}(z)}{\eta}\ge\sum_{v\in B}|G^C_{uv}(z)|^2 =|G^C_{ux}(z)|^2|h_{xy}|^2\sum_{v\in B}|G^B_{yv}(z)|^2 =|G^C_{ux}(z)|^2|h_{xy}|^2\,\frac{\Im\Gamma^C_{y\to x}(z)}{\eta}. \] For the last statement, a child $y$ of $x$ is never on the path from $o$ to $x$, and $\Gamma_{y\to x}=\Gamma_y$. \end{proof} \section{Gauge and parity symmetries}\label{sec:tree-symmetry} The phases of the edge weights can be removed by a diagonal unitary transformation, because a tree has no cycles. For $\zeta\in\C$, let $\sigma(\zeta)=\zeta/|\zeta|$ if $\zeta\ne0$ and $\sigma(0)=1$. \begin{lemma}\label{lem:tree-gauge} On $\Omega_{\mathrm{sa}}$, let $C$ be a component of a finite removal and fix $r\in C$. For $v\in C$, let $r=v_0,v_1,\dots,v_k=v$ be the path from $r$ to $v$ and set $\varphi_v=\prod_{j=1}^k\sigma(h_{v_{j-1}v_j})$, and let $U$ be the unitary operator $(Uf)(v)=\varphi_vf(v)$ on $\ell^2(C)$. Then the weighted adjacency operator of $C$ with the nonnegative weights $|h_{uv}|$ is essentially self-adjoint on $c_c(C)$, and its closure $\mathcal H^{\mathrm{abs}}_C$ satisfies $U\mathcal H_CU^*=\mathcal H^{\mathrm{abs}}_C$. Further, for all $u,v\in C$ and $z\in\C\setminus\R$, writing $G^{\mathrm{abs}}$ for the resolvent entries of $\mathcal H^{\mathrm{abs}}_C$, \[ G^C_{uv}(z)=\overline{\varphi_u}\,\varphi_v\,G^{\mathrm{abs}}_{uv}(z), \qquad G^{\mathrm{abs}}_{uv}(z)=G^{\mathrm{abs}}_{vu}(z), \qquad |G^C_{uv}(z)|=|G^C_{vu}(z)|. \] The same identities hold for the entries of $G^{F,D}$, which vanish between different components. \end{lemma} \begin{proof} Let $u,v\in C$ be adjacent, and suppose that $u$ lies on the path from $r$ to $v$. Then $\varphi_v=\varphi_u\sigma(h_{uv})$, and \[ \varphi_u\,h_{uv}\,\overline{\varphi_v}=h_{uv}\overline{\sigma(h_{uv})}=|h_{uv}|, \qquad \varphi_v\,h_{vu}\,\overline{\varphi_u}=\sigma(h_{uv})\overline{h_{uv}}=|h_{uv}|, \] where we used $|\varphi_u|=1$ and $h_{vu}=\overline{h_{uv}}$. Every pair of adjacent vertices is of this form after possibly exchanging $u$ and $v$. Since $U$ is diagonal, it maps $c_c(C)$ onto itself and $U\mathcal H_{C,0}U^*$ is the weighted adjacency operator with the weights $|h_{uv}|$ on $c_c(C)$. Since conjugation by a unitary operator commutes with taking closures and adjoints, this operator is essentially self-adjoint with closure $U\mathcal H_CU^*$. Then $U(\mathcal H_C-z)^{-1}U^*=(\mathcal H^{\mathrm{abs}}_C-z)^{-1}$, and taking the $(u,v)$ entry gives the first identity. For the second identity, let $K$ be complex conjugation, $(Kf)(v)=\overline{f(v)}$. It is antiunitary, maps $c_c(C)$ onto itself, and commutes with the minimal operator with real weights. Taking closures, we have $K\mathcal H^{\mathrm{abs}}_CK=\mathcal H^{\mathrm{abs}}_C$. If $f=(\mathcal H^{\mathrm{abs}}_C-z)^{-1}Kg$, then $Kf\in\Dom(\mathcal H^{\mathrm{abs}}_C)$ and $(\mathcal H^{\mathrm{abs}}_C-\bar z)Kf=K(\mathcal H^{\mathrm{abs}}_C-z)f=g$. This proves that $K(\mathcal H^{\mathrm{abs}}_C-z)^{-1}K=(\mathcal H^{\mathrm{abs}}_C-\bar z)^{-1}$. Using \Cref{lem:tree-herglotz}(i) and $\langle\delta_u,Kf\rangle=\overline{\langle\delta_u,f\rangle}$, we have \[ G^{\mathrm{abs}}_{vu}(z)=\overline{\bigl\langle\delta_u,(\mathcal H^{\mathrm{abs}}_C-\bar z)^{-1} \delta_v\bigr\rangle} =\overline{\bigl\langle\delta_u,K(\mathcal H^{\mathrm{abs}}_C-z)^{-1}K\delta_v\bigr\rangle} =G^{\mathrm{abs}}_{uv}(z). \] The third identity follows from the first two, since $|\varphi_u|=|\varphi_v|=1$. The statement for $G^{F,D}$ follows by applying the lemma to each component. \end{proof} Recall that $V_0$ and $V_1$ are the sets of vertices of even and odd generation. Every edge of $\mathcal T$ joins $V_0$ and $V_1$. \begin{proposition}\label{prop:tree-symmetry} \leavevmode \begin{enumerate}[label=\textup{(\roman*)}] \item On $\Omega_{\mathrm{sa}}$, let $C$ be a component of a finite removal, and let $J$ be the unitary operator $(Jf)(v)=(-1)^{|v|}f(v)$ on $\ell^2(C)$. Then $J\mathcal H_CJ=-\mathcal H_C$. Further, $J\one_B(\mathcal H_C)J=\one_{-B}(\mathcal H_C)$ for every Borel set $B\subset\R$; the spectral measure of every $\delta_v$, $v\in C$, is invariant under $\lambda\mapsto-\lambda$; \[ G^C_{uv}(-z)=-(-1)^{|u|+|v|}\,G^C_{uv}(z),\qquad u,v\in C,\ z\in\C\setminus\R; \] and for every Borel set $I\subset\R$, the operator $\mathcal H_C$ has pure point spectrum in $I$ if and only if it has pure point spectrum in $-I$, and it is purely absolutely continuous in $I$ if and only if it is purely absolutely continuous in $-I$. In particular, this holds for $\mathcal H$ under $\P$ and under $\P^\infty$. \item For the finite model of \Cref{def:tree-model}, the pairs $(G_n,H_n)$ and $(G_n,-H_n)$ have the same law. In particular $H_n\eqdist-H_n$ and $H_n^{\gc}\eqdist-H_n^{\gc}$. \end{enumerate} \end{proposition} \begin{proof} For (i), the operator $J$ is a self-adjoint unitary which maps $c_c(C)$ onto itself. For $f\in c_c(C)$ and $v\in C$, we have \[ (J\mathcal H_{C,0}Jf)(v)=(-1)^{|v|}\sum_{u\sim v}h_{vu}(-1)^{|u|}f(u)=-(\mathcal H_{C,0}f)(v), \] since adjacent vertices have generations differing by one. Taking closures, we find $J\mathcal H_CJ=-\mathcal H_C$. Since the functional calculus is invariant under unitary conjugation, we have $J\phi(\mathcal H_C)J=\phi(-\mathcal H_C)$ for every bounded Borel function $\phi$. With $\phi=\one_B$, this gives $J\one_B(\mathcal H_C)J=\one_{-B}(\mathcal H_C)$, and with $\phi(\lambda)=(\lambda-z)^{-1}$ it gives $J(\mathcal H_C-z)^{-1}J=-(\mathcal H_C+z)^{-1}$, whose $(u,v)$ entry is the displayed identity. For every $f\in\ell^2(C)$ and Borel $B$, we find \[ \mu^{\mathcal H_C}_{Jf}(B)=\langle f,J\one_B(\mathcal H_C)Jf\rangle=\mu^{\mathcal H_C}_f(-B). \] Since $J\delta_v=\pm\delta_v$, the spectral measure of $\delta_v$ is reflection invariant. If $\mathcal H_C$ has pure point spectrum in $I$, then for every $f$ the restriction of $\mu^{\mathcal H_C}_{Jf}$ to $I$ is purely atomic, and by the last display the restriction of $\mu^{\mathcal H_C}_f$ to $-I$ is purely atomic, and $\mathcal H_C$ has pure point spectrum in $-I$. The same argument applies to absolute continuity, since reflection preserves Lebesgue null sets. For (ii), since the variables $-w_{ij}$ are independent standard complex Gaussians, independent of $G_n$, the pairs $(G_n,(w_{ij}))$ and $(G_n,(-w_{ij}))$ have the same law. The matrix $H_n$ is a function of $(G_n,(w_{ij}))$, and replacing $(w_{ij})$ by $(-w_{ij})$ replaces $H_n$ by $-H_n$. The set $\gc$ is a function of $G_n$. \end{proof} \section{The bipartite factorization}\label{sec:tree-bipartite} Since every edge joins $V_0$ and $V_1$, the operator $\mathcal H$ maps functions on $V_0$ to functions on $V_1$ and conversely, and its square splits into two nonnegative blocks. In this section, we make this precise for the unbounded operator, and we write each block as a sum of rank-one terms, one for each vertex of the opposite parity, with independent Gamma-distributed squared norms. We begin with an abstract lemma. \begin{lemma}\label{lem:tree-block} Let $\mathfrak h_0,\mathfrak h_1$ be Hilbert spaces and let $A$ be a closed, densely defined operator from $\Dom(A)\subset\mathfrak h_0$ to $\mathfrak h_1$. Define the operator $T_A$ on $\mathfrak h_0\oplus\mathfrak h_1$ by \[ \Dom(T_A)=\Dom(A)\oplus\Dom(A^*),\qquad T_A(f,g)=(A^*g,Af). \] \begin{enumerate}[label=\textup{(\roman*)}] \item The operator $T_A$ is self-adjoint, $\Dom(T_A^2)=\Dom(A^*A)\oplus\Dom(AA^*)$, and $T_A^2=A^*A\oplus AA^*$. The operators $A^*A$ and $AA^*$ are nonnegative and self-adjoint. Further, $\Dom((A^*A)^{1/2})=\Dom(A)$ and $\|(A^*A)^{1/2}f\|=\|Af\|$ for $f\in\Dom(A)$. \item Let $z\in\C\setminus\R$. Then $z^2\notin[0,\infty)$, and with $Q_0=(A^*A-z^2)^{-1}$ and $Q_1=(AA^*-z^2)^{-1}$, \[ (T_A-z)^{-1}(f,g)=\bigl(zQ_0f+A^*Q_1g,\ AQ_0f+zQ_1g\bigr), \qquad f\in\mathfrak h_0,\ g\in\mathfrak h_1. \] In particular, $P_0(T_A-z)^{-1}P_0=zQ_0$ and $P_1(T_A-z)^{-1}P_1=zQ_1$, where $P_0,P_1$ are the projections onto $\mathfrak h_0,\mathfrak h_1$. \item Let $z\in\C\setminus\R$ and $g\in\Dom(A^*)$. Then $Q_0A^*g=A^*Q_1g$, and \[ \bigl\langle A^*g,(A^*A-z^2)^{-1}A^*g\bigr\rangle=\|g\|^2+z\,\bigl\langle g,(T_A-z)^{-1}g\bigr\rangle. \] \item Let $B$ be a densely defined operator from $\Dom(B)\subset\mathfrak h_1$ to $\mathfrak h_0$. If the operator $(f,g)\mapsto(Bg,Af)$ with domain $\Dom(A)\oplus\Dom(B)$ is self-adjoint, then $B=A^*$, including domains. \end{enumerate} \end{lemma} \begin{proof} We begin with (iv) and a computation of adjoints. Let $T'$ be the operator in (iv). A pair $(\varphi,\psi)$ is in $\Dom(T'^*)$ exactly when the map $(f,g)\mapsto\langle(\varphi,\psi),T'(f,g)\rangle=\langle\varphi,Bg\rangle+\langle\psi,Af\rangle$ is bounded on $\Dom(A)\oplus\Dom(B)$. Taking $g=0$ and $f=0$ separately, we see that this holds exactly when $\psi\in\Dom(A^*)$ and $\varphi\in\Dom(B^*)$, and then $T'^*(\varphi,\psi)=(A^*\psi,B^*\varphi)$. If $T'=T'^*$, then comparing domains, we find $\Dom(B)=\Dom(A^*)$, and comparing first components, we find $B\psi=A^*\psi$ for $\psi\in\Dom(B)$. This proves (iv). For (i), the same computation with $B=A^*$ gives $\Dom(T_A^*)=\Dom(A^{**})\oplus\Dom(A^*)$ and $T_A^*(\varphi,\psi)=(A^*\psi,A^{**}\varphi)$. Since $A$ is closed, we have $A^{**}=A$ and $T_A^*=T_A$. A pair $(f,g)\in\Dom(T_A)$ belongs to $\Dom(T_A^2)$ exactly when $(A^*g,Af)\in\Dom(A)\oplus\Dom(A^*)$, that is, when $f\in\Dom(A^*A)$ and $g\in\Dom(AA^*)$, and then $T_A^2(f,g)=(A^*Af,AA^*g)$. The operator $T_A^2$ is self-adjoint by the functional calculus. As shown in the proof of \Cref{thm:tree-esa}, the adjoint of an orthogonal sum is the orthogonal sum of the adjoints. Hence $A^*A$ and $AA^*$ are self-adjoint; they are nonnegative because $\langle f,A^*Af\rangle=\|Af\|^2$. By the functional calculus, $|T_A|=(T_A^2)^{1/2}=(A^*A)^{1/2}\oplus(AA^*)^{1/2}$, and $\Dom(|T_A|)=\Dom(T_A)$ with $\||T_A|\varphi\|=\|T_A\varphi\|$, since both sides equal $\bigl(\int\lambda^2\dd\mu^{T_A}_\varphi\bigr)^{1/2}$. Applying this to $\varphi=(f,0)$ gives the last assertion of (i). For (ii), since $z^2\in[0,\infty)$ only if $z$ is real, the operators $Q_0$ and $Q_1$ are bounded. Let $f\in\mathfrak h_0$ and $\phi=Q_0f\in\Dom(A^*A)$. Then $\phi\in\Dom(A)$ and $A\phi\in\Dom(A^*)$, which gives $(z\phi,A\phi)\in\Dom(T_A)$ and \[ (T_A-z)(z\phi,A\phi)=\bigl(A^*A\phi-z^2\phi,\ zA\phi-zA\phi\bigr)=(f,0). \] Similarly, for $g\in\mathfrak h_1$ and $\psi=Q_1g\in\Dom(AA^*)$, we have $(A^*\psi,z\psi)\in \Dom(T_A)$ and $(T_A-z)(A^*\psi,z\psi)=(0,g)$. Adding these gives (ii). For (iii), let $\psi=Q_1g$. Then $A^*\psi\in\Dom(A)$, and $AA^*\psi=g+z^2\psi$ belongs to $\Dom(A^*)$, because $g$ and $\psi$ do. We conclude that $A^*\psi\in\Dom(A^*A)$ and \[ (A^*A-z^2)A^*\psi=A^*(AA^*\psi)-z^2A^*\psi=A^*(g+z^2\psi)-z^2A^*\psi=A^*g, \] so $Q_0A^*g=A^*\psi$. Since $A^*\psi\in\Dom(A)$ and $g\in\Dom(A^*)$, we obtain \begin{align*} \langle A^*g,Q_0A^*g\rangle=\langle A^*g,A^*\psi\rangle=\langle g,AA^*\psi\rangle &=\|g\|^2+z^2\langle g,Q_1g\rangle\\ &=\|g\|^2+z\langle g,(T_A-z)^{-1}g\rangle, \end{align*} where in the last step we used (ii). \end{proof} We now apply this to the tree. For a component $C$ of a finite removal, set $C_0=C\cap V_0$ and $C_1=C\cap V_1$, identify $\ell^2(C)=\ell^2(C_0)\oplus\ell^2(C_1)$, and write $P_0$ and $P_1$ for the corresponding projections. For $y\in C_1$, let $d_y$ be the number of neighbors of $y$ in $C$. For $y\in C_1$, the \emph{row vector}, the \emph{squared row norm} and, when $U_y>0$, the \emph{row direction} are \begin{equation}\label{eq:tree-rows} a_y=\sum_{x\in C,\,x\sim y}h_{xy}\,\delta_x\in\ell^2(C_0), \qquad U_y=\|a_y\|^2=\sum_{x\in C,\,x\sim y}|h_{xy}|^2, \qquad v_y=\frac{a_y}{\sqrt{U_y}}. \end{equation} These depend on $C$; when $C=V$ we have $d_y=\deg(y)=N_y+1$. Almost surely all edge weights are nonzero, and then $U_y>0$ exactly when $d_y\ge1$. For $y\in C_1$, let $\mathcal H^{(y)}_C=\bigoplus_{x\in C,\,x\sim y}\mathcal H_{C_{x\to y}}$ be the operator on $\ell^2(C\setminus\{y\})$ with the vertex $y$ deleted; by \Cref{lem:tree-branch}(ii), the branches $C_{x\to y}$ are the components of $C\setminus\{y\}$. For $C=V$, this is $\mathcal H^{(y)}$. \begin{proposition}\label{prop:tree-bipartite} On $\Omega_{\mathrm{sa}}$, let $C$ be a component of a finite removal $(F,D)$; for instance $C=V$. Write $\mathcal H_C$ and $G^C$ for its operator and resolvent. \begin{enumerate}[label=\textup{(\roman*)}] \item There exists a unique closed, densely defined operator $A$ from $\Dom(A)\subset\ell^2(C_0)$ to $\ell^2(C_1)$ such that $\mathcal H_C=T_A$, that is, \[ \mathcal H_C=\begin{pmatrix}0&A^*\\A&0\end{pmatrix}, \qquad \Dom(\mathcal H_C)=\Dom(A)\oplus\Dom(A^*). \] Its domain is $\Dom(A)=\{f\in\ell^2(C_0):\mathcal H_{C,\mathrm{form}}f\in\ell^2(C_1)\}$. The space $c_c(C_0)$ is a core for $A$, and $c_c(C_1)$ is a core for $A^*$. Further, $\mathcal H_C^2=A^*A\oplus AA^*$, with $\Dom(\mathcal H_C^2)=\Dom(A^*A)\oplus\Dom(AA^*)$. \item For $y\in C_1$, we have $\delta_y\in\Dom(A^*)$ and $A^*\delta_y=a_y$. For $f\in\Dom(A)$ and $y\in C_1$, we have $(Af)(y)=\langle a_y,f\rangle$. Further, \[ \Dom(A)=\Bigl\{f\in\ell^2(C_0):\sum_{y\in C_1:\,U_y>0}U_y|\langle v_y,f\rangle|^2<\infty\Bigr\}, \] and $\|Af\|^2=\sum_{y\in C_1:\,U_y>0}U_y|\langle v_y,f\rangle|^2$ for $f\in\Dom(A)$, so the closed quadratic form $f\mapsto\|(A^*A)^{1/2}f\|^2$ of $A^*A$, with form domain $\Dom(A)$, is the sum of the rank-one forms $f\mapsto U_y|\langle v_y,f\rangle|^2$. The closed linear span of $\{v_y:y\in C_1,\ U_y>0\}$ is $\overline{\Ran A^*}=(\ker A)^\perp$. \item Suppose that $(F,D)$ is selected without inspecting the retained marks, with inspected set $Q$, and that $C$ is determined by $\mathcal G_Q$ (for instance, the component containing a fixed word, on the event that this word is in $V\setminus F$). Conditionally on $\mathcal G_Q$, the pairs $(U_y,v_y)$, for $y\in C_1$ with $d_y\ge1$, are independent; the direction $v_y$ is uniformly distributed on the unit sphere of the $d_y$-dimensional space $\spn\{\delta_x:x\in C,\ x\sim y\}$ and is independent of $U_y$; and $U_y$ has the density \[ \frac{p^{d}}{\Gamma(d)}\,u^{d-1}e^{-pu}\,\one_{(0,\infty)}(u),\qquad d=d_y. \] The same holds under $\P^\infty$. For $C=V$, the hypothesis holds with $Q=\emptyset$ and $\mathcal G_Q=\mathcal G_0$. \item Let $y\in C_1$ with $U_y>0$. The operator $A^{(y)}=A-|\delta_y\rangle\langle a_y|$ with domain $\Dom(A)$ is closed, and $T_{A^{(y)}}=0\oplus\mathcal H^{(y)}_C$ with respect to $\ell^2(C)=\C\delta_y\oplus\ell^2(C\setminus\{y\})$. The square block \[ K_0^{(y)}=(A^{(y)})^*A^{(y)}=A^*A-U_y|v_y\rangle\langle v_y|, \qquad \Dom(K_0^{(y)})=\Dom(A^*A), \] is nonnegative and self-adjoint, and for $z\in\C\setminus\R$, \begin{equation}\label{eq:tree-square-resolvent} P_0\bigl(\mathcal H^{(y)}_C-z\bigr)^{-1}P_0=z\bigl(K_0^{(y)}-z^2\bigr)^{-1}, \end{equation} and \begin{equation}\label{eq:tree-square-resolvent-entry} z\bigl\langle v_y,(K_0^{(y)}-z^2)^{-1}v_y\bigr\rangle =\sum_{x\in C,\,x\sim y}|v_y(x)|^2\,\Gamma^C_{x\to y}(z). \end{equation} Under the hypotheses of \textup{(iii)}, let $y$ be determined by $\mathcal G_Q$, let $R_y$ be the set of retained edges which are not incident to $y$, and let $\mathcal G^y$ be the $\sigma$-algebra generated by $\mathcal G_Q$, by $v_y$, and by the variables $w_e\one_{\{e\in R_y\}}$, $e\in\mathbb U_*$. Then $v_y$, the operator $\mathcal H^{(y)}_C$ and the operator $K^{(y)}_0$ are determined by $\mathcal G^y$, and conditionally on $\mathcal G^y$ the squared row norm $U_y$ has the density in \textup{(iii)} with $d=d_y$, under $\P$ and under $\P^\infty$. \item For $z\in\C\setminus\R$, we have $P_0(\mathcal H_C-z)^{-1}P_0=z(A^*A-z^2)^{-1}$ and $P_1(\mathcal H_C-z)^{-1}P_1=z(AA^*-z^2)^{-1}$, and for every $y\in C_1$ with $U_y>0$, \begin{equation}\label{eq:tree-intertwining} U_y\,\bigl\langle v_y,(A^*A-z^2)^{-1}v_y\bigr\rangle=1+z\,G^C_{yy}(z). \end{equation} \item For every Borel set $B\subset(0,\infty)$, with $B^2=\{b^2:b\in B\}$, \[ \one_B(\mathcal H_C)=\tfrac12\bigl(I+\sgn(\mathcal H_C)\bigr)\one_{B^2}(\mathcal H_C^2), \qquad \one_{-B}(\mathcal H_C)=\tfrac12\bigl(I-\sgn(\mathcal H_C)\bigr)\one_{B^2}(\mathcal H_C^2), \] and $\one_{B^2}(\mathcal H_C^2)=\one_{B^2}(A^*A)\oplus\one_{B^2}(AA^*)$. Further, $\mathcal H_C$ has pure point spectrum in $B\cup(-B)$ if and only if both $A^*A$ and $AA^*$ have pure point spectrum in $B^2$, and the same holds with ``pure point'' replaced by ``purely absolutely continuous''. \item All statements hold with the roles of $C_0$ and $C_1$ exchanged and $A$ replaced by $A^*$; the rows are then indexed by $x\in C_0$, with $a_x=A\delta_x$. \end{enumerate} \end{proposition} \begin{proof} The proof proceeds part by part. \proofpart{Proof of (i).} By \Cref{prop:tree-symmetry}(i), we have $J\mathcal H_CJ=-\mathcal H_C$. It follows that $J$ maps $\Dom(\mathcal H_C)$ onto itself, and so do $P_0=(I+J)/2$ and $P_1=(I-J)/2$, which gives $\Dom(\mathcal H_C)=\mathcal D_0\oplus\mathcal D_1$ with $\mathcal D_i=\Dom(\mathcal H_C)\cap \ell^2(C_i)$. For $f\in\mathcal D_0$, the vector $\mathcal H_Cf=\mathcal H_{C,\mathrm{form}}f$ is supported on $C_1$, because every edge joins the two parity classes; similarly, $\mathcal H_C$ maps $\mathcal D_1$ into $\ell^2(C_0)$. Define $A$ on $\Dom(A)=\mathcal D_0$ and $\widetilde A$ on $\mathcal D_1$ as the restrictions of $\mathcal H_C$. Then $\mathcal H_C(f,g)=(\widetilde Ag,Af)$ on $\Dom(A)\oplus\Dom(\widetilde A)$. Both operators are densely defined, since they are defined on the finitely supported vectors, and $A$ is closed: if $f_n\to f$ and $Af_n\to g$, then $\mathcal H_Cf_n\to g$, and since $\mathcal H_C$ is closed, $f\in\Dom(\mathcal H_C)\cap\ell^2(C_0)$ and $Af=g$. By \Cref{lem:tree-block}(iv), we have $\widetilde A=A^*$, that is, $\mathcal H_C=T_A$. If $\mathcal H_C=T_{A'}$ for another closed operator $A'$, then $\Dom(A')=\{f:(f,0)\in\Dom(\mathcal H_C)\}=\Dom(A)$ and $A'f=P_1\mathcal H_C(f,0)=Af$, which proves uniqueness. The description of $\Dom(A)$ follows from \Cref{thm:tree-esa}(ii). The graph norm of $\mathcal H_C$ splits as $\|(f,g)\|^2+\|\mathcal H_C(f,g)\|^2=\|f\|^2+\|Af\|^2+\|g\|^2+\|A^*g\|^2$. Let $f\in\Dom(A)$. Since $c_c(C)$ is a core for $\mathcal H_C$, there exist $\varphi_n\in c_c(C)$ converging to $(f,0)$ in the graph norm of $\mathcal H_C$. Then $P_0\varphi_n\in c_c(C_0)$ converges to $f$ in the graph norm of $A$, by the splitting. This shows that $c_c(C_0)$ is a core for $A$, and the same argument applied to $(0,g)$ proves that $c_c(C_1)$ is a core for $A^*$. The statement about $\mathcal H_C^2$ is \Cref{lem:tree-block}(i). \proofpart{Proof of (ii).} Since $\delta_y\in c_c(C_1)\subset\Dom(A^*)$, we have $A^*\delta_y=\mathcal H_{C,\mathrm{form}}\delta_y=\sum_{x\sim y}h_{xy}\delta_x=a_y$. For $f\in\Dom(A)$, by \Cref{thm:tree-esa}(ii), \[ (Af)(y)=(\mathcal H_{C,\mathrm{form}}f)(y)=\sum_{x\sim y}h_{yx}f(x) =\sum_{x\sim y}\overline{h_{xy}}f(x)=\langle a_y,f\rangle. \] The same computation applies to every $f\in\ell^2(C_0)$, with $\mathcal H_{C,\mathrm{form}}$ in place of $A$. Together with \Cref{thm:tree-esa}(ii), this implies that $f\in\Dom(A)$ exactly when $\sum_y|\langle a_y,f\rangle|^2<\infty$, and then $\|Af\|^2=\sum_y|\langle a_y,f\rangle|^2$. The terms with $U_y=0$ vanish, and the others equal $U_y|\langle v_y,f\rangle|^2$. By \Cref{lem:tree-block}(i), the form domain of $A^*A$ is $\Dom(A)$ and $\|(A^*A)^{1/2}f\|^2=\|Af\|^2$, which gives the form identity. For the span, $f\perp\Ran A^*$ means that $\langle A^*g,f\rangle=0$ for all $g\in\Dom(A^*)$, which holds exactly when $f\in\Dom(A^{**})=\Dom(A)$ and $Af=0$. We conclude that $(\Ran A^*)^\perp=\ker A$ and $\overline{\Ran A^*}=(\ker A)^\perp$. Since $c_c(C_1)$ is a core for $A^*$, the set $A^*c_c(C_1)=\spn\{a_y:y\in C_1\}$ is dense in $\Ran A^*$. The nonzero vectors among the $a_y$ are multiples of the $v_y$ with $U_y>0$. \proofpart{Proof of (iii).} By \Cref{lem:tree-depletion-marks}, conditionally on $\mathcal G_Q$, the marks of the retained edges are independent standard complex Gaussians. Fix $y\in C_1$ with $d_y=d\ge1$. The coordinates of $\sqrt p\,a_y$ are $h_{xy}\sqrt p$ for the $d$ neighbors $x$ of $y$ in $C$; each of them is the mark of a retained edge or its complex conjugate. Since complex conjugation preserves the standard complex Gaussian law, the vector $\zeta=\sqrt p\,a_y\in\C^d$ has, conditionally on $\mathcal G_Q$, the density $\pi^{-d}e^{-|\zeta|^2}$ on $\C^d\cong\R^{2d}$. In polar coordinates $\zeta=r\vartheta$, with $r>0$ and $\vartheta$ in the unit sphere $\mathbb S^{2d-1}$, Lebesgue measure is $r^{2d-1}\dd r\,\sigma(\dd\vartheta)$, where $\sigma$ is the surface measure, of total mass $2\pi^d/\Gamma(d)$. Then $(r,\vartheta)$ has the product density $\pi^{-d}e^{-r^2}r^{2d-1}\dd r\,\sigma(\dd\vartheta)$; in particular, $\vartheta$ is uniform on the sphere and independent of $r$, and $r$ has the density $2\Gamma(d)^{-1}r^{2d-1}e^{-r^2}$. The change of variables $u=r^2/p$ shows that $U_y=\|a_y\|^2=r^2/p$ has the density in (iii), and $v_y=\zeta/|\zeta|=\vartheta$. Since each edge has exactly one endpoint in $V_1$, the vectors $a_y$ for distinct $y\in C_1$ are built from disjoint sets of retained marks, and they are independent conditionally on $\mathcal G_Q$. The statement under $\P^\infty$ follows in the same way from the $\P^\infty$ version of \Cref{lem:tree-depletion-marks}. \proofpart{Proof of (iv).} By \eqref{eq:tree-star}, we have $\mathcal H_C=(0\oplus\mathcal H^{(y)}_C)+W_y$, where \[ W_y=\sum_{x\in C,\,x\sim y}\bigl(h_{yx}|\delta_y\rangle\langle\delta_x|+h_{xy}|\delta_x\rangle \langle\delta_y|\bigr)=|\delta_y\rangle\langle a_y|+|a_y\rangle\langle\delta_y|, \] and the domains agree. With respect to $\ell^2(C_0)\oplus\ell^2(C_1)$, the operator $W_y$ maps $(f,g)$ to $(g(y)a_y,\langle a_y,f\rangle\delta_y)$. Since $|\delta_y\rangle\langle a_y|$ is bounded, the operator $A^{(y)}$ is closed with domain $\Dom(A)$, and its adjoint is $(A^{(y)})^*=A^*-|a_y\rangle\langle\delta_y|$ with domain $\Dom(A^*)$; here we used that the adjoint of the sum of a densely defined operator and a bounded everywhere defined operator is the sum of the adjoints. We conclude that \[ 0\oplus\mathcal H^{(y)}_C=\mathcal H_C-W_y=\begin{pmatrix}0&A^*-|a_y\rangle\langle\delta_y|\\ A-|\delta_y\rangle\langle a_y|&0\end{pmatrix}=T_{A^{(y)}}. \] By \Cref{lem:tree-block}(i), $K_0^{(y)}$ is nonnegative and self-adjoint. A vector $f\in\Dom(A)$ belongs to $\Dom(K_0^{(y)})$ exactly when $A^{(y)}f=Af-\langle a_y,f\rangle\delta_y$ belongs to $\Dom(A^*)$, and since $\delta_y\in\Dom(A^*)$, this holds exactly when $Af\in\Dom(A^*)$. This proves that $\Dom(K_0^{(y)})=\Dom(A^*A)$, and for such $f$, using $\langle\delta_y,Af\rangle=\langle a_y,f\rangle$, \[ K_0^{(y)}f=\bigl(A^*-|a_y\rangle\langle\delta_y|\bigr)\bigl(Af-\langle a_y,f\rangle\delta_y\bigr) =A^*Af-\langle a_y,f\rangle a_y=A^*Af-U_y\langle v_y,f\rangle v_y. \] By \Cref{lem:tree-block}(ii) applied to $A^{(y)}$, we have $P_0(T_{A^{(y)}}-z)^{-1}P_0=z(K_0^{(y)}-z^2)^{-1}$. Since $(T_{A^{(y)}}-z)^{-1}=(-z)^{-1}\oplus(\mathcal H^{(y)}_C-z)^{-1}$ and $\ell^2(C_0)\subset\ell^2(C\setminus\{y\})$, this is \eqref{eq:tree-square-resolvent}. For \eqref{eq:tree-square-resolvent-entry}, the neighbors of $y$ in $C$ lie in distinct branches $C_{x\to y}$, which are distinct components of $C\setminus\{y\}$. We obtain \[ \bigl\langle v_y,(\mathcal H^{(y)}_C-z)^{-1}v_y\bigr\rangle =\sum_{x\sim y}|v_y(x)|^2\bigl\langle\delta_x,(\mathcal H_{C_{x\to y}}-z)^{-1}\delta_x\bigr\rangle =\sum_{x\sim y}|v_y(x)|^2\,\Gamma^C_{x\to y}(z). \] For the last assertion, since the components $C_{x\to y}$ are determined by $\mathcal G_Q$ and the weights of their edges are the marks in $R_y$, the operator $\mathcal H^{(y)}_C$ is determined by $\mathcal G^y$; so is $K_0^{(y)}$, which by \Cref{lem:tree-block}(i) is the restriction of $(0\oplus\mathcal H^{(y)}_C)^2$ to $\ell^2(C_0)$. Conditionally on $\mathcal G_Q$, the marks of the $d_y$ retained edges at $y$ are independent of the marks in $R_y$, and by the proof of (iii) the squared row norm $U_y$ is independent of the direction $v_y$, with the stated density. Therefore, conditionally on $\mathcal G^y$, the squared row norm $U_y$ has the stated density. Since the conditional statements given $\mathcal G_Q$ hold under $\P^\infty$ as well, by \Cref{lem:tree-depletion-marks}, the same argument applies under $\P^\infty$. \proofpart{Proof of (v).} The first two identities are \Cref{lem:tree-block}(ii). For the third, we apply \Cref{lem:tree-block}(iii) with $g=\delta_y$, using $A^*\delta_y=a_y=\sqrt{U_y}v_y$, $\|\delta_y\|=1$ and $\langle\delta_y,(\mathcal H_C-z)^{-1}\delta_y\rangle=G^C_{yy}(z)$. \proofpart{Proof of (vi).} Since the square map $s(\lambda)=\lambda^2$ is a homeomorphism of $(0,\infty)$, the set $B^2$ is Borel. For every $\lambda\in\R$, with $\sgn(0)=0$, we have \[ \one_B(\lambda)=\one_{B^2}(\lambda^2)\,\tfrac12\bigl(1+\sgn\lambda\bigr), \qquad \one_{-B}(\lambda)=\one_{B^2}(\lambda^2)\,\tfrac12\bigl(1-\sgn\lambda\bigr). \] Indeed, for $\lambda>0$ we have $\lambda^2\in B^2$ exactly when $\lambda\in B$; for $\lambda<0$ the first right side vanishes, and $\lambda^2\in B^2$ exactly when $-\lambda\in B$; and $0\notin B^2$. The functional calculus is multiplicative and satisfies $\phi(\mathcal H_C^2)=(\phi\circ s)(\mathcal H_C)$, which gives the two identities. Since $\mathcal H_C^2=A^*A\oplus AA^*$ is an orthogonal sum of self-adjoint operators, its functional calculus acts blockwise. For the spectral types, let $\varphi=(f_0,f_1)\in\ell^2(C)$. The same composition rule shows that the spectral measure of $\varphi$ for $\mathcal H_C^2$ is the image of $\mu^{\mathcal H_C}_\varphi$ under $s$, and the block structure shows that it equals $\mu^{A^*A}_{f_0}+\mu^{AA^*}_{f_1}$. The restriction of $\mu^{\mathcal H_C^2}_\varphi$ to $B^2$ is the image under $s$ of the restriction of $\mu^{\mathcal H_C}_\varphi$ to $B\cup(-B)$. The map $s$ sends countable sets to countable sets, and the preimage under $s$ of a countable subset of $B^2$ is countable. Further, $s$ and its local inverses $\pm\sqrt{\cdot}$ are locally Lipschitz on $\R\setminus\{0\}$ and $(0,\infty)$, and therefore map Lebesgue null sets to Lebesgue null sets. We conclude that the restriction of $\mu^{\mathcal H_C}_\varphi$ to $B\cup(-B)$ is purely atomic (respectively absolutely continuous) if and only if the restriction of $\mu^{\mathcal H_C^2}_\varphi$ to $B^2$ is. Taking $\varphi=(f_0,0)$ and $\varphi=(0,f_1)$, and using that a sum of two purely atomic (absolutely continuous) measures is purely atomic (absolutely continuous), we obtain the last assertion. \proofpart{Proof of (vii).} With respect to $\ell^2(C_1)\oplus\ell^2(C_0)$, we have $\mathcal H_C(g,f)=(Af,A^*g)=T_{A^*}(g,f)$, since $A^{**}=A$. All the arguments above apply to $A^*$ in place of $A$. \end{proof} \section{Compactly supported eigenvectors}\label{sec:tree-compact} Let $C$ be a component of a finite removal. A finitely supported vector $\psi\in c_c(C)$ lies in the domain of $\mathcal H_C$ and satisfies $\mathcal H_C\psi=\mathcal H_{C,\mathrm{form}}\psi$. In particular, $\psi$ is an eigenvector of $\mathcal H_C$ with eigenvalue $E$ exactly when $(\mathcal H_{C,\mathrm{form}}\psi)(v)=E\psi(v)$ for every $v\in C$. In this section, we show that such eigenvectors do not exist for $E\ne0$ on infinite components. For $E=0$, they occur infinitely often on the surviving tree. For a finite set $T$ of vertices, we write $M_T=(h_{uv})_{u,v\in T}$ for the corresponding finite Hermitian matrix. \begin{proposition}\label{prop:tree-compact-modes} \leavevmode \begin{enumerate}[label=\textup{(\roman*)}] \item Let $\mathcal T'$ be a deterministic countable, infinite, connected, locally finite tree with vertex set $V'$. Let $(h_e)_e$ be independent complex random variables indexed by the edges of $\mathcal T'$, each with a law that is absolutely continuous with respect to Lebesgue measure on $\C$, and set $h_{vu}=\overline{h_{uv}}$, with an arbitrary fixed orientation of each edge (for example, independent nondegenerate complex Gaussian variables). Then almost surely there exist no $E\in\R\setminus\{0\}$ and no nonzero finitely supported $\psi\colon V'\to\C$ with $\sum_{u\sim v}h_{vu}\psi(u)=E\psi(v)$ for every $v\in V'$. \item Let $p>0$. There exists an event $\Omega_{\mathrm{cm}}$ with $\P(\Omega_{\mathrm{cm}})=1$ such that on $\Omega_{\mathrm{sa}}\cap\Omega_{\mathrm{cm}}$, for every finite removal of $\mathcal T$ and every infinite component $C$ of it, the operator $\mathcal H_C$ has no finitely supported eigenvector with a nonzero eigenvalue. In particular, if $p>1$, then $\P^\infty$-almost surely $\mathcal H$ has no finitely supported eigenvector with a nonzero eigenvalue. \end{enumerate} \end{proposition} The restriction to infinite components cannot be removed: on the extinction event $\mathcal S^c$, which has probability $1-\varrho_p>0$, the operator $\mathcal H$ is a finite Hermitian matrix, and it has nonzero eigenvalues as soon as $\mathcal T$ has an edge. The conclusion for $\mathcal H$ holds $\P$-almost surely on $\mathcal S$, but not $\P$-almost surely. \begin{proof} The proof shows that a finitely supported eigenvector produces two disjoint finite subtrees whose matrices share a nonzero eigenvalue, and that this has probability zero for each fixed pair. \proofstep{1}{Two disjoint subtrees with a common eigenvalue.} We work on the event that all weights are nonzero, which has probability one. Suppose that $E\ne0$ and $\psi\ne0$ are as in (i), and let $S=\supp\psi$, a finite nonempty set. Since $\mathcal T'$ is infinite and connected, there exists an edge $\{a,b\}$ with $a\notin S$ and $b\in S$. The equation at $a$ reads $0=E\psi(a)=\sum_{u\sim a}h_{au}\psi(u)$. Since the term $h_{ab}\psi(b)$ is nonzero, there exists a second neighbor $b'\ne b$ of $a$ with $\psi(b')\ne0$. Let $T_1$ and $T_2$ be the vertex sets of the connected components of the subgraph of $\mathcal T'$ induced on $S$ which contain $b$ and $b'$, respectively. Both sets are connected and do not contain $a$, so in the graph obtained from $\mathcal T'$ by removing $a$, the set $T_1$ lies in the component of $b$ and $T_2$ lies in the component of $b'$. These components are different, because the only path from $b$ to $b'$ in $\mathcal T'$ is $b,a,b'$. In particular, $T_1\cap T_2=\emptyset$. For $v\in T_i$, every neighbor $u$ of $v$ with $\psi(u)\ne0$ lies in $S$, and therefore in $T_i$. It follows that \[ \bigl(M_{T_i}\psi|_{T_i}\bigr)(v)=\sum_{u\in T_i}h_{vu}\psi(u)=\sum_{u\sim v}h_{vu}\psi(u) =E\psi(v),\qquad v\in T_i, \] and $\psi|_{T_i}\ne0$. We conclude that $E$ is a common nonzero eigenvalue of $M_{T_1}$ and $M_{T_2}$, where $T_1$ and $T_2$ are disjoint finite sets which induce connected subgraphs. \proofstep{2}{A fixed pair has no common nonzero eigenvalue.} Fix disjoint finite sets $T_1,T_2\subset V'$ which induce connected subgraphs. We show that, almost surely, $M_{T_1}$ and $M_{T_2}$ have no common nonzero eigenvalue. (The event is Borel, because the eigenvalues of a Hermitian matrix, listed in increasing order, depend continuously on its entries.) If $T_2$ is a single vertex, then $M_{T_2}=0$. Otherwise, let $e_1,\dots,e_k$, with $k\ge1$, be the edges with both endpoints in $T_2$, and let $\zeta=(h_{e_1},\dots,h_{e_k})\in\C^k$. Since $T_1\cap T_2=\emptyset$, the vector $\zeta$ is independent of the weights $X_1$ of the edges inside $T_1$, and it has a density $\rho$ on $\C^k\cong\R^{2k}$, the product of the densities of its coordinates. We write $\zeta=r\vartheta$ with $r=|\zeta|$ and $\vartheta\in\mathbb S^{2k-1}$. Then $M_{T_2}=rM(\vartheta)$, where $M(\vartheta)$ is the Hermitian matrix built in the same way from $\vartheta$, and the nonzero eigenvalues of $M_{T_2}$ are the numbers $r\lambda$ with $\lambda$ a nonzero eigenvalue of $M(\vartheta)$. Let $\Sigma(\vartheta,X_1)$ be the finite set of quotients $\kappa/\lambda$, with $\kappa$ a nonzero eigenvalue of $M_{T_1}$ and $\lambda$ a nonzero eigenvalue of $M(\vartheta)$. A common nonzero eigenvalue exists only if $r\in\Sigma(\vartheta,X_1)$. In polar coordinates, the law of $(r,\vartheta)$ has the density $\rho(r\vartheta)r^{2k-1}$ with respect to $\dd r\,\sigma(\dd\vartheta)$. By Fubini's theorem and the independence of $\zeta$ and $X_1$, we have \[ \P\bigl(r\in\Sigma(\vartheta,X_1)\bigr) =\E\Bigl[\int_{\mathbb S^{2k-1}}\int_0^\infty\one_{\{r\in\Sigma(\vartheta,X_1)\}}\, \rho(r\vartheta)\,r^{2k-1}\dd r\,\sigma(\dd\vartheta)\Bigr]=0, \] where the expectation is over $X_1$. The inner integral vanishes because $\Sigma(\vartheta,X_1)$ is a finite set. \proofstep{3}{Conclusion.} Since there are countably many pairs of finite subsets of $V'$, almost surely no pair as in Step~2 has a common nonzero eigenvalue. By Step~1, this proves (i). For (ii), for disjoint finite sets $T_1,T_2\subset\mathbb U$, let $A(T_1,T_2)$ be the event that $T_1,T_2\subset V$, that both induce connected subgraphs of $\mathcal T$, and that $M_{T_1}$ and $M_{T_2}$ have a common nonzero eigenvalue. The first two conditions are $\mathcal G_0$-measurable. Conditionally on $\mathcal G_0$, the marks are independent standard complex Gaussians. In particular, the weights of the edges inside $T_1$ and inside $T_2$ are independent and have densities, and Step~2 implies that $\P(A(T_1,T_2)\mid\mathcal G_0)=0$. Let $\Omega_{\mathrm{cm}}$ be the event that all marks are nonzero and that no event $A(T_1,T_2)$ occurs; then $\P(\Omega_{\mathrm{cm}})=1$. On $\Omega_{\mathrm{sa}}\cap\Omega_{\mathrm{cm}}$, let $C$ be an infinite component of a finite removal, and suppose that $\mathcal H_C$ has a finitely supported eigenvector with eigenvalue $E\ne0$. The network $(C,h)$ is an infinite, connected, locally finite tree with nonzero weights. Step~1 applies to it and produces disjoint finite sets $T_1,T_2\subset C$ which induce connected subgraphs of $C$ and whose matrices share the eigenvalue $E$. By \Cref{lem:tree-branch}(i), $C$ is an induced subtree of $\mathcal T$, and $T_1$ and $T_2$ therefore induce connected subgraphs of $\mathcal T$ with the same matrices, and $A(T_1,T_2)$ occurs. This contradiction proves (ii). The last assertion is the case of the empty finite removal on $\mathcal S$, where $C=V$ is infinite. \end{proof} At energy zero, the situation is different. Call a vertex $v\in V$ a \emph{twin-leaf vertex} if $N_v=2$ and $N_{v1}=N_{v2}=0$, that is, if $v$ has exactly two children and both are leaves. \begin{proposition}\label{prop:tree-zero-modes} \leavevmode \begin{enumerate}[label=\textup{(\roman*)}] \item On $\Omega_{\mathrm{sa}}$, let $C$ be a component of a finite removal, let $v\in C$, and let $\ell_1\ne\ell_2$ be neighbors of $v$ in $C$ whose only neighbor in $C$ is $v$. Then \[ \psi=h_{v\ell_2}\,\delta_{\ell_1}-h_{v\ell_1}\,\delta_{\ell_2} \] satisfies $\mathcal H_C\psi=0$, and $\psi\ne0$ if the weights are nonzero. \item Let $p>1$. Then $\P^\infty$-almost surely there are infinitely many twin-leaf vertices. Consequently, $\P^\infty$-almost surely, the vectors $\psi_v=h_{v,v2}\delta_{v1}-h_{v,v1}\delta_{v2}$, for $v$ a twin-leaf vertex, form an infinite family of nonzero, pairwise orthogonal, finitely supported eigenvectors of $\mathcal H$ with eigenvalue $0$. In particular, $0$ is an eigenvalue of $\mathcal H$ of infinite multiplicity. \end{enumerate} \end{proposition} \begin{proof} For (i), since the vector $\psi$ is finitely supported, it suffices to check the pointwise equation. At $v$, we have $(\mathcal H_{C,\mathrm{form}}\psi)(v)=h_{v\ell_1}h_{v\ell_2}-h_{v\ell_2}h_{v\ell_1}=0$. At $\ell_1$ and $\ell_2$, the only neighbor is $v$, and $\psi(v)=0$. Every other vertex $u$ is not adjacent to $\ell_1$ or $\ell_2$, since their only neighbor is $v$, and $(\mathcal H_{C,\mathrm{form}}\psi)(u)=0$. For (ii), let $\mathcal V_n=\{v\in V:|v|=n\}$. We first recall that the generation sizes $|\mathcal V_n|$ tend to infinity almost surely on $\mathcal S$ (see \cite[Section~4.3]{Durrett2019probability} for background on Galton--Watson processes); for completeness, we give the argument. Let $\mathscr N_n=\sigma(N_u:|u|\tau_j$ and $\tau_{j+1}=\infty$. Therefore $\P(\tau_{j+1}<\infty)\le(1-e^{-pK})\P(\tau_j<\infty)$, and $\P(\tau_j<\infty)\to0$ as $j\to\infty$, and almost surely $1\le |\mathcal V_n|\le K$ for only finitely many $n$. On $\mathcal S$, we have $|\mathcal V_n|\ge1$ for all $n$, and it follows that $|\mathcal V_n|>K$ for all large $n$. Since $K$ is arbitrary, $|\mathcal V_n|\to\infty$ almost surely on $\mathcal S$. For a generation-$n$ vertex $v$, the event that $v$ is a twin-leaf vertex depends on $N_v,N_{v1},N_{v2}$, which are independent of $\mathscr N_n$, and these variables are distinct for distinct $v$. Conditionally on $\mathscr N_n$, these events are independent with probability \[ a_p=\P(N_o=2)\,\P(N_o=0)^2=\frac{p^2e^{-p}}2\,e^{-2p}=\frac{p^2e^{-3p}}2>0 \] each. Since $\mathcal S\subset\{|\mathcal V_n|\ge1\}$, we obtain \[ \P\bigl(\mathcal S\cap\{\text{no twin-leaf vertex in generation }n\}\bigr) \le\E\bigl[\one_{\{|\mathcal V_n|\ge1\}}(1-a_p)^{|\mathcal V_n|}\bigr]. \] As $n\to\infty$, the integrand tends to zero almost surely: on $\mathcal S$ because $|\mathcal V_n|\to\infty$, and on $\mathcal S^c$ because $|\mathcal V_n|=0$ for all large $n$. It is bounded by one, so the right side tends to zero by dominated convergence. For every $m$, the event that $\mathcal S$ occurs and no generation $n\ge m$ contains a twin-leaf vertex is contained in each of the events on the left with $n\ge m$, and it has probability zero. Taking the union over $m$, we find that $\P$-almost surely on $\mathcal S$ there are infinitely many twin-leaf vertices. The two children of a twin-leaf vertex are leaves of $\mathcal T$, whose only neighbor is their parent. Then (i) applies with $C=V$ and shows that $\mathcal H\psi_v=0$; on the event that all marks are nonzero, $\psi_v\ne0$. Since the supports $\{v1,v2\}$ of the vectors $\psi_v$ are pairwise disjoint for distinct $v$, these vectors are pairwise orthogonal, and in particular linearly independent. \end{proof} \chapter{Spine decompositions and fresh squared moduli}\label{sec:palm} This section collects the probabilistic tools that we use to compute expectations of sums over the vertices of the tree. They are of three kinds. The first are spine decompositions. If $F(v)$ is a nonnegative functional of the marked tree and of a vertex $v$ of generation $L$, then the expectation of $\sum_{|v|=L}F(v)$ equals $p^L$ times the expectation of $F$ evaluated at the endpoint of a spine: a path of length $L$ from the root whose vertices carry independent Poisson families of further children, and whose endpoint carries an independent copy of the whole tree. This identity is the many-to-one formula. We prove it, its analogue for ordered pairs of vertices, which is the many-to-two formula, and their versions under the survival-conditioned law $\P^\infty$. The second kind concerns endpoints: the law of an endpoint functional under $\P^\infty$, the children of an endpoint classified by the fate of their forward subtrees, and the growth of the set of vertices with infinite forward subtree. The third is an anti-concentration estimate for affine combinations of independent exponential variables, together with a definition of fresh squared moduli, which says when this estimate may be applied conditionally on a $\sigma$-field. In \Cref{sec:palm-construction}, we realize the marked tree on the space of finite words and record its branching property. \Cref{sec:palm-mecke} recalls the Mecke formula and splits the children of the root according to the survival of their forward subtrees. \Cref{sec:palm-spines} defines the spine laws and proves the many-to-one and many-to-two formulas. \Cref{sec:palm-endpoints} compares endpoint laws under $\P$ and $\P^\infty$. \Cref{sec:palm-fresh} defines fresh squared moduli and proves the affine negative-moment lemma. \Cref{sec:palm-decorations} treats the children of a single vertex and the growth of the backbone. Throughout this section, $p>0$ is fixed. Whenever the survival event enters, we assume that $p>1$, so that $\varrho_p>0$. \section{The Ulam--Harris construction}\label{sec:palm-construction} Let $\mathbb U_0=\{o\}$, where $o$ denotes the empty word, let $\mathbb U_n=\{1,2,\dots\}^n$ for $n\ge1$, and set $\mathbb U=\bigcup_{n\ge0}\mathbb U_n$ and $\mathbb U_*=\mathbb U\setminus\{o\}$. For a word $u=i_1\cdots i_n$ and a word $v$, we write $|u|=n$ for the length of $u$ and $uv$ for the concatenation of $u$ and $v$. For $0\le l\le n$, we let $u_l=i_1\cdots i_l$ denote the prefix of $u$ of length $l$, so that $u_0=o$ and $u_n=u$, and for $u\neq o$ we write $u^-=u_{n-1}$. For $u,v\in\mathbb U$, we let $u\wedge v$ denote the longest common prefix of $u$ and $v$. Let $\gamma$ denote the standard complex Gaussian law, which has density $\pi^{-1}e^{-|w|^2}$ with respect to Lebesgue measure on $\C$. If $w$ has law $\gamma$, then $|w|^2$ has law $\Exp(1)$, and $w/|w|$ is uniform on the unit circle and independent of $|w|^2$. Let $\mathsf Z=[0,1]^{\{1,2,\dots\}}$ with its product $\sigma$-field, and let $\kappa$ be the product of the Lebesgue measures on the factors. \begin{definition}\label{def:palm-config} The configuration space is \[ \Omega=\{0,1,2,\dots\}^{\mathbb U}\times\mathsf Z^{\mathbb U}\times\C^{\mathbb U_*}, \] with the product $\sigma$-field $\mathscr F$. Its elements are written \[ \omega=\bigl((N_v)_{v\in\mathbb U},(\zeta_v)_{v\in\mathbb U},(w_v)_{v\in\mathbb U_*}\bigr). \] We let $\P$ be the probability measure on $(\Omega,\mathscr F)$ under which all coordinates are independent, each $N_v$ has law $\Pois(p)$, each $\zeta_v$ has law $\kappa$, and each $w_v$ has law $\gamma$. The \emph{rank} of $v\in\mathbb U$ is the first coordinate $\zeta_v(1)$ of $\zeta_v=(\zeta_v(1),\zeta_v(2),\dots)$; the variables $\zeta_v$ are called auxiliary marks. \begin{enumerate}[label=\textup{(\alph*)}] \item The vertex set is \[ V(\omega)=\bigl\{i_1\cdots i_n\in\mathbb U:\ n\ge0,\ i_l\le N_{i_1\cdots i_{l-1}}\text{ for }1\le l\le n\bigr\}, \] and $\mathcal T=\mathcal T(\omega)$ is the tree with vertex set $V(\omega)$, root $o$, and edges $\{v^-,v\}$ for $v\in V(\omega)\setminus\{o\}$. The children of $v\in V(\omega)$ are $v1,\dots,vN_v$. \item For $v\in V(\omega)\setminus\{o\}$, the edge $\{v^-,v\}$ carries the mark $w_{v^-v}=w_v$. We set $h_{v^-v}=p^{-1/2}w_v$, $h_{vv^-}=\overline{h_{v^-v}}$, and $Y_v=Y_{v^-v}=|w_v|^2$. \item For $u\in\mathbb U$, the shift $\theta_u\colon\Omega\to\Omega$ is defined by \[ \theta_u\omega=\bigl((N_{uv})_{v\in\mathbb U},(\zeta_{uv})_{v\in\mathbb U},(w_{uv})_{v\in\mathbb U_*}\bigr). \] \item The survival event is $\mathcal S=\{\omega\in\Omega:|V(\omega)|=\infty\}$, and $\varrho_p=\P(\mathcal S)$. For $p>1$, we set $\P^\infty=\P(\,\cdot\mid\mathcal S)$ and write $\E^\infty$ for the corresponding expectation. \item For $n\ge0$, we set $\mathcal V_n=V(\omega)\cap\mathbb U_n$, $\mathcal V_n^\infty=\{v\in\mathcal V_n:\theta_v\omega\in\mathcal S\}$, and $Z_n=|\mathcal V_n^\infty|$. The \emph{backbone} is the set $\bigcup_{n\ge0}\mathcal V_n^\infty$ of vertices whose forward subtree is infinite. \end{enumerate} \end{definition} Under $\P$, the offspring numbers $N_v$ are independent with law $\Pois(p)$, and $\mathcal T$ is a Poisson--Galton--Watson tree with mean offspring $p$. The marks $w_v$, $v\in\mathbb U_*$, are independent of $(N_v)_{v\in\mathbb U}$. This implies that, conditionally on $\mathcal T$, the marks of its edges are independent with law $\gamma$; in particular, the squared moduli $Y_v=|w_v|^2$ are independent $\Exp(1)$ variables, and they are independent of the phases $w_v/|w_v|$, which are uniform on the unit circle. We conclude that $\mathcal T$, together with the weights $h_e$, is a realization of the marked tree of \Cref{def:tree-pgw}, with the auxiliary element $\mathfrak a=(\zeta_v)_{v\in\mathbb U}$ of the standard Borel space $\mathsf Z^{\mathbb U}$, which is independent of the marked tree. In particular, the $\sigma$-algebra $\mathcal G_0$ of \Cref{def:tree-pgw} is generated by the offspring numbers $N_v$ and the auxiliary marks $\zeta_v$, $v\in\mathbb U$. From now on we work with this realization: $\P$ and $\E$ refer to the law of $\omega$, and $\mathcal S$, $\varrho_p$, $\P^\infty$ are the objects of \Cref{def:tree-pgw}. The event $\mathcal S$ is measurable, since every vertex has finitely many children and therefore, by K\"onig's lemma, $\mathcal S=\bigcap_{n\ge0}\{\mathcal V_n\neq\emptyset\}$. Recall that $0<\varrho_p<1$ for $p>1$. The coordinates of $\omega$ at labels outside $V(\omega)$ do not affect the marked tree. We keep them because they make the shifts measure preserving and the spine laws of \Cref{sec:palm-spines} simple to write down. Note that each shift $\theta_u$ preserves $\P$, since it moves the coordinates at the labels $uv$ to the labels $v$, and all coordinates of the same type have the same law. If $u\in V(\omega)$, then $V(\theta_u\omega)=\{v\in\mathbb U:uv\in V(\omega)\}$, and the map $v\mapsto uv$ is an isomorphism from $\mathcal T(\theta_u\omega)$ onto the forward subtree $\mathcal T_u$. The edge $\{v^-,v\}$ of $\mathcal T(\theta_u\omega)$ carries the mark $w_{uv}$, which is also the mark of the corresponding edge $\{(uv)^-,uv\}$ of $\mathcal T_u$. We use this correspondence throughout: if a quantity $f(\omega)$ is defined in terms of the marked tree $\mathcal T(\omega)$ and its root $o$, then the same quantity defined in terms of the marked forward subtree $\mathcal T_u$ and its root $u$ equals $f(\theta_u\omega)$. For example, $\{\mathcal T_u\text{ is infinite}\}=\{\theta_u\omega\in\mathcal S\}$, so that $Z_n$ is the number of vertices of generation $n$ whose forward subtree is infinite. Similarly, the forward cavity variable of \Cref{def:tree-resolvent} satisfies $\Gamma_u(z)=G_{oo}(z)\circ\theta_u$ for every $u\in V(\omega)\setminus\{o\}$ and $z\in\C\setminus\R$. Here, and in the rest of this section, a function such as $G_{oo}(z)$ which is defined only $\P$-almost surely is extended by $0$ to the exceptional null set; since every shift preserves $\P$, the identity $\Gamma_u(z)=G_{oo}(z)\circ\theta_u$ then holds for all $u$ simultaneously, $\P$-almost surely. \begin{lemma}\label{lem:palm-branching} Let $p>0$, let $\mathsf M=\C\times\Omega$ with the product $\sigma$-field, and let $\mathfrak m=\gamma\otimes\P$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $n\ge0$, the configurations $\theta_v\omega$, $v\in\mathbb U_n$, are independent, each has law $\P$, and they are independent of the $\sigma$-field \[ \mathscr F_n=\sigma\bigl(N_u,\zeta_u:\ u\in\mathbb U,\ |u|0$, and let $\Pi$ be a Poisson process on $\mathsf X$ with intensity measure $p\mu$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every measurable $F\colon\mathsf X\times\mathbf N(\mathsf X)\to[0,\infty]$, \begin{equation}\label{eq:palm-mecke} \E\int F(x,\Pi-\delta_x)\,\Pi(\dd x)=p\int\E F(x,\Pi)\,\mu(\dd x). \end{equation} \item For every $k\ge1$ and every measurable $F\colon\mathsf X^k\times\mathbf N(\mathsf X)\to[0,\infty]$, \begin{multline}\label{eq:palm-factorial-mecke} \E\int F\Bigl(x_1,\dots,x_k,\Pi-\sum_{i=1}^k\delta_{x_i}\Bigr)\,\Pi^{(k)}(\dd x_1\cdots\dd x_k)\\ =p^k\int\E F(x_1,\dots,x_k,\Pi)\,\mu^{\otimes k}(\dd x_1\cdots\dd x_k). \end{multline} \end{enumerate} \end{lemma} Identity \eqref{eq:palm-mecke} is the Mecke equation \cite{Mecke1967stationare}; see \cite[Theorem 4.1]{LastPenrose2018lectures}, and \eqref{eq:palm-factorial-mecke} is its multivariate form \cite[Theorem 4.4]{LastPenrose2018lectures}. In words, \eqref{eq:palm-mecke} states that the measure $B\mapsto\E\int\one_B(x,\Pi-\delta_x)\,\Pi(\dd x)$ on $\mathsf X\times\mathbf N(\mathsf X)$ equals $p\mu\otimes\Law(\Pi)$. Summing a functional over the points of $\Pi$ amounts to drawing the selected point from $\mu$, with total weight $p$, while the remaining points form a Poisson process with the original intensity measure which is independent of the selected point (Slivnyak's theorem). We include the short proof for finite intensity measures. \begin{proof}[Proof of \Cref{lem:palm-mecke}] Since both sides of \eqref{eq:palm-factorial-mecke} depend only on the law of $\Pi$, we may assume that $\Pi=\sum_{i=1}^N\delta_{X_i}$, where $N$ has law $\Pois(p)$ and $X_1,X_2,\dots$ are independent with law $\mu$ and independent of $N$. Then the left side of \eqref{eq:palm-factorial-mecke} equals \[ \E\sum_{(i_1,\dots,i_k)}F\Bigl(X_{i_1},\dots,X_{i_k},\sum_{i\le N,\ i\notin\{i_1,\dots,i_k\}}\delta_{X_i}\Bigr), \] where the sum runs over ordered $k$-tuples of pairwise distinct indices in $\{1,\dots,N\}$. Conditionally on $N=n\ge k$, there are $n!/(n-k)!$ such tuples, and by exchangeability of $(X_1,\dots,X_n)$, each term has the same conditional expectation as $F(X_1,\dots,X_k,\sum_{i=k+1}^n\delta_{X_i})$. Since $\P(N=n)\,n!/(n-k)!=p^k\P(N=n-k)$, we obtain \[ p^k\sum_{n\ge k}\P(N=n-k)\,\E F\Bigl(X_1,\dots,X_k,\sum_{i=k+1}^{n}\delta_{X_i}\Bigr)=p^k\,\E F\Bigl(X_1,\dots,X_k,\sum_{i=k+1}^{k+N'}\delta_{X_i}\Bigr), \] where $N'$ is a $\Pois(p)$ variable independent of $(X_i)_{i\ge1}$. The process $\sum_{i=k+1}^{k+N'}\delta_{X_i}$ is a Poisson process with intensity $p\mu$ which is independent of $(X_1,\dots,X_k)$. Then \eqref{eq:palm-factorial-mecke} follows from Tonelli's theorem. The case $k=1$ is \eqref{eq:palm-mecke}. \end{proof} We now split the children of the root according to the survival of their forward subtrees. Recall the space $\mathsf M=\C\times\Omega$, the law $\mathfrak m=\gamma\otimes\P$, and the child process $\Pi_o$ from \Cref{lem:palm-branching}. \begin{lemma}\label{lem:palm-survival} Let $p>1$. Set $\mathsf M^\infty=\C\times\mathcal S$ and $\mathsf M^{\mathrm f}=\C\times(\Omega\setminus\mathcal S)$, let $\mathfrak m^\infty=\gamma\otimes\P^\infty$ and $\mathfrak m^{\mathrm f}=\gamma\otimes\P(\,\cdot\mid\Omega\setminus\mathcal S)$, and let $\Pi^\infty$ and $\Pi^{\mathrm f}$ be the restrictions of $\Pi_o$ to $\mathsf M^\infty$ and $\mathsf M^{\mathrm f}$. Let $N^\infty=\Pi^\infty(\mathsf M)$ and $N^{\mathrm f}=\Pi^{\mathrm f}(\mathsf M)$ be the numbers of children of the root with infinite and with finite forward subtree. \begin{enumerate}[label=\textup{(\roman*)}] \item Under $\P$, the random elements $\Pi^\infty$, $\Pi^{\mathrm f}$, and $\zeta_o$ are independent, $\Pi^\infty$ is a Poisson process with intensity measure $p\varrho_p\mathfrak m^\infty$, and $\Pi^{\mathrm f}$ is a Poisson process with intensity measure $p(1-\varrho_p)\mathfrak m^{\mathrm f}$. In particular, $N^\infty$ and $N^{\mathrm f}$ are independent with laws $\Pois(p\varrho_p)$ and $\Pois(p(1-\varrho_p))$. Moreover, $\mathcal S=\{N^\infty\ge1\}$ and $\varrho_p=1-e^{-p\varrho_p}$. \item Under $\P^\infty$, the random elements $\Pi^\infty$, $\Pi^{\mathrm f}$, and $\zeta_o$ are independent; $\Pi^{\mathrm f}$ and $\zeta_o$ have the same laws as under $\P$, and $\Pi^\infty$ has the law of a Poisson process with intensity measure $p\varrho_p\mathfrak m^\infty$ conditioned to have at least one point. \item For every measurable $Z\colon\Omega\to[0,\infty]$, \begin{equation}\label{eq:palm-survival-domination} \E^\infty Z=\varrho_p^{-1}\E\bigl[Z\one_{\mathcal S}\bigr]\le\varrho_p^{-1}\E Z. \end{equation} \item For every $k\ge1$ and every measurable $F\colon\mathsf M^k\times\mathbf N(\mathsf M)\to[0,\infty]$, \begin{multline}\label{eq:palm-conditioned-mecke} \E^\infty\int F\Bigl(m_1,\dots,m_k,\Pi_o-\sum_{i=1}^k\delta_{m_i}\Bigr)\,(\Pi^\infty)^{(k)}(\dd m_1\cdots\dd m_k)\\ =p^k\varrho_p^{k-1}\int\E F(m_1,\dots,m_k,\Pi_o)\,(\mathfrak m^\infty)^{\otimes k}(\dd m_1\cdots\dd m_k). \end{multline} \end{enumerate} \end{lemma} The left side of \eqref{eq:palm-conditioned-mecke} is the sum, over ordered $k$-tuples of distinct children of the root with infinite forward subtrees, of $F$ evaluated at their marks and at the process of all remaining children. On the right side, the selected marks are independent with law $\mathfrak m^\infty$, so that their edge marks have law $\gamma$ and their forward configurations have law $\P^\infty$, while the residual process is the unconditioned child process $\Pi_o$ under $\P$, independent of the selected marks. For $k=1$ the factor is $p$. \begin{proof} We begin with (i). Let $u_1,u_2\colon\mathsf M\to[0,\infty)$ be measurable and set $u=u_1\one_{\mathsf M^\infty}+u_2\one_{\mathsf M^{\mathrm f}}$. Using the representation of $\Pi_o$ from \Cref{lem:palm-branching}, we have \begin{align*} \E\exp\bigl(-\Pi^\infty(u_1)-\Pi^{\mathrm f}(u_2)\bigr)&=\E\Bigl(\int e^{-u}\,\dd\mathfrak m\Bigr)^{N_o}=\exp\Bigl(-p\int(1-e^{-u})\,\dd\mathfrak m\Bigr)\\ &=\exp\Bigl(-p\int_{\mathsf M^\infty}(1-e^{-u_1})\,\dd\mathfrak m\Bigr)\exp\Bigl(-p\int_{\mathsf M^{\mathrm f}}(1-e^{-u_2})\,\dd\mathfrak m\Bigr). \end{align*} The right side is the joint Laplace functional of two independent Poisson processes with intensity measures $p\,\mathfrak m(\,\cdot\cap\mathsf M^\infty)=p\varrho_p\mathfrak m^\infty$ and $p\,\mathfrak m(\,\cdot\cap\mathsf M^{\mathrm f})=p(1-\varrho_p)\mathfrak m^{\mathrm f}$. Since the joint Laplace functional determines the joint law of $(\Pi^\infty,\Pi^{\mathrm f})$ (see \cite[Chapter 2]{LastPenrose2018lectures}; alternatively, this is the restriction theorem \cite[Theorem 5.2]{LastPenrose2018lectures}), the first claim follows; $\zeta_o$ is independent of $\Pi_o$ by \Cref{lem:palm-branching}. Since $|V(\omega)|=1+\sum_{k=1}^{N_o}|V(\theta_k\omega)|$ and $N_o<\infty$, the tree is infinite if and only if the forward subtree of some child of the root is infinite, that is, $\mathcal S=\{N^\infty\ge1\}$. Then $\varrho_p=\P(N^\infty\ge1)=1-e^{-p\varrho_p}$. For (ii), let $A$, $B$, $C$ be measurable sets. Since $\mathcal S=\{\Pi^\infty\neq0\}$, part (i) implies that \[ \P^\infty(\Pi^\infty\in A,\ \Pi^{\mathrm f}\in B,\ \zeta_o\in C)=\varrho_p^{-1}\P(\Pi^\infty\in A,\ \Pi^\infty\neq0)\,\P(\Pi^{\mathrm f}\in B)\,\P(\zeta_o\in C), \] and $\varrho_p=\P(\Pi^\infty\neq0)$. This is (ii). Part (iii) is the definition of $\P^\infty$ together with $Z\one_{\mathcal S}\le Z$. For (iv), we note that the integrand on the left side vanishes unless $N^\infty\ge k\ge1$, that is, unless $\omega\in\mathcal S$. By (iii), the left side equals $\varrho_p^{-1}$ times the corresponding expectation under $\P$. The measure $(\Pi^\infty)^{(k)}$ is the restriction of $\Pi_o^{(k)}$ to $(\mathsf M^\infty)^k$. Applying \eqref{eq:palm-factorial-mecke} to $\Pi_o$ and to the function $F(m_1,\dots,m_k,\pi)\prod_{i=1}^k\one_{\mathsf M^\infty}(m_i)$, we find that the left side of \eqref{eq:palm-conditioned-mecke} equals \[ \varrho_p^{-1}p^k\int\E F(m_1,\dots,m_k,\Pi_o)\prod_{i=1}^k\one_{\mathsf M^\infty}(m_i)\,\mathfrak m^{\otimes k}(\dd m_1\cdots\dd m_k). \] Since $\one_{\mathsf M^\infty}\mathfrak m=\varrho_p\mathfrak m^\infty$, this is the right side of \eqref{eq:palm-conditioned-mecke}. \end{proof} \section{Spine decompositions}\label{sec:palm-spines} We now define the laws of the spine trees and prove the many-to-one and many-to-two formulas. They are instances of the Palm (Mecke) formula for the Poisson process; see \Cref{rem:palm-labels}. The idea is the classical size-biasing of the offspring numbers along a path. In a sum over the vertices $v$ of generation $L$, the vertex $v$ exists only if each of its ancestors has enough children; weighting by this event replaces the law $\Pois(p)$ of each ancestor's offspring number by the law of $1+\Pois(p)$, at the cost of a factor $p$ per generation, and makes the position of the next ancestor among its siblings uniform. \begin{definition}\label{def:palm-spine-laws} Let $L\ge0$. \begin{enumerate}[label=\textup{(\alph*)}] \item \emph{One-spine law.} Let $\omega'$ be a random configuration with law $\P$, let $N_0',\dots,N_{L-1}'$ be random variables with law $\Pois(p)$, and let $I_1,\dots,I_L$ be random variables such that $\omega',N_0',\dots,N_{L-1}'$ are independent and, conditionally on them, $I_1,\dots,I_L$ are independent and $I_{l+1}$ is uniformly distributed on $\{1,\dots,N_l'+1\}$. Set $\mathsf v=I_1I_2\cdots I_L\in\mathbb U_L$, so that $\mathsf v_l=I_1\cdots I_l$, and let $\omega$ be the configuration obtained from $\omega'$ by setting $N_{\mathsf v_l}=N_l'+1$ for $0\le l1$. We set \[ \mathbb Q_L^\infty=\mathbb Q_L(\,\cdot\mid\theta_{\mathsf v}\omega\in\mathcal S),\qquad\mathbb Q_{L,j}^\infty=\mathbb Q_{L,j}(\,\cdot\mid\theta_{\mathsf u}\omega\in\mathcal S,\ \theta_{\mathsf v}\omega\in\mathcal S). \] \end{enumerate} We write $\E_{\mathbb Q}$ for the expectation with respect to one of these laws $\mathbb Q$, and we use $(\omega,\mathsf v)$, respectively $(\omega,\mathsf u,\mathsf v)$, for the canonical random elements; the labels $I_l$, $J_i$, $I^{(i)}_l$ are the letters of $\mathsf u$ and $\mathsf v$. \end{definition} Note that under $\mathbb Q_L$ the vertex $\mathsf v$ belongs to $V(\omega)$, because $I_{l+1}\le N_{\mathsf v_l}$ for every $l0$. \begin{lemma}\label{lem:palm-spine-structure} Let $L\ge0$, and let $\mathsf M$ and $\mathfrak m=\gamma\otimes\P$ be as in \Cref{lem:palm-branching}. Under $\mathbb Q_L$, for $0\le l1$. Under $\mathbb Q_L^\infty$, assertions \textup{(i)} and \textup{(ii)} hold with the single change that $\theta_{\mathsf v}\omega$ has law $\P^\infty$. \item Let $0\le j1$), the same holds, except that the two endpoint configurations have law $\P^\infty$. \end{enumerate} \end{lemma} The marked spine tree is assembled from the random elements in (i) and the labels: the spine edges carry the marks $w_{\mathsf v_l}$, the side children of $\mathsf v_l$ carry the entries of $S_l$ (edge mark and forward configuration), and the endpoint carries the forward configuration $\theta_{\mathsf v}\omega$. In particular, under $\mathbb Q_L$ the squared moduli $Y_{\mathsf v_1},\dots,Y_{\mathsf v_L}$ of the spine marks are independent $\Exp(1)$ variables, every side child has a forward subtree which is an independent Poisson--Galton--Watson tree with independent marks, and the endpoint has an independent forward subtree with the law of the whole tree. \begin{proof} Fix $n=(n_0,\dots,n_{L-1})\in\{0,1,\dots\}^L$ and $i=(i_1,\dots,i_L)$ with $1\le i_{l+1}\le n_l+1$, and let $v=i_1\cdots i_L$. On the event $E_{n,i}=\{N'=n,\ I=i\}$, which has probability $\prod_{l0$ and $L\ge0$. All functions below are measurable; a function on $\Omega\times\mathbb U$ is measurable if it is measurable in $\omega$ for each fixed word. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $F\colon\Omega\times\mathbb U\to[0,\infty]$, \begin{equation}\label{eq:palm-one-spine} \E\sum_{v\in\mathcal V_L}F(\omega,v)=p^L\,\E_{\mathbb Q_L}F(\omega,\mathsf v). \end{equation} \item Let $0\le j1$. For every $F$ as in \textup{(i)}, \begin{align} \E^\infty\sum_{v\in\mathcal V_L}F(\omega,v)&=\varrho_p^{-1}p^L\,\E_{\mathbb Q_L}\bigl[F(\omega,\mathsf v)\one_{\mathcal S}(\omega)\bigr]\le\varrho_p^{-1}p^L\,\E_{\mathbb Q_L}F(\omega,\mathsf v),\label{eq:palm-one-spine-conditioned}\\ \E^\infty\sum_{v\in\mathcal V_L^\infty}F(\omega,v)&=p^L\,\E_{\mathbb Q_L^\infty}F(\omega,\mathsf v)\le\varrho_p^{-1}p^L\,\E_{\mathbb Q_L}F(\omega,\mathsf v).\label{eq:palm-backbone} \end{align} \item Let $p>1$ and $0\le j1$. Applying \eqref{eq:palm-survival-domination} and then \eqref{eq:palm-one-spine} to $F(\omega,v)\one_{\mathcal S}(\omega)$, we have the identity in \eqref{eq:palm-one-spine-conditioned}; the inequality follows from $\one_{\mathcal S}\le1$. If $v\in\mathcal V_L^\infty$, then the forward subtree of $v$ is infinite, and in particular $\omega\in\mathcal S$. By \eqref{eq:palm-survival-domination} and \eqref{eq:palm-one-spine}, we have \[ \E^\infty\sum_{v\in\mathcal V_L^\infty}F(\omega,v)=\varrho_p^{-1}\E\sum_{v\in\mathcal V_L}F(\omega,v)\one_{\mathcal S}(\theta_v\omega)=\varrho_p^{-1}p^L\,\E_{\mathbb Q_L}\bigl[F(\omega,\mathsf v)\one_{\mathcal S}(\theta_{\mathsf v}\omega)\bigr]. \] By \Cref{lem:palm-spine-structure}, we have $\mathbb Q_L(\theta_{\mathsf v}\omega\in\mathcal S)=\varrho_p$. Then the right side equals $p^L\E_{\mathbb Q_L^\infty}F(\omega,\mathsf v)$, and it is at most $\varrho_p^{-1}p^L\E_{\mathbb Q_L}F(\omega,\mathsf v)$. This proves \eqref{eq:palm-backbone}. \proofstep{4}{Conditioned pairs.} The identity and inequality in \eqref{eq:palm-two-spine-conditioned} follow from \eqref{eq:palm-two-spine} exactly as in Step~3. For \eqref{eq:palm-two-backbone}, we argue as in Step~3 and obtain \[ \E^\infty\sum_{\substack{u,v\in\mathcal V_L^\infty\\|u\wedge v|=j}}F(\omega,u,v)=\varrho_p^{-1}p^{2L-j}\,\E_{\mathbb Q_{L,j}}\bigl[F(\omega,\mathsf u,\mathsf v)\one_{\mathcal S}(\theta_{\mathsf u}\omega)\one_{\mathcal S}(\theta_{\mathsf v}\omega)\bigr]. \] By \Cref{lem:palm-spine-structure}(iv), the two endpoint configurations are independent with law $\P$ under $\mathbb Q_{L,j}$, and the probability that both survive is $\varrho_p^2$. Then the right side equals $\varrho_p\,p^{2L-j}\E_{\mathbb Q_{L,j}^\infty}F(\omega,\mathsf u,\mathsf v)$, and it is at most $\varrho_p^{-1}p^{2L-j}\E_{\mathbb Q_{L,j}}F(\omega,\mathsf u,\mathsf v)$. This completes the proof. \end{proof} \begin{remark}\label{rem:palm-labels} The identity $n\,\P(N=n)=p\,\P(N=n-1)$, used at each ancestor in Step~1, is the Mecke formula \eqref{eq:palm-mecke} for the number of points of a Poisson process. If $F(\omega,v)$ depends on $\omega$ and $v$ only through the marked tree and the auxiliary marks, and is invariant under relabelings of siblings, then \eqref{eq:palm-one-spine} can also be obtained by applying \eqref{eq:palm-mecke} to the child process at each of the $L$ ancestors, by induction on $L$; the labels $I_l$ then play no role, and only \Cref{lem:palm-spine-structure}(i) is needed to evaluate the right side. We have stated the formulas for labelled configurations so that they apply without a separate invariance check, for example to functionals that select children by their labels. \end{remark} The spine laws are absolutely continuous with respect to $\P$. In particular, every almost sure property of the tree holds along spines. \begin{corollary}\label{cor:palm-abscont} Let $p>0$, $L\ge0$, and $0\le j1$, the laws of $\omega$ under $\mathbb Q_L^\infty$ and $\mathbb Q_{L,j}^\infty$ are absolutely continuous with respect to $\P^\infty$. Further, if $A\in\mathscr F$ satisfies $\P(A)=1$, then almost surely under each of the laws $\P$, $\P^\infty$, $\mathbb Q_L$, $\mathbb Q_{L,j}$, $\mathbb Q_L^\infty$, and $\mathbb Q_{L,j}^\infty$, we have $\theta_u\omega\in A$ for all $u\in\mathbb U$ simultaneously. \end{corollary} \begin{proof} Applying \eqref{eq:palm-one-spine} to $F(\omega,v)=\one_B(\omega)$, we find that $\mathbb Q_L(\omega\in B)=p^{-L}\E[|\mathcal V_L|\one_B]$ for $B\in\mathscr F$, which is the first claim. The second follows in the same way from \eqref{eq:palm-two-spine}. For $p>1$, \eqref{eq:palm-backbone} implies that $\mathbb Q_L^\infty(\omega\in B)=p^{-L}\E^\infty[|\mathcal V_L^\infty|\one_B]$, and \eqref{eq:palm-two-backbone} gives the analogous formula for $\mathbb Q_{L,j}^\infty$. Finally, $\P(\theta_u^{-1}A)=\P(A)=1$ for every $u$, because $\theta_u$ preserves $\P$. The countable intersection $\bigcap_{u\in\mathbb U}\theta_u^{-1}A$ therefore has $\P$-probability one, and by absolute continuity it has probability one under the other laws. \end{proof} For instance, the conclusions of \Cref{thm:tree-esa}, \Cref{prop:tree-schur}, and \Cref{prop:tree-path} hold almost surely under all spine laws. We next describe recursively defined variables along a spine. This is the form in which the many-to-one formula enters the computation of path moments. \begin{corollary}\label{cor:palm-chain} Let $p>0$ and $L\ge1$. Let $(\mathsf X,\mathscr X)$ be a measurable space, let $f_0,\dots,f_L\colon\Omega\to\mathsf X$ be measurable, and let $\Psi_0,\dots,\Psi_{L-1}\colon\mathsf Z\times\mathbf N(\C\times\mathsf X)\to\mathsf X$ be measurable maps such that, for $\P$-almost every $\omega$ and every $0\le l1$. Under $\mathbb Q_L^\infty$, \textup{(i)} and \textup{(ii)} hold with the single change that $X_L$ has law $\P^\infty\circ f_L^{-1}$. \end{enumerate} Further, for every measurable $\Phi\colon\mathsf X^{L+1}\times\C^L\to[0,\infty]$, \begin{equation}\label{eq:palm-chain} \E\sum_{v\in\mathcal V_L}\Phi\Bigl(\bigl(f_l(\theta_{v_l}\omega)\bigr)_{l=0}^L,\bigl(w_{v_l}\bigr)_{l=1}^L\Bigr)=p^L\,\E_{\mathbb Q_L}\Phi\Bigl((X_l)_{l=0}^L,\bigl(w_{\mathsf v_l}\bigr)_{l=1}^L\Bigr), \end{equation} and the right side is the expectation of a function of the independent random elements in \textup{(i)}, computed through the backward recursion in \textup{(ii)}. The same holds under $\P^\infty$, with the sum over $\mathcal V_L^\infty$ and $\mathbb Q_L^\infty$ in place of $\mathbb Q_L$. \end{corollary} \begin{proof} The process $\Sigma_l^f$ is the image of the process $\Sigma_l$ of \Cref{lem:palm-spine-structure} under the measurable map $(w,\omega'')\mapsto(w,f_{l+1}(\omega''))$, and $X_L=f_L(\theta_{\mathsf v}\omega)$. Then (i) follows from \Cref{lem:palm-spine-structure}(i) and the fact that the image of a Poisson process under a measurable map is a Poisson process with the image intensity measure \cite[Theorem 5.1]{LastPenrose2018lectures}; for mixed binomial processes this is immediate, because the images of independent points are independent. For (ii), let $A$ be the set of $\omega$ for which \eqref{eq:palm-recursion} holds for every $l$. By \Cref{cor:palm-abscont}, almost surely under $\mathbb Q_L$, we have $\theta_{\mathsf v_l}\omega\in A$ for every $l$. The configuration $\theta_{\mathsf v_l}\omega$ has root offspring number $N_{\mathsf v_l}$, root auxiliary mark $\zeta_{\mathsf v_l}$, and child points $(w_{\mathsf v_lk},\theta_{\mathsf v_lk}\omega)$, $1\le k\le N_{\mathsf v_l}$. The child with $k=I_{l+1}$ is $\mathsf v_{l+1}$, and its point contributes $\delta_{(w_{\mathsf v_{l+1}},X_{l+1})}$ in \eqref{eq:palm-recursion}; the other children contribute $\Sigma_l^f$. This proves (ii). Part (iii) follows in the same way from \Cref{lem:palm-spine-structure}(iii). Finally, \eqref{eq:palm-chain} is \eqref{eq:palm-one-spine} applied to $F(\omega,v)=\Phi((f_l(\theta_{v_l}\omega))_l,(w_{v_l})_l)$, and its conditioned version is \eqref{eq:palm-backbone}. \end{proof} The same argument applies to the two-spine laws. Under $\mathbb Q_{L,j}$, the variables along the two arms are computed from the two endpoint variables $f_L(\theta_{\mathsf u}\omega)$ and $f_L(\theta_{\mathsf v}\omega)$, which are independent with law $\P\circ f_L^{-1}$, by the recursion of \Cref{cor:palm-chain}(ii). At the fork, $\Psi_j$ receives the side process of the fork together with the two points $(w_{\mathsf u_{j+1}},f_{j+1}(\theta_{\mathsf u_{j+1}}\omega))$ and $(w_{\mathsf v_{j+1}},f_{j+1}(\theta_{\mathsf v_{j+1}}\omega))$, and along the common part the recursion is that of (ii). All spine edge marks, auxiliary marks, and side processes are independent, by \Cref{lem:palm-spine-structure}(iv). \begin{remark}\label{rem:palm-green} As an example, fix $z\in\C\setminus\R$ and $s>0$, let $f_l=G_{oo}(z)$ for every $l$, and let $\Psi_l(\zeta,\pi)=-\bigl(z+p^{-1}\int|w|^2x\,\pi(\dd w\,\dd x)\bigr)^{-1}$. By \Cref{prop:tree-schur} at the root, \eqref{eq:palm-recursion} holds $\P$-almost surely, and $X_0=G_{oo}(z)$ and $X_l=\Gamma_{\mathsf v_l}(z)$ for $1\le l\le L$. By \Cref{prop:tree-path}, for $v\in\mathcal V_L$ we have \[ |G_{ov}(z)|^s=|G_{oo}(z)|^s\prod_{l=1}^Lp^{-s/2}Y_{v_l}^{s/2}|\Gamma_{v_l}(z)|^s. \] By \eqref{eq:palm-chain}, we have \[ \E\sum_{v\in\mathcal V_L}|G_{ov}(z)|^s=p^{L(1-s/2)}\,\E_{\mathbb Q_L}\Bigl[|X_0|^s\prod_{l=1}^LY_{\mathsf v_l}^{s/2}|X_l|^s\Bigr]. \] On the right, $X_L$ has the law of $G_{oo}(z)$, the squared moduli $Y_{\mathsf v_l}$ of the spine marks are independent $\Exp(1)$ variables, and $X_l=-(z+p^{-1}Y_{\mathsf v_{l+1}}X_{l+1}+\Xi_l)^{-1}$ for $l0$ and let $\pi$ be a real solution at energy $E$ in the sense of \Cref{def:rde-real}. Let $q_\pi$ be a measurable map from $[0,1]$ to $\R\setminus\{0\}$ which pushes Lebesgue measure forward to $\pi$, and define $f^{(0)}(\omega)=q_\pi(\zeta_o(2))$, where $\zeta_o(2)$ is the second coordinate of $\zeta_o$, and, for $m\ge1$, \[ f^{(m)}(\omega)=E-p^{-1}\sum_{k=1}^{N_o}\frac{Y_k}{f^{(m-1)}(\theta_k\omega)} \] (we set $f^{(m)}(\omega)=E$ if some denominator vanishes, and we use the same convention for $\Psi_l$ below). By induction on $m$, $f^{(m)}$ has law $\pi$ under $\P$ for every $m\ge0$. Indeed, by \Cref{lem:palm-branching}(ii), the pairs $(Y_k,f^{(m-1)}(\theta_k\omega))$ are independent with law $\Exp(1)\otimes\pi$ and independent of $N_o$. Then $f^{(m)}$ has the law of $E-p^{-1}\sum_{k\le N}Y_k/D_k$ with independent $D_k\sim\pi$, which is $\pi$; in particular the denominators vanish with probability zero. Fix $m\ge L$ and apply \Cref{cor:palm-chain} with $f_l=f^{(m-l)}$ and $\Psi_l(\zeta,\pi')=E-p^{-1}\int|w|^2x^{-1}\,\pi'(\dd w\,\dd x)$. In this case, $X_L$ has law $\pi$ and the side processes are Poisson with intensity measure $p\,\gamma\otimes\pi$, for every $m\ge L$. The family $D_v=f^{(m-|v|)}(\theta_v\omega)$, $|v|\le m$, satisfies $D_v=E-p^{-1}\sum_kY_{vk}/D_{vk}$ for $|v|1$, let $(\mathsf X,\mathscr X)$ be a measurable space, and let $X\colon\Omega\to\mathsf X$ be measurable. \begin{enumerate}[label=\textup{(\roman*)}] \item As measures on $\mathsf X$, $\P^\infty\circ X^{-1}\le\varrho_p^{-1}\,\P\circ X^{-1}$. Equivalently, $\E^\infty\varphi(X)\le\varrho_p^{-1}\E\varphi(X)$ for every measurable $\varphi\colon\mathsf X\to[0,\infty]$. In particular, if $C\in\mathscr X$ and $\P(X\notin C)\le\delta$, then $\P^\infty(X\in C)\ge1-\varrho_p^{-1}\delta$. \item Let $L\ge0$ and $0\le j0$, we have \[ \sup_n\E^\infty\bigl[|Z_{\eta_n}|\one\{|Z_{\eta_n}|>K\}\bigr]\le\varrho_p^{-1}\sup_n\E\bigl[|Z_{\eta_n}|\one\{|Z_{\eta_n}|>K\}\bigr], \] and the right side tends to zero as $K\to\infty$. This shows that the sequence is uniformly integrable under $\P^\infty$ as well. By Fatou's lemma, $Z_0$ is integrable under $\P$. Then Vitali's convergence theorem (see \cite[Section 4.6]{Durrett2019probability}), applied under $\P^\infty$, implies that $Z_{\eta_n}\to Z_0$ in $L^1(\P^\infty)$. The last statement follows from (ii), since $\E_{\mathbb Q_L^\infty}Z(\theta_{\mathsf v}\omega)=\E^\infty Z$ for every integrable $Z$. For (iv), $\varphi(X_\eta)\to\varphi(X_0)$ on the event $\{X_\eta\to X_0\}\cap\{X_0\notin D\}$, which has $\P$-probability one and therefore $\P^\infty$-probability one. The bounded convergence theorem concludes the proof. \end{proof} \begin{remark}\label{rem:palm-endpoint-atom} The domination in \Cref{prop:palm-endpoints}(i) cannot be reversed with a constant independent of $X$. For instance, $\P(N_o=0)=e^{-p}$, while $\P^\infty(N_o=0)=0$, because a root without children has a finite tree. If a functional $X$ equals a fixed value $x_0$ on $\{N_o=0\}$, as the real boundary denominator does (it equals $E$ there), then the part of the atom of $\P\circ X^{-1}$ at $x_0$ coming from the event $\{N_o=0\}$ is absent from $\P^\infty\circ X^{-1}$. In particular, conditioned endpoint laws are in general not equal to the corresponding unconditioned laws, and only upper bounds transfer from $\P$ to $\P^\infty$ through (i). \end{remark} \section{Fresh squared moduli}\label{sec:palm-fresh} Many estimates in this paper bound negative moments of quantities of the form $c+\sum_ja_jY_j$, where the $Y_j$ are squared moduli of edge marks and $c$ and the $a_j$ depend on the rest of the tree. The following definition makes precise when such an estimate may be applied conditionally. \begin{definition}\label{def:palm-fresh} Let $(\widehat\Omega,\mathscr A,\mathbb P)$ be a probability space and let $\mathscr G\subset\mathscr A$ be a $\sigma$-field. Let $M$ be a $\mathscr G$-measurable random variable with values in $\{0,1,2,\dots\}$, and let $Y_1,Y_2,\dots$ be random variables with values in $[0,\infty)$. We say that $Y_1,\dots,Y_M$ are \emph{fresh squared moduli given $\mathscr G$}, or fresh given $\mathscr G$, if for every $m\ge0$ and every bounded Borel function $\varphi\colon[0,\infty)^m\to\R$, \begin{equation}\label{eq:palm-fresh} \mathbb E\bigl[\varphi(Y_1,\dots,Y_m)\one\{M=m\}\bigm|\mathscr G\bigr]=\one\{M=m\}\int_{[0,\infty)^m}\varphi(y)\,e^{-(y_1+\dots+y_m)}\,\dd y \end{equation} almost surely. Random variables that are $\mathscr G$-measurable are called revealed. \end{definition} In words, on the event $\{M=m\}$, the variables $Y_1,\dots,Y_m$ are independent $\Exp(1)$ variables conditionally on $\mathscr G$. The number $M$ of fresh variables is revealed, while their values are not; the variables $Y_j$ with $j>M$ play no role. In the applications, $\widehat\Omega$ is $\Omega$ with one of the laws $\P$, $\P^\infty$, $\mathbb Q_L$, $\mathbb Q_L^\infty$, $\mathbb Q_{L,j}$, $\mathbb Q_{L,j}^\infty$, and the $Y_j$ are the squared moduli of the marks of a random finite set of edges, listed in a revealed order. We record three elementary consequences of the definition. \begin{enumerate}[label=\textup{(\alph*)}] \item If $Y_1,\dots,Y_M$ are fresh given $\mathscr G$, and $\mathscr G'\subset\mathscr G$ is a $\sigma$-field with respect to which $M$ is measurable, then $Y_1,\dots,Y_M$ are fresh given $\mathscr G'$. This follows from \eqref{eq:palm-fresh} by the tower property of conditional expectation. \item Revealing some of the fresh variables leaves the others fresh. Precisely, let $M'\le M$ be $\mathscr G$-measurable, and let $\mathscr G'$ be generated by $\mathscr G$ and the variables $Y_j\one\{M'0$, and let $A$ be a random finite subset of $\mathbb U_*$ such that $\{A=a\}\in\mathscr G_a$ for every finite set $a\subset\mathbb U_*$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every measurable $F\colon\Omega\to[0,\infty]$, \begin{equation}\label{eq:palm-fresh-edges} \E F(\omega)=\E\int_{\C^A}F\bigl(\omega^{A\leftarrow x}\bigr)\,\gamma^{\otimes A}(\dd x). \end{equation} If $p>1$, the same identity holds with $\E^\infty$ in place of $\E$ on both sides. \item Let $L\ge0$, $0\le j1$. The analogous identities hold under $\mathbb Q_{L,j}$ and $\mathbb Q_{L,j}^\infty$ for $A=A(\omega,\mathsf u,\mathsf v)$ such that $A(\cdot,u,v)$ satisfies the hypothesis above for all $u,v\in\mathbb U_L$. \item In the setting of \textup{(i)} or \textup{(ii)}, list the elements of $A$ as $v^{(1)},\dots,v^{(M)}$, where $M=|A|$, in a fixed total order of $\mathbb U$. Let $\mathscr H_A$ be the $\sigma$-field generated by $A$, by the configuration $\omega^{A\leftarrow\mathbf 1}$ in which all selected marks are replaced by $1$, by the phases $w_v/|w_v|$, $v\in A$ (set to $1$ if $w_v=0$), and, in the setting of \textup{(ii)}, by the endpoint $\mathsf v$, respectively $(\mathsf u,\mathsf v)$. Then $Y_{v^{(1)}},\dots,Y_{v^{(M)}}$ are fresh squared moduli given $\mathscr H_A$. \end{enumerate} \end{lemma} \begin{proof} For (i), fix a finite set $a\subset\mathbb U_*$. The indicator $\one\{A=a\}$ and the configuration $\omega^{a\leftarrow x}$, for fixed $x$, are functions of the coordinates of $\omega$ other than $(w_v)_{v\in a}$, and under $\P$ the coordinates $(w_v)_{v\in a}$ are independent of the other coordinates and have law $\gamma^{\otimes a}$. Since $\omega=\omega^{a\leftarrow(w_v)_{v\in a}}$, we have by Fubini's theorem that \[ \E\bigl[\one\{A=a\}F(\omega)\bigr]=\E\Bigl[\one\{A=a\}\int_{\C^a}F\bigl(\omega^{a\leftarrow x}\bigr)\,\gamma^{\otimes a}(\dd x)\Bigr]. \] Summing over the countably many finite sets $a$ gives \eqref{eq:palm-fresh-edges}. Under $\P^\infty$, we apply \eqref{eq:palm-fresh-edges} to $F\one_{\mathcal S}$ and use that $\one_{\mathcal S}(\omega^{A\leftarrow x})=\one_{\mathcal S}(\omega)$, since $\mathcal S$ depends only on the offspring numbers. For (ii), we use \eqref{eq:palm-one-spine} to write $\E_{\mathbb Q_L}F(\omega,\mathsf v)=p^{-L}\sum_{v\in\mathbb U_L}\E[\one\{v\in V(\omega)\}F(\omega,v)]$. For each fixed $v$, we apply (i) with the random set $A(\cdot,v)$ to the function $\omega\mapsto\one\{v\in V(\omega)\}F(\omega,v)$, and we use that $V(\omega^{a\leftarrow x})=V(\omega)$. Then we apply \eqref{eq:palm-one-spine} backwards to the function $(\omega,v)\mapsto\int F(\omega^{A(\omega,v)\leftarrow x},v)\,\gamma^{\otimes A(\omega,v)}(\dd x)$. This proves the identity under $\mathbb Q_L$. Under $\mathbb Q_L^\infty$, we argue in the same way with \eqref{eq:palm-backbone} and the second part of (i), using that $\one_{\mathcal S}(\theta_v\omega)$ depends only on the offspring numbers. The two-spine identities follow in the same way from \eqref{eq:palm-two-spine} and \eqref{eq:palm-two-backbone}. For (iii), we first note that changing the selected marks does not change $A$: if $\omega\in\{A=a\}$, then $\omega^{a\leftarrow x}\in\{A=a\}$ for every $x\in\C^a$, since $\{A=a\}\in\mathscr G_a$. Fix $m\ge0$, a bounded Borel function $\varphi\ge0$ on $[0,\infty)^m$, and a bounded measurable function $g\ge0$ of the generators of $\mathscr H_A$. We write $x_v=y_v^{1/2}e^{\iu\phi_v}$; under $\gamma$, $y_v$ has law $\Exp(1)$, $e^{\iu\phi_v}$ is uniform on the unit circle, and the two are independent. Applying (i) to the function $F=g\cdot\one\{M=m\}\varphi(Y_{v^{(1)}},\dots,Y_{v^{(m)}})$, we obtain \begin{multline*} \E\bigl[g\,\one\{M=m\}\varphi(Y_{v^{(1)}},\dots,Y_{v^{(m)}})\bigr]\\ =\E\Bigl[\one\{M=m\}\int g\bigl(\omega^{A\leftarrow x}\bigr)\,\gamma^{\otimes A}(\dd x)\Bigr]\int\varphi(y)e^{-(y_1+\dots+y_m)}\,\dd y. \end{multline*} Here we used that $g(\omega^{A\leftarrow x})$ depends on $x$ only through the phases of the $x_v$, while $\varphi$ depends only on their moduli. Applying (i) once more, to the function $g\,\one\{M=m\}$, shows that the first factor on the right equals $\E[g\,\one\{M=m\}]$. This is \eqref{eq:palm-fresh} for $\mathscr G=\mathscr H_A$. The same argument applies in the setting of (ii). \end{proof} For example, take $A=\{1,\dots,N_o\}$, the set of children of the root. We have $\{A=a\}\in\sigma(N_o)\subset\mathscr G_a$, and $\mathscr H_A$ contains $N_o$, $\zeta_o$, all forward configurations $\theta_k\omega$, $k\ge1$, and the phases of the $w_k$. By \Cref{lem:palm-fresh-edges}(iii) and consequence (a) after \Cref{def:palm-fresh}, the squared moduli $Y_1,\dots,Y_{N_o}$ of the root edge marks are fresh given the $\sigma$-field generated by $N_o$, $\zeta_o$, the forward configurations of the children, and the phases. This is the situation of the cavity equation. Similarly, under $\mathbb Q_L$, the choice $A(\omega,v)=\{v_1,\dots,v_L\}$ in \Cref{lem:palm-fresh-edges}(ii) shows that the squared moduli $Y_{\mathsf v_1},\dots,Y_{\mathsf v_L}$ of the spine marks are fresh given the $\sigma$-field generated by the endpoint $\mathsf v$, the phases of the spine edge marks, and all coordinates of $\omega$ other than the spine edge marks. We now state the affine negative-moment lemma. \begin{lemma}\label{lem:palm-affine} Let $00$, and set $C_{r,d}=(1-r)^{-1}(2/d)^r$. Let $m\ge0$, let $Y_1,\dots,Y_m$ be independent random variables with law $\Exp(1)$, let $a_1,\dots,a_m\in\C$ and $c\in\C$, and set $S=c+\sum_{j=1}^ma_jY_j$. \begin{enumerate}[label=\textup{(\roman*)}] \item If $c\neq0$, then for every $\varepsilon>0$, \begin{equation}\label{eq:palm-small-ball} \P(|S|\le\varepsilon)\le\min\Bigl\{1,\frac{2\varepsilon\sqrt{m+1}}{|c|}\Bigr\}. \end{equation} \item If $c\neq0$, then \begin{equation}\label{eq:palm-affine-moment} \E|S|^{-r}\le\frac1{1-r}\Bigl(\frac{2\sqrt{m+1}}{|c|}\Bigr)^r. \end{equation} In particular, if $|c|\ge d$, then $\E|S|^{-r}\le C_{r,d}(m+1)^{r/2}$. \item Suppose that $c,a_1,\dots,a_m$ are real. If $a_1=\dots=a_m=0$, then the law of $S$ is $\delta_c$. Otherwise, the law of $S$ is absolutely continuous, and it has a density $\rho$ such that \begin{equation}\label{eq:palm-affine-density} \rho(x)\le\frac{\sqrt m}{\sqrt{2\pi}\,|x-c|}\qquad\text{for every }x\in\R\setminus\{c\}. \end{equation} In either case, the absolutely continuous part of the law of $S$ has a density which is at most $\sqrt{m+1}/(2d)$ at every $x$ with $|x-c|\ge d$. \item Let $\mathscr G$, $M$, and $Y_1,Y_2,\dots$ be as in \Cref{def:palm-fresh}, with $Y_1,\dots,Y_M$ fresh given $\mathscr G$. Let $c,a_1,a_2,\dots$ be $\mathscr G$-measurable complex random variables, and set $S=c+\sum_{j=1}^Ma_jY_j$. Then, for every $\varepsilon>0$, almost surely on the event $\{c\neq0\}$, \[ \mathbb P(|S|\le\varepsilon\mid\mathscr G)\le\min\Bigl\{1,\frac{2\varepsilon\sqrt{M+1}}{|c|}\Bigr\},\qquad\mathbb E\bigl[|S|^{-r}\bigm|\mathscr G\bigr]\le\frac1{1-r}\Bigl(\frac{2\sqrt{M+1}}{|c|}\Bigr)^r. \] If $|c|\ge d$ almost surely, then $\mathbb E|S|^{-r}\le C_{r,d}\,\mathbb E(M+1)^{r/2}\le C_{r,d}(1+\mathbb EM)^{r/2}$. If $c$ and all $a_j$ are real, then for every Borel set $B\subset\R$, almost surely, \[ \mathbb P\bigl(S\in B,\ |S-c|\ge d\bigm|\mathscr G\bigr)\le\frac{\sqrt{M+1}}{2d}\,|B|, \] where $|B|$ denotes the Lebesgue measure of $B$. \end{enumerate} \end{lemma} The bounds in (i)--(iii) depend on the coefficients $a_j$ only through their number. This is what makes the lemma applicable when the coefficients are cavity variables of subtrees, about which nothing is known except that they are revealed. \begin{proof} We first prove the density bound (iii) for real coefficients using the Gamma--Dirichlet decomposition of $(Y_1,\dots,Y_m)$. Then we deduce the small-ball bound (i) for complex coefficients by rotating $c$ to the positive real axis and projecting onto the real axis, and we integrate it to obtain (ii). Finally, we derive the conditional version (iv) by treating the revealed variables as constants. \proofstep{1}{The Gamma--Dirichlet decomposition.} Let $m\ge1$, set $T=Y_1+\dots+Y_m$, and set $U_j=Y_j/T$ for $1\le j\le m$. Then $T$ has the Gamma density \[ f_m(t)=\frac{t^{m-1}e^{-t}}{(m-1)!},\qquad t>0, \] and the vector $(U_1,\dots,U_m)$ is independent of $T$ (it is uniformly distributed on the simplex). Indeed, the joint density of $(Y_1,\dots,Y_m)$ is $e^{-(y_1+\dots+y_m)}=e^{-t}$, and the change of variables $(y_1,\dots,y_m)\mapsto(t,u_1,\dots,u_{m-1})$ has Jacobian $t^{m-1}$; it follows that the joint density of $(T,U_1,\dots,U_{m-1})$ is a product of a function of $t$ and a function of $(u_1,\dots,u_{m-1})$. We also need the bound \begin{equation}\label{eq:palm-gamma-sup} \sigma_m=\sup_{t>0}t\,f_m(t)=\frac{m^me^{-m}}{(m-1)!}=\frac{m^{m+1}e^{-m}}{m!}\le\frac{\sqrt m}{\sqrt{2\pi}}. \end{equation} The supremum is attained at $t=m$, since $t\mapsto t^me^{-t}$ is maximal there, and in the last inequality we used Stirling's formula in the form $m!\ge\sqrt{2\pi m}\,(m/e)^m$, valid for every $m\ge1$. \proofstep{2}{Real coefficients.} Let $c,a_1,\dots,a_m$ be real. If all $a_j$ vanish (including the case $m=0$), then $S=c$. Otherwise $m\ge1$, and we may assume that $a_1\neq0$. Then $S=a_1Y_1+(c+\sum_{j\ge2}a_jY_j)$ is the sum of an absolutely continuous variable and an independent variable, and its law is therefore absolutely continuous. With the notation of Step~1, set $A=\sum_{j=1}^ma_jU_j$, so that $S=c+AT$, where $A$ is independent of $T$. Since $\{A=0\}\subset\{S=c\}$ and $\P(S=c)=0$, we have $\P(A=0)=0$. For a Borel set $B\subset\R$, Tonelli's theorem and the change of variables $x=c+At$ give \[ \P(S\in B)=\E\Bigl[\int_0^\infty\one_B(c+At)f_m(t)\,\dd t\Bigr]=\int_B\rho(x)\,\dd x, \] where \[ \rho(x)=\E\Bigl[\one\Bigl\{\frac{x-c}A>0\Bigr\}\frac{1}{|A|}\,f_m\Bigl(\frac{x-c}A\Bigr)\Bigr], \] and where the integrand in the definition of $\rho$ is set to zero on the null event $\{A=0\}$. The function $\rho$ is Borel by Tonelli's theorem. Let $x\neq c$, and suppose that $t=(x-c)/A>0$. Then $|A|=|x-c|/t$, so that \[ \frac1{|A|}f_m\Bigl(\frac{x-c}A\Bigr)=\frac{t\,f_m(t)}{|x-c|}\le\frac{\sigma_m}{|x-c|}. \] Taking expectations and using \eqref{eq:palm-gamma-sup}, we arrive at \eqref{eq:palm-affine-density}. The last assertion of (iii) follows because $\sqrt m/\sqrt{2\pi}\le\sqrt{m+1}/2$, and because the absolutely continuous part of $\delta_c$ is zero. \proofstep{3}{The small-ball bound.} Let $c\neq0$, let $\theta$ be an argument of $c$, and set \[ Q=\Re\bigl(e^{-\iu\theta}S\bigr)=|c|+\sum_{j=1}^mb_jY_j,\qquad b_j=\Re\bigl(e^{-\iu\theta}a_j\bigr)\in\R. \] Then $|S|=|e^{-\iu\theta}S|\ge|Q|$, which implies that $\P(|S|\le\varepsilon)\le\P(|Q|\le\varepsilon)$. If $\varepsilon>|c|/2$, then the right side of \eqref{eq:palm-small-ball} equals $1$ and there is nothing to prove. Let $0<\varepsilon\le|c|/2$. If all $b_j$ vanish, then $Q=|c|>\varepsilon$, and $\P(|Q|\le\varepsilon)=0$. Otherwise, by Step~2 applied to $Q$, whose constant is $|c|$, the law of $Q$ has a density bounded by $\sigma_m/|x-|c||$. Every $x\in[-\varepsilon,\varepsilon]$ satisfies $|x-|c||\ge|c|/2$, so \[ \P(|Q|\le\varepsilon)\le2\varepsilon\cdot\frac{2\sigma_m}{|c|}\le\frac{4}{\sqrt{2\pi}}\cdot\frac{\varepsilon\sqrt m}{|c|}\le\frac{2\varepsilon\sqrt{m+1}}{|c|}. \] This proves (i). \proofstep{4}{The negative moment.} By (i), $\P(S=0)=0$. For $t>0$, we have $t^{-r}=\int_t^\infty r\varepsilon^{-r-1}\,\dd\varepsilon$, and Tonelli's theorem implies that \[ \E|S|^{-r}=\int_0^\infty r\varepsilon^{-r-1}\,\P(|S|\le\varepsilon)\,\dd\varepsilon. \] Set $K=2\sqrt{m+1}/|c|$. By (i), the right side is at most \[ \int_0^{1/K}rK\varepsilon^{-r}\,\dd\varepsilon+\int_{1/K}^\infty r\varepsilon^{-r-1}\,\dd\varepsilon=\frac{r}{1-r}K^r+K^r=\frac{K^r}{1-r}, \] which is \eqref{eq:palm-affine-moment}. If $|c|\ge d$, then $K^r\le(2/d)^r(m+1)^{r/2}$, which gives the second assertion of (ii). \proofstep{5}{The conditional version.} Fix $m\ge0$. By consequence (c) after \Cref{def:palm-fresh}, applied with $X=(c,a_1,\dots,a_m)$, the substitution identity \eqref{eq:palm-freeze} holds for every Borel function $\Phi\colon\C^{1+m}\times[0,\infty)^m\to[0,\infty]$. We now apply \eqref{eq:palm-freeze} with $\Phi(c,a,y)=\one\{|c+\sum_ja_jy_j|\le\varepsilon\}\one\{c\neq0\}$. By (i), applied for fixed values of $c\neq0$ and $a_1,\dots,a_m$, the right side of \eqref{eq:palm-freeze} is at most $\one\{M=m\}\one\{c\neq0\}\min\{1,2\varepsilon\sqrt{m+1}/|c|\}$. Summing over $m$ gives the small-ball bound in (iv). The bound on $\mathbb E[|S|^{-r}\mid\mathscr G]$ follows in the same way from (ii), with $\Phi(c,a,y)=|c+\sum_ja_jy_j|^{-r}\one\{c\neq0\}$. If $|c|\ge d$ almost surely, then taking expectations, we have $\mathbb E|S|^{-r}\le C_{r,d}\,\mathbb E(M+1)^{r/2}$, and Jensen's inequality for the concave function $t\mapsto t^{r/2}$ gives $\mathbb E(M+1)^{r/2}\le(1+\mathbb EM)^{r/2}$. Finally, if $c$ and the $a_j$ are real, we apply \eqref{eq:palm-freeze} with $\Phi(c,a,y)=\one\{c+\sum_ja_jy_j\in B,\ |\sum_ja_jy_j|\ge d\}$. For fixed real $c,a_1,\dots,a_m$, the integral of $\Phi$ vanishes if all $a_j$ vanish, and otherwise it is at most $\int_{B\cap\{|x-c|\ge d\}}\rho(x)\,\dd x\le\sqrt{m+1}\,|B|/(2d)$ by (iii). Summing over $m$ completes the proof. \end{proof} \begin{remark}\label{rem:palm-not-fresh} Freshness is a property of the variables together with the $\sigma$-field, and it is lost as soon as the $\sigma$-field carries information about their values. We give an example. Let $Y_1$ have law $\Exp(1)$, let $0<\delta\le1/2$, and let $\mathscr G=\sigma(A_\delta)$, where $A_\delta=\{|Y_1-1|\le\delta\}$ is a truncation event for $Y_1$. The variable $Y_1$ is not fresh given $\mathscr G$: on $A_\delta$, its conditional law is supported in $[1-\delta,1+\delta]$. The conclusion of \Cref{lem:palm-affine}(iv) fails as well. Let $M=1$, $c=1$, and $a_1=-1$, which are constants and hence $\mathscr G$-measurable, so that $S=1-Y_1$. On $A_\delta$, we have \[ \mathbb E\bigl[|S|^{-r}\bigm|\mathscr G\bigr]=\frac{\int_{1-\delta}^{1+\delta}|1-y|^{-r}e^{-y}\,\dd y}{\int_{1-\delta}^{1+\delta}e^{-y}\,\dd y}\ge\frac{e^{-1-\delta}\cdot2\delta^{1-r}(1-r)^{-1}}{2\delta e^{-1+\delta}}=\frac{e^{-2\delta}\delta^{-r}}{1-r}, \] which tends to infinity as $\delta\to0$, while the bound $(1-r)^{-1}2^{3r/2}$ of \Cref{lem:palm-affine}(iv) does not depend on $\delta$. The same failure occurs if $\mathscr G$ contains $\sigma(Y_1)$, that is, if $Y_1$ is conditioned on its own value. In this case $Y_1$ is revealed, the term $a_1Y_1$ must be absorbed into the constant as in (b) after \Cref{def:palm-fresh}, and the new constant $1-Y_1$ is not bounded away from zero. In particular, whenever an event defined in terms of a squared modulus is included in the conditioning $\sigma$-field, that squared modulus can no longer be treated as fresh. \end{remark} As a first application, we bound fractional moments of the diagonal resolvent at the root, uniformly in the imaginary part of the spectral parameter. \begin{corollary}\label{cor:palm-root-moment} Let $p>0$, $01$, then $\E^\infty|G_{oo}(z)|^r$ is at most $\varrho_p^{-1}$ times the same bound. \end{corollary} \begin{proof} By \Cref{prop:tree-schur} at the root and the identity $\Gamma_k(z)=G_{oo}(z)\circ\theta_k$, we have, $\P$-almost surely, \[ G_{oo}(z)=-\Bigl(z+\frac1p\sum_{k=1}^{N_o}Y_k\,G_{oo}(z)\circ\theta_k\Bigr)^{-1}. \] Let $\mathscr G$ be the $\sigma$-field generated by $N_o$, $\zeta_o$, the configurations $\theta_k\omega$ for $k\ge1$, and the phases $w_k/|w_k|$ for $k\ge1$. By the example following the proof of \Cref{lem:palm-fresh-edges}, together with consequence (a) after \Cref{def:palm-fresh}, the squared moduli $Y_1,\dots,Y_{N_o}$ are fresh given $\mathscr G$. The coefficients $a_k=p^{-1}G_{oo}(z)\circ\theta_k$ are $\mathscr G$-measurable, and the constant $c=z$ is deterministic and nonzero. Then \Cref{lem:palm-affine}(iv) implies that $\E[|G_{oo}(z)|^r\mid\mathscr G]\le(1-r)^{-1}(2\sqrt{N_o+1}/|z|)^r$. Taking expectations and using Jensen's inequality, $\E(N_o+1)^{r/2}\le(p+1)^{r/2}$, we find the first claim. The second follows from \eqref{eq:palm-survival-domination}. \end{proof} Since $|z|\ge|\Re z|$, the bound is uniform over all $z$ with real part bounded away from zero, which is the uniformity required at energies $E\neq0$. \section{Selections at a vertex and the backbone}\label{sec:palm-decorations} We now classify the children of a vertex by the shape of their forward subtrees. For a vertex $x\in V(\omega)$, a child $xk$ of $x$ is a \emph{leaf child} if $N_{xk}=0$, so that its forward subtree consists of $xk$ alone; it is a \emph{surviving child} if $\theta_{xk}\omega\in\mathcal S$; and it is an \emph{other child} if neither holds. We write $N^{\mathrm{leaf}}_x$, $N^{\mathrm{surv}}_x$, and $N^{\mathrm{oth}}_x$ for the numbers of these children. For a vertex $x$ with $N^{\mathrm{leaf}}_x\ge1$, we let $\ell(x)$ denote the leaf child $xk$ of $x$ for which the pair $(\zeta_{xk}(1),k)$ is lexicographically smallest; thus $\ell(x)$ is the leaf child with the smallest rank, ties (which have probability zero) being broken by the label. Recall $\mathsf M=\C\times\Omega$ and $\mathfrak m=\gamma\otimes\P$, and set \[ \mathfrak m^{\mathrm{leaf}}=\gamma\otimes\P(\,\cdot\mid N_o=0),\qquad\mathfrak m^\infty=\gamma\otimes\P^\infty,\qquad\mathfrak m^{\mathrm{oth}}=\gamma\otimes\P(\,\cdot\mid\Omega\setminus(\mathcal S\cup\{N_o=0\})). \] The last conditioning event has probability $1-\varrho_p-e^{-p}\ge\P(N_o=1,\ N_1=0)=pe^{-2p}>0$. Finally, let \[ \mathscr G^{\mathrm{leaf}}=\sigma\bigl(N_v,\zeta_v:\ v\in\mathbb U;\ \ w_v\one\{N_v\ge1\}:\ v\in\mathbb U_*\bigr) \] be the $\sigma$-field generated by the configuration in which the marks of all edges leading to leaves have been erased. \begin{lemma}\label{lem:palm-decorations} Let $p>1$. \begin{enumerate}[label=\textup{(\roman*)}] \item Under $\P$, the processes $\Pi^{\mathrm{leaf}}$, $\Pi^{\mathrm{surv}}$, and $\Pi^{\mathrm{oth}}$ formed by the points $(w_k,\theta_k\omega)$ of the leaf children, the surviving children, and the other children of the root are independent Poisson processes with intensity measures $pe^{-p}\mathfrak m^{\mathrm{leaf}}$, $p\varrho_p\mathfrak m^\infty$, and $p(1-\varrho_p-e^{-p})\mathfrak m^{\mathrm{oth}}$, and they are independent of $\zeta_o$. In particular, $N^{\mathrm{leaf}}_o$, $N^{\mathrm{surv}}_o$, and $N^{\mathrm{oth}}_o$ are independent Poisson variables with means $pe^{-p}$, $p\varrho_p$, and $p(1-\varrho_p-e^{-p})$. Under $\P^\infty$, the processes $\Pi^{\mathrm{leaf}}$, $\Pi^{\mathrm{oth}}$, and the mark $\zeta_o$ have the same laws as under $\P$ and are independent of $\Pi^{\mathrm{surv}}$, which has the law of a Poisson process with intensity measure $p\varrho_p\mathfrak m^\infty$ conditioned to have at least one point. In particular, for every measurable set $B\subset\mathsf M^\infty=\C\times\mathcal S$, \begin{equation}\label{eq:palm-decoration-probability} \P^\infty\bigl(N^{\mathrm{leaf}}_o\ge1,\ \Pi^{\mathrm{surv}}(B)\ge1\bigr)=\varrho_p^{-1}\bigl(1-e^{-pe^{-p}}\bigr)\bigl(1-e^{-p\,\mathfrak m(B)}\bigr). \end{equation} \item Let $\mathcal R=\{N^{\mathrm{leaf}}_o=1,\ N^{\mathrm{surv}}_o=1,\ N^{\mathrm{oth}}_o=0\}$ be the event that the root has exactly one leaf child, exactly one surviving child, and no other child, and on $\mathcal R$ let $\ell,y\in\{1,2\}$ be the labels of the leaf child and of the surviving child. Then $\mathcal R\subset\mathcal S$, $\P(\mathcal R)=p^2\varrho_pe^{-2p}$, and $\P^\infty(\mathcal R)=p^2e^{-2p}$. Moreover, for all measurable $F_1,F_2\colon\mathsf M\to[0,\infty]$ and $F_3\colon\mathsf Z\to[0,\infty]$, \begin{equation}\label{eq:palm-rigid} \E^\infty\bigl[\one_{\mathcal R}\,F_1(w_y,\theta_y\omega)\,F_2(w_\ell,\theta_\ell\omega)\,F_3(\zeta_o)\bigr]=p^2e^{-2p}\int F_1\,\dd\mathfrak m^\infty\int F_2\,\dd\mathfrak m^{\mathrm{leaf}}\int F_3\,\dd\kappa. \end{equation} In particular, conditionally on $\mathcal R$ under $\P^\infty$, the squared moduli $Y_\ell$ and $Y_y$ are independent $\Exp(1)$ variables, the forward configuration $\theta_y\omega$ of the surviving child has law $\P^\infty$, and these three random elements are independent of each other and of the phases $w_\ell/|w_\ell|$, $w_y/|w_y|$. \item The root has exactly one child with probability $\P(N_o=1)=\P^\infty(N_o=1)=pe^{-p}$. Conditionally on $\{N_o=1\}$, the point $(w_1,\theta_1\omega)$ has law $\mathfrak m$ under $\P$ and law $\mathfrak m^\infty$ under $\P^\infty$. Along a spine, let $L\ge1$, let $J\subset\{0,\dots,L-1\}$, and let $\mathcal C_J=\{N_{\mathsf v_l}=1\text{ for all }l\in J\}$ be the event that the spine vertices $\mathsf v_l$, $l\in J$, have no side children. Then $\mathbb Q_L(\mathcal C_J)=\mathbb Q_L^\infty(\mathcal C_J)=e^{-p|J|}$; conditionally on $\mathcal C_J$, the conclusions of \Cref{lem:palm-spine-structure} hold with $S_l$ empty for $l\in J$; and for every measurable $F\colon\Omega\times\mathbb U\to[0,\infty]$, \[ \E\sum_{v\in\mathcal V_L}F(\omega,v)\prod_{l\in J}\one\{N_{v_l}=1\}=p^Le^{-p|J|}\,\E_{\mathbb Q_L}\bigl[F(\omega,\mathsf v)\bigm|\mathcal C_J\bigr]. \] The same identity holds with $\E^\infty$, $\mathcal V_L^\infty$, and $\mathbb Q_L^\infty$ in place of $\E$, $\mathcal V_L$, and $\mathbb Q_L$. \item Let $X$ be a random finite subset of $\mathbb U$ which is $\mathscr G^{\mathrm{leaf}}$-measurable, and let $A=\{\ell(x):x\in X\cap V(\omega),\ N^{\mathrm{leaf}}_x\ge1\}$ be the set of selected leaf children of the vertices in $X$. Then $\{A=a\}\in\mathscr G_a$ for every finite $a\subset\mathbb U_*$, and the conclusions of \Cref{lem:palm-fresh-edges} hold for $A$ under $\P$ and $\P^\infty$. In particular, the selected squared moduli $(Y_v)_{v\in A}$ are fresh given $\mathscr H_A$: conditionally on $\mathscr H_A$, they are independent $\Exp(1)$ variables. The same holds under the spine laws if $X=X(\omega,\mathsf v)$, respectively $X(\omega,\mathsf u,\mathsf v)$, is $\mathscr G^{\mathrm{leaf}}$-measurable for each fixed value of the endpoints; for example, $X=\{\mathsf v\}$ or $X=\{\mathsf u,\mathsf v\}$. \item Under $\mathbb Q_L$, respectively $\mathbb Q_L^\infty$, the endpoint configuration $\theta_{\mathsf v}\omega$ has law $\P$, respectively $\P^\infty$, and is independent of the spine edge marks, the auxiliary marks of the spine vertices, and the side sequences. In particular, \textup{(i)}--\textup{(iii)} hold for the children of the endpoint $\mathsf v$, with $\theta_{\mathsf v}\omega$ in place of $\omega$; for example, $\mathbb Q_L^\infty(\mathcal R\circ\theta_{\mathsf v})=p^2e^{-2p}$, and \eqref{eq:palm-rigid} holds conditionally on the spine data. Under $\mathbb Q_{L,j}^\infty$, the events $\mathcal R\circ\theta_{\mathsf u}$ and $\mathcal R\circ\theta_{\mathsf v}$ are independent, each has probability $p^2e^{-2p}$, and \eqref{eq:palm-rigid} holds jointly at the two endpoints, with independent factors. \end{enumerate} \end{lemma} \begin{proof} For (i), the sets $\mathsf M^{\mathrm{leaf}}=\C\times\{N_o=0\}$, $\mathsf M^\infty=\C\times\mathcal S$, and $\mathsf M^{\mathrm{oth}}=\C\times(\Omega\setminus(\mathcal S\cup\{N_o=0\}))$ partition $\mathsf M$, and their $\mathfrak m$-masses are $e^{-p}$, $\varrho_p$, and $1-\varrho_p-e^{-p}$. For measurable $u_1,u_2,u_3\colon\mathsf M\to[0,\infty)$, the computation in the proof of \Cref{lem:palm-survival}(i) gives \[ \E\exp\bigl(-\Pi^{\mathrm{leaf}}(u_1)-\Pi^{\mathrm{surv}}(u_2)-\Pi^{\mathrm{oth}}(u_3)\bigr)=\prod_{i=1}^3\exp\Bigl(-p\int_{\mathsf M_i}(1-e^{-u_i})\,\dd\mathfrak m\Bigr), \] where $(\mathsf M_1,\mathsf M_2,\mathsf M_3)=(\mathsf M^{\mathrm{leaf}},\mathsf M^\infty,\mathsf M^{\mathrm{oth}})$. As there, this shows that the three processes are independent Poisson processes with intensity measures $p\,\mathfrak m(\,\cdot\cap\mathsf M_i)$, which are the stated ones; they are independent of $\zeta_o$ by \Cref{lem:palm-branching}. The statement under $\P^\infty$ follows as in \Cref{lem:palm-survival}(ii), since $\mathcal S=\{N^{\mathrm{surv}}_o\ge1\}$. For \eqref{eq:palm-decoration-probability}, we note that $\{\Pi^{\mathrm{surv}}(B)\ge1\}\subset\mathcal S$, that $\Pi^{\mathrm{surv}}(B)=\Pi_o(B)$ has law $\Pois(p\,\mathfrak m(B))$, and that it is independent of $N^{\mathrm{leaf}}_o$, which has law $\Pois(pe^{-p})$; dividing by $\varrho_p$ gives \eqref{eq:palm-decoration-probability}. For (ii), write $M_k=(w_k,\theta_k\omega)$. On $\mathcal R$, we have $N_o=2$, and exactly one of $M_1,M_2$ lies in $\mathsf M^\infty$ while the other lies in $\mathsf M^{\mathrm{leaf}}$. By \Cref{lem:palm-branching}(ii), $N_o$, $\zeta_o$, $M_1$, $M_2$ are independent, and $M_1,M_2$ have law $\mathfrak m$. Using also $\P(N_o=2)=p^2e^{-p}/2$, we obtain \begin{align*} \E\bigl[\one_{\mathcal R}F_1(M_y)F_2(M_\ell)F_3(\zeta_o)\bigr]&=\P(N_o=2)\cdot2\int_{\mathsf M^\infty}F_1\,\dd\mathfrak m\int_{\mathsf M^{\mathrm{leaf}}}F_2\,\dd\mathfrak m\int F_3\,\dd\kappa\\ &=p^2e^{-p}\cdot\varrho_p\int F_1\,\dd\mathfrak m^\infty\cdot e^{-p}\int F_2\,\dd\mathfrak m^{\mathrm{leaf}}\int F_3\,\dd\kappa, \end{align*} where the factor $2$ accounts for the two possible values of $(y,\ell)$. The surviving child has an infinite forward subtree, which gives $\mathcal R\subset\mathcal S$. Dividing by $\varrho_p$ gives \eqref{eq:palm-rigid}, and taking $F_1=F_2=F_3=1$ gives the two probabilities. For the last assertion, we take $F_1(w,\omega'')=g_1(|w|^2)g_2(w/|w|)g_3(\omega'')$ and $F_2(w,\omega'')=g_4(|w|^2)g_5(w/|w|)$, and use that under $\gamma$ the modulus squared and the phase are independent with laws $\Exp(1)$ and uniform. For (iii), $\P(N_o=1)=pe^{-p}$, and $\{N_o=1\}\cap\mathcal S=\{N_o=1,\ \theta_1\omega\in\mathcal S\}$ has probability $pe^{-p}\varrho_p$ by \Cref{lem:palm-branching}(ii); dividing by $\varrho_p$ gives $\P^\infty(N_o=1)=pe^{-p}$. The conditional laws follow in the same way. Along a spine, we have $N_{\mathsf v_l}=N_l'+1$ and therefore $\mathcal C_J=\{|S_l|=0\text{ for }l\in J\}$, which has $\mathbb Q_L$-probability $e^{-p|J|}$ by \Cref{lem:palm-spine-structure}(i). Since $\mathcal C_J$ is an event in terms of the sequences $S_l$, conditioning on it preserves the independence in \Cref{lem:palm-spine-structure} and only changes the laws of $S_l$, $l\in J$, to the point mass at the empty sequence. Under $\mathbb Q_L^\infty$ the event $\mathcal C_J$ has the same probability, because it is independent of $\theta_{\mathsf v}\omega$. The displayed identity is \eqref{eq:palm-one-spine} applied to $F(\omega,v)\prod_{l\in J}\one\{N_{v_l}=1\}$, and its conditioned version is \eqref{eq:palm-backbone}. For (iv), fix a finite set $a\subset\mathbb U_*$, and let $\tilde\omega=\omega^{a\leftarrow0}$ be the configuration in which the marks $w_v$, $v\in a$, are replaced by $0$. Every element of $A$ is a leaf, and therefore $\{A=a\}\subset\{N_v=0\text{ for all }v\in a\}$. On the latter event, the generators $w_v\one\{N_v\ge1\}$ of $\mathscr G^{\mathrm{leaf}}$ take the same values at $\omega$ and at $\tilde\omega$, and so do all offspring numbers and auxiliary marks. Since $X$ is $\mathscr G^{\mathrm{leaf}}$-measurable and the selection $x\mapsto\ell(x)$ depends only on the offspring numbers and the ranks, we have $\one\{A=a\}(\omega)=\one\{A=a\}(\tilde\omega)$ on this event. Off this event, both sides vanish, because $\omega$ and $\tilde\omega$ have the same offspring numbers. Since the right side does not depend on $(w_v)_{v\in a}$, we conclude that $\{A=a\}\in\mathscr G_a$. The remaining assertions are \Cref{lem:palm-fresh-edges}; the independence of the selected leaf marks at distinct vertices of $X$ is part of \eqref{eq:palm-fresh-edges}, which identifies their joint conditional law as $\gamma^{\otimes A}$. Under the spine laws, we argue in the same way for each fixed value of the endpoints and apply \Cref{lem:palm-fresh-edges}(ii). Part (v) follows from \Cref{lem:palm-spine-structure}(i), (iii), and (iv). \end{proof} We emphasize that the structural events in \Cref{lem:palm-decorations} are defined in terms of offspring numbers and survival, and that the selections use only offspring numbers and ranks; neither inspects an edge mark. This is why the selected squared moduli remain fresh; compare \Cref{rem:palm-not-fresh}, where the conditioning event is defined in terms of the squared modulus itself. We conclude with the growth of the backbone. By \Cref{lem:palm-survival}(ii), the probability under $\P^\infty$ that the root has exactly one surviving child is \[ q=\frac{p\varrho_pe^{-p\varrho_p}}{1-e^{-p\varrho_p}}=pe^{-p\varrho_p}, \] where we used $\varrho_p=1-e^{-p\varrho_p}$. With $\lambda=p\varrho_p>0$, we have $q=\lambda/(e^\lambda-1)<1$, since $e^\lambda-1>\lambda$. \begin{lemma}\label{lem:palm-backbone} Let $p>1$ and $q=pe^{-p\varrho_p}$. \begin{enumerate}[label=\textup{(\roman*)}] \item On $\mathcal S$, we have $Z_0=1$ and $Z_{n+1}\ge Z_n$ for every $n\ge0$. \item For all $n\ge0$, $m\ge1$, and $K\ge0$, $\P^\infty(Z_n=m,\ Z_{n+K}=m)=q^{mK}\,\P^\infty(Z_n=m)$. \item $Z_n\to\infty$ as $n\to\infty$, $\P^\infty$-almost surely. \end{enumerate} \end{lemma} \begin{proof} The proof proceeds in four steps. After the elementary monotonicity, we compute the probability that the backbone of a surviving tree consists of a single line for $K$ generations, using the Mecke formula. Then we use the branching property at generation $n$ to obtain (ii), from which (iii) follows. \proofstep{1}{Monotonicity.} On $\mathcal S$, the root has an infinite forward subtree, and $Z_0=1$. If $v\in\mathcal V_n^\infty$, then the forward subtree of $v$ is infinite and $v$ has finitely many children, so at least one child of $v$ has an infinite forward subtree and belongs to $\mathcal V_{n+1}^\infty$. Since distinct vertices have distinct children, we have $Z_{n+1}\ge Z_n$. This proves (i). Further, for all $n,K\ge0$ and $\omega\in\Omega$, we have \begin{equation}\label{eq:palm-backbone-decomposition} Z_{n+K}(\omega)=\sum_{v\in\mathcal V_n^\infty}Z_K(\theta_v\omega). \end{equation} Indeed, for $v\in\mathcal V_n$, the vertices of $\mathcal V_{n+K}^\infty$ that descend from $v$ are exactly the vertices $vu$ with $u\in\mathcal V_K^\infty(\theta_v\omega)$, and there are none unless the forward subtree of $v$ is infinite, that is, unless $v\in\mathcal V_n^\infty$. \proofstep{2}{A single line.} For $K\ge0$, let $\mathcal A_K=\mathcal S\cap\{Z_K=1\}$. We claim that $\P^\infty(\mathcal A_K)=q^K$. For $K=0$ this holds since $\mathcal A_0=\mathcal S$. Let $K\ge1$. By Step~1, we have $Z_1=1$ on $\mathcal A_K$, that is, the root has exactly one child $k$ with $\theta_k\omega\in\mathcal S$, and \eqref{eq:palm-backbone-decomposition} with $n=1$ gives $Z_{K-1}(\theta_k\omega)=1$. Conversely, if the root has exactly one surviving child $k$ and $\theta_k\omega\in\mathcal A_{K-1}$, then $\omega\in\mathcal A_K$. In terms of the child process, this reads \[ \one_{\mathcal A_K}(\omega)=\sum_{k=1}^{N_o}\one_{\mathcal A_{K-1}}(\theta_k\omega)\prod_{k'\le N_o,\ k'\neq k}\one_{\Omega\setminus\mathcal S}(\theta_{k'}\omega)=\int F(m,\Pi_o-\delta_m)\,\Pi_o(\dd m), \] where $F((w,\omega''),\pi)=\one_{\mathcal A_{K-1}}(\omega'')\one\{\pi(\mathsf M^\infty)=0\}$ and $\mathsf M^\infty=\C\times\mathcal S$. In the first equality we used $\mathcal A_{K-1}\subset\mathcal S$. By \Cref{lem:palm-branching}(iii) and \eqref{eq:palm-mecke}, we obtain \[ \P(\mathcal A_K)=p\,\P(\mathcal A_{K-1})\,\P\bigl(\Pi_o(\mathsf M^\infty)=0\bigr)=p\,e^{-p\varrho_p}\,\P(\mathcal A_{K-1}). \] Dividing by $\varrho_p$, we find $\P^\infty(\mathcal A_K)=q\,\P^\infty(\mathcal A_{K-1})$, and the claim follows by induction. \proofstep{3}{Proof of \textup{(ii)}.} Let $n\ge0$, $m\ge1$, and $K\ge0$. On $\{Z_n=m\}$, each term in \eqref{eq:palm-backbone-decomposition} is at least $1$ by Step~1, and $Z_{n+K}=m$ holds if and only if $\theta_v\omega\in\mathcal A_K$ for every $v\in\mathcal V_n^\infty$. We obtain \[ \one\{Z_n=m,\ Z_{n+K}=m\}=\sum_{\substack{a\subset\mathbb U_n\\|a|=m}}\one\{a\subset\mathcal V_n\}\prod_{v\in a}\one_{\mathcal A_K}(\theta_v\omega)\prod_{v\in\mathcal V_n\setminus a}\one_{\Omega\setminus\mathcal S}(\theta_v\omega), \] since only the term with $a=\mathcal V_n^\infty$ can be nonzero. By \Cref{lem:palm-branching}(i), the configurations $\theta_v\omega$, $v\in\mathbb U_n$, are independent with law $\P$ and independent of $\mathscr F_n$, and the finite set $\mathcal V_n$ is $\mathscr F_n$-measurable. Conditioning on $\mathscr F_n$, we find that \[ \P\bigl(Z_n=m,\ Z_{n+K}=m\bigm|\mathscr F_n\bigr)=\sum_{\substack{a\subset\mathbb U_n\\|a|=m}}\one\{a\subset\mathcal V_n\}\,\P(\mathcal A_K)^m(1-\varrho_p)^{|\mathcal V_n|-m}. \] The same computation with $\mathcal S$ in place of $\mathcal A_K$ gives the conditional probability of $\{Z_n=m\}$, with $\varrho_p^m$ in place of $\P(\mathcal A_K)^m$. By Step~2, we have $\P(\mathcal A_K)=\varrho_pq^K$, and the first conditional probability is therefore $q^{mK}$ times the second. Taking expectations, and using that both events are contained in $\mathcal S$ because $m\ge1$, we deduce (ii) after division by $\varrho_p$. \proofstep{4}{Proof of \textup{(iii)}.} By (i), the limit $Z_\infty=\lim_nZ_n\in\{1,2,\dots\}\cup\{\infty\}$ exists on $\mathcal S$. For $m\ge1$, we have \[ \{Z_\infty=m\}\subset\bigcup_{n\ge0}\bigcap_{K\ge0}\{Z_n=m,\ Z_{n+K}=m\}. \] By (ii), each event in the union has $\P^\infty$-probability at most $\inf_Kq^{mK}=0$, since $q<1$. We conclude that $\P^\infty(Z_\infty<\infty)=0$, which is (iii). \end{proof} \chapter{Real solutions of the cavity equation}\label{sec:rde} At an energy $E>0$ where the imaginary parts of the cavity variables vanish at the real axis, the cavity equation of \Cref{prop:tree-schur} becomes an equation for a probability measure on the real line. This section studies that equation and its relation with the tree. In \Cref{sec:rde-def}, we define real solutions and the normalizations used in the rest of the paper. In \Cref{sec:rde-properties}, we prove properties shared by all real solutions: the structure of their atoms, a density bound near zero, fractional moments of both signs, full support, a fractional cancellation estimate, and a bound on the probability that the denominator is small on the scale $p^{-1/2}$. These properties hold for every $p>1$, every $E>0$ and every real solution. The atom structure, the moments, the full support and the fractional cancellation estimate are the hypotheses under which the free energy is defined and identified in \Cref{sec:press}. The rest of the section connects real solutions with the tree. In \Cref{sec:rde-regular}, we define a deterministic set $\mathcal E_p$ of regular energies, of full Lebesgue measure, at which all boundary values used in this paper exist, and we prove a zero--one law for the imaginary part of the root cavity variable. If this imaginary part vanishes under $\P^\infty$, then the denominators at the real axis are finite, real and nonzero, and the law of the root denominator is a real solution, which we call the real boundary law. In \Cref{sec:rde-rtp}, we construct, for an arbitrary real solution, a recursive tree process with that solution as its marginal. \Cref{sec:rde-moments} records fractional moment bounds at $z=E+\iu\eta$ with $\eta>0$ that are uniform in $\eta$. Finally, \Cref{sec:rde-fixed} proves that when the boundary is real, every cavity variable in a fixed neighborhood of the root, of a spine, or of two spines converges as the imaginary part $\eta$ tends to zero, and identifies the limit with the real recursive tree process. Throughout this section, $p>1$ is fixed. Dependence of constants on parameters is indicated explicitly. \section{Real solutions and their normalizations}\label{sec:rde-def} By \Cref{prop:tree-schur}, the root cavity variable $\Gamma_o(z)$ satisfies the cavity equation \[ \Gamma_o(z)\eqdist-\Big(z+\frac1p\sum_{j=1}^NY_j\Gamma_j(z)\Big)^{-1}, \] where $N\sim\Pois(p)$, the $Y_j$ are $\Exp(1)$ variables, the $\Gamma_j(z)$ are copies of $\Gamma_o(z)$, and all these variables are independent. If the variables are real at $z=E$, then the inverse cavity variable $D=-1/\Gamma$, which we call the denominator, satisfies the equation in the following definition. \begin{definition}\label{def:rde-real} Let $E>0$. A \emph{real solution at energy $E$} is a Borel probability measure $\pi$ on $\R\setminus\{0\}$ with the following property. Let $N$, $(Y_j)_{j\ge1}$ and $(D_j)_{j\ge1}$ be independent, with $N\sim\Pois(p)$, $Y_j\sim\Exp(1)$ and $D_j\sim\pi$ for every $j$. Then the random variable \begin{equation}\label{eq:rde-real} E-\frac1p\sum_{j=1}^N\frac{Y_j}{D_j} \end{equation} has law $\pi$. We abbreviate this property by \[ D\eqdist E-\frac1p\sum_{j=1}^N\frac{Y_j}{D_j}, \] where $D\sim\pi$ and the variables on the right are as above, and we call $D$ the \emph{denominator}. \end{definition} Since $D_j\neq0$ almost surely and $N$ is finite, the sum in \eqref{eq:rde-real} is a real number almost surely, and the definition makes sense. Since $\pi$ is carried by $\R\setminus\{0\}$, the definition includes the requirement that the variable \eqref{eq:rde-real} is nonzero almost surely. We only consider positive energies. By \Cref{prop:tree-symmetry}, spectral statements at negative energies follow from those at positive energies, and if $\pi$ is a real solution at energy $E$, then the law of $-D$ satisfies the same equation with $E$ replaced by $-E$. The following normalizations are used throughout the paper. \begin{definition}\label{def:rde-normalizations} Let $E>0$, let $\pi$ be a real solution at energy $E$, and let $D\sim\pi$. Let $Y\sim\Exp(1)$ be independent of $D$. \begin{enumerate}[label=\textup{(\roman*)}] \item For a \emph{scaling} $g>0$, set $B=gD$ and $\beta=g^2Y/p$. The \emph{transfer matrix} is \[ M(B,\beta)=\beta^{-1/2}\begin{pmatrix}0&1\\-\beta&B\end{pmatrix}. \] \item For $E>2$, set \[ g(E)=\frac{E-\sqrt{E^2-4}}2,\qquad\theta=g(E)^2,\qquad K=\frac p\theta, \] and, with the scaling $g=g(E)$, set $B=g(E)D$ and $X=1/B$. We call this the \emph{$B$-normalization}. \item The \emph{scaled energy} is $\xi=E/\sqrt p$. For $\xi>2/\sqrt p$, set \[ a_p(\xi)=\frac{1}{\sqrt p\,g(\xi\sqrt p)}. \] \item The scaling $g=1/E$ gives $\widetilde B=D/E$, in which the atom of the law sits at $1$ (see \Cref{prop:rde-properties}(i)). We use the parameter $\tau=1/(pE^2)$. \end{enumerate} \end{definition} We record the identities behind these normalizations. \begin{lemma}\label{lem:rde-normalizations} Let $E>0$, let $\pi$ be a real solution at energy $E$, and let $N$, $(Y_j)$, $(D_j)$ be as in \Cref{def:rde-real}. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $g>0$, the matrix $M(B,\beta)$ has determinant one, and with $B_j=gD_j$ and $\beta_j=g^2Y_j/p$ we have \[ B\eqdist gE-\sum_{j=1}^N\frac{\beta_j}{B_j}. \] \item Let $E>2$. Then $Eg(E)=1+\theta$, $\beta=Y/K$, $X=-\Gamma/g(E)$ when $D=-1/\Gamma$, and \[ B\eqdist1+\theta-\frac\theta p\sum_{j=1}^N\frac{Y_j}{B_j}. \] Further, $E^{-1}2$. Then \[ a_p(\xi)=\frac{\xi+\sqrt{\xi^2-4/p}}2,\qquad K=p^2a_p(\xi)^2,\qquad \frac\xi2\le a_p(\xi)<\xi. \] \item With $\widetilde B_j=D_j/E$, we have \[ \widetilde B\eqdist1-\tau\sum_{j=1}^N\frac{Y_j}{\widetilde B_j}. \] If $E>2$, then $\widetilde B=B/(1+\theta)$ and $\tau=\theta/(p(1+\theta)^2)$. \end{enumerate} \end{lemma} \begin{proof} The determinant of $M(B,\beta)$ is $\beta^{-1}(0\cdot B+\beta)=1$. Multiplying \eqref{eq:rde-real} by $g$, we obtain \[ gD\eqdist gE-\frac gp\sum_{j=1}^N\frac{Y_j}{D_j}=gE-\sum_{j=1}^N\frac{g^2Y_j/p}{gD_j}, \] which is (i). For (ii), the numbers $(E\pm\sqrt{E^2-4})/2$ are the roots of $t^2-Et+1=0$. In particular, $g(E)^2-Eg(E)+1=0$, that is, $Eg(E)=1+\theta$, and since the product of the two roots is one, we have \begin{equation}\label{eq:rde-ginverse} \frac1{g(E)}=\frac{E+\sqrt{E^2-4}}2. \end{equation} Inserting $gE=1+\theta$ and $g^2/p=\theta/p$ into (i), we obtain the equation for $B$. The identities $\beta=g(E)^2Y/p=Y/K$ and $X=1/(g(E)D)=-\Gamma/g(E)$ follow from the definitions. Since $E\le E+\sqrt{E^2-4}<2E$, \eqref{eq:rde-ginverse} implies that $E^{-1}2$. For (iii), using \eqref{eq:rde-ginverse}, we find \[ a_p(\xi)=\frac{E+\sqrt{E^2-4}}{2\sqrt p}=\frac{\xi\sqrt p+\sqrt{\xi^2p-4}}{2\sqrt p} =\frac{\xi+\sqrt{\xi^2-4/p}}2, \] and $K=p/g(E)^2=p\cdot pa_p(\xi)^2$. The bounds on $a_p(\xi)$ follow from $0\le\sqrt{\xi^2-4/p}<\xi$. For (iv), we divide \eqref{eq:rde-real} by $E$ and use $Y_j/(pED_j)=\tau Y_j/\widetilde B_j$. If $E>2$, then $B/(1+\theta)=g(E)D/(g(E)E)=\widetilde B$, and $\theta/(p(1+\theta)^2)=g(E)^2/(pg(E)^2E^2)=\tau$. \end{proof} The following table summarizes the normalizations. The last column is the atom produced by the event $\{N=0\}$, on which the denominator equals the constant term of its equation; by \Cref{prop:rde-properties}, this is the only atom of a real solution. \begin{center} \renewcommand{\arraystretch}{1.35} \begin{tabular}{lll} \hline variable & equation in law & atom\\ \hline $D=-1/\Gamma$ & $D\eqdist E-p^{-1}\sum_{j\le N}Y_j/D_j$ & $e^{-p}\delta_E$\\ $B=gD$, scaling $g>0$ & $B\eqdist gE-\sum_{j\le N}\beta_j/B_j$, \ $\beta_j=g^2Y_j/p$ & $e^{-p}\delta_{gE}$\\ $B=g(E)D$, $X=1/B$, $E>2$ & $B\eqdist1+\theta-(\theta/p)\sum_{j\le N}Y_j/B_j$ & $e^{-p}\delta_{1+\theta}$\\ $\widetilde B=D/E$ & $\widetilde B\eqdist1-\tau\sum_{j\le N}Y_j/\widetilde B_j$ & $e^{-p}\delta_1$\\ \hline \end{tabular} \end{center} \section{A priori properties of real solutions}\label{sec:rde-properties} We use the density bound of \Cref{lem:palm-affine}(iii) for affine combinations of independent exponential variables, which is proved there by the Gamma--Dirichlet decomposition, and the following bound on translated Gamma integrals. \begin{lemma}\label{lem:rde-translated} Let $q\in(0,1)$, and let $T_q$ be a random variable with density \[ \omega_q(t)=\frac{t^{q/2}e^{-t}}{\Gamma(1+q/2)},\qquad t>0. \] Then for every $u\in\R$, \[ \E|T_q-u|^{-q}\le\frac{70}{1-q}(1+|u|)^{-q}. \] \end{lemma} \begin{proof} Let $I(u)=\E|T_q-u|^{-q}$. Since $t^{q/2}e^{-t/2}\le(q/e)^{q/2}\le1$ for $t>0$ and $\Gamma(1+q/2)\ge1/2$, we have \begin{equation}\label{eq:rde-omega-bound} \omega_q(t)\le2e^{-t/2}\le2,\qquad t>0. \end{equation} We consider three cases. Let $|u|\le2$. Splitting the integral according to whether $|t-u|\le1$ and using \eqref{eq:rde-omega-bound} on the first part, we have \[ I(u)\le2\int_{|t-u|\le1}|t-u|^{-q}\dd t+\int_0^\infty\omega_q(t)\dd t=\frac4{1-q}+1\le\frac5{1-q}. \] Since $(1+|u|)^{-q}\ge3^{-q}\ge1/3$, this gives $I(u)\le15(1-q)^{-1}(1+|u|)^{-q}$. Let $u<-2$. Since $|t-u|=t+|u|\ge|u|$ for $t>0$, we have $I(u)\le|u|^{-q}\le2(1+|u|)^{-q}$, where we used $1+|u|\le2|u|$. Let $u>2$. We split $(0,\infty)$ into $(0,u/2]$, the set of $t>u/2$ with $|t-u|\le1$, and the set of $t>u/2$ with $|t-u|>1$. On the first set, we have $|t-u|\ge u/2$, and its contribution is at most $(u/2)^{-q}\le2u^{-q}$. On the second set, we have $\omega_q(t)\le2e^{-u/4}$ by \eqref{eq:rde-omega-bound}, and its contribution is at most $4e^{-u/4}/(1-q)$. The contribution of the third set is at most $\int_{u/2}^\infty2e^{-t/2}\dd t=4e^{-u/4}$. Since the maximum of $u^qe^{-u/4}$ over $u>0$ is $(4q/e)^q\le4$, we have $e^{-u/4}\le4u^{-q}$. Combining these bounds, we obtain \[ I(u)\le\Big(2+\frac{16}{1-q}+16\Big)u^{-q}\le\frac{34}{1-q}u^{-q}\le\frac{68}{1-q}(1+u)^{-q}, \] where in the last step we used $1+u\le2u$. The three cases prove the lemma. \end{proof} \begin{proposition}\label{prop:rde-properties} Let $E>0$, let $\pi$ be a real solution at energy $E$, and let $D\sim\pi$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\pi=e^{-p}\delta_E+\pi_{\mathrm{ac}}$, where $\pi_{\mathrm{ac}}$ is absolutely continuous with respect to Lebesgue measure. In particular, $\pi$ has an atom of mass $e^{-p}$ at $E$ and no other atom. \item The density $f$ of $\pi_{\mathrm{ac}}$ can be chosen so that \[ f(x)\le\sqrt{\frac p{2\pi}}\,\frac1{|x-E|}\qquad\text{for all }x\neq E. \] In particular, $f(x)\le\sqrt p/E$ for $|x|\le E/2$, and $\P(|D|\le t)\le2\sqrt p\,t/E$ for $0\le t\le E/2$. \item For every $q\in(0,1)$, \[ \E|D|^{-q}\le2\Big(1+\frac{\sqrt p}{1-q}\Big)E^{-q},\qquad \E|D|^{q}\le E^q+p^{1-q}\Gamma(1+q)\,\E|D|^{-q}. \] \item The support of $\pi$ is $\R$. \item Let $q\in(0,1)$ and $g>0$, set $B=gD$, and let $T_q$ be as in \Cref{lem:rde-translated} and independent of $D$. Then for every $a\ge0$ and $\sigma\in\{-1,1\}$, \[ \E|B-\sigma aT_q|^{-q}\le\frac{70}{1-q}\,\E(a+|B|)^{-q}\le C\min\{1,a^{-q}\}, \] where $C=70(1-q)^{-1}\max\{1,g^{-q}\E|D|^{-q}\}$ depends only on $p$, $E$, $g$ and $q$, and $\min\{1,a^{-q}\}=1$ for $a=0$. \end{enumerate} \end{proposition} By (i), (iii) and (iv), for every $g>0$, the law of $B=gD$ has no atom at zero, has full support, and satisfies $\E|B|^q<\infty$ for every $q\in(0,1)$. These are the hypotheses under which the free energy is defined in \Cref{sec:press}. \begin{proof} Throughout the proof, $N$, $(Y_j)$ and $(D_j)$ are as in \Cref{def:rde-real}, and $D$ has the law of \begin{equation}\label{eq:rde-D-rep} E-R,\qquad R=\frac1p\sum_{j=1}^N\frac{Y_j}{D_j}. \end{equation} \proofstep{1}{Atoms and density.} On the event $\{N=0\}$, which has probability $e^{-p}$, the variable \eqref{eq:rde-D-rep} equals $E$. Fix $n\ge1$ and condition on $N=n$ and on $D_1,\dots,D_n$. Then $E-R=E+\sum_{j\le n}a_jY_j$ with real nonzero coefficients $a_j=-1/(pD_j)$, and $Y_1,\dots,Y_n$ are independent $\Exp(1)$ variables independent of the conditioning. By \Cref{lem:palm-affine}(iii), the conditional law of $E-R$ has a density $\rho_n(x)=\rho_n(x;D_1,\dots,D_n)$ with \begin{equation}\label{eq:rde-cond-density} \rho_n(x)\le\sqrt{\frac n{2\pi}}\,\frac1{|x-E|},\qquad x\neq E, \end{equation} and the formula for $\rho_n$ in the proof of that lemma is jointly measurable in $(x,D_1,\dots,D_n)$. Using Tonelli's theorem, we find that for every Borel set $A\subset\R$, \[ \begin{split} \pi(A)&=e^{-p}\one_A(E)+\sum_{n\ge1}\P(N=n)\,\E\int_A\rho_n(x;D_1,\dots,D_n)\dd x\\ &=e^{-p}\one_A(E)+\int_Af(x)\dd x, \end{split} \] with \[ f(x)=\sum_{n\ge1}\P(N=n)\,\E\rho_n(x;D_1,\dots,D_n). \] This proves (i). For $x\neq E$, \eqref{eq:rde-cond-density} implies that \[ f(x)\le\frac{\E\sqrt N}{\sqrt{2\pi}\,|x-E|}\le\sqrt{\frac p{2\pi}}\,\frac1{|x-E|}, \] where we used Jensen's inequality $\E\sqrt N\le\sqrt{\E N}=\sqrt p$. If $|x|\le E/2$, then $|x-E|\ge E/2$ and $f(x)\le2\sqrt{p/(2\pi)}/E\le\sqrt p/E$. Since the interval $[-t,t]$ does not contain the atom $E$ for $0\le t\le E/2$, we have $\P(|D|\le t)=\int_{-t}^tf\le2\sqrt p\,t/E$. This proves (ii). \proofstep{2}{Moments.} By (ii), \[ \E|D|^{-q}\le\Big(\frac E2\Big)^{-q}+\int_{-E/2}^{E/2}|x|^{-q}f(x)\dd x \le\Big(\frac2E\Big)^q+\frac{\sqrt p}E\cdot\frac{2(E/2)^{1-q}}{1-q} \le2E^{-q}+\frac{2\sqrt p}{1-q}E^{-q}. \] In the first inequality, we bounded $|D|^{-q}$ by $(E/2)^{-q}$ on the event $\{|D|>E/2\}$. For the positive moment, we use \eqref{eq:rde-D-rep} and the inequality $|a+b|^q\le|a|^q+|b|^q$, valid for $q\in(0,1]$, to obtain \[ |D|^q\le E^q+p^{-q}\sum_{j=1}^NY_j^q|D_j|^{-q}. \] Taking expectations and using the independence of $N$, $(Y_j)$ and $(D_j)$, we find \[ \E|D|^q\le E^q+p^{-q}\,\E N\,\E Y_1^q\,\E|D|^{-q}=E^q+p^{1-q}\Gamma(1+q)\,\E|D|^{-q}, \] which proves (iii). \proofstep{3}{Full support.} It suffices to show that every open interval $I$ has $\pi(I)>0$. By (i), the event $\{N=1,\ D_1=E\}$ has probability $pe^{-p}\pi(\{E\})=pe^{-2p}>0$. On this event, the variable \eqref{eq:rde-D-rep} equals $E-Y_1/(pE)$, which has the density $pEe^{-pE(E-x)}$ on $(-\infty,E)$. This density is strictly positive on $(-\infty,E)$, and we conclude that $\pi(I)>0$ for every open interval $I$ that meets $(-\infty,E)$. Since $E>0$, this applies to $I=(-\infty,0)$, and the event $\{N=1,\ D_1<0\}$ has probability $pe^{-p}\pi((-\infty,0))>0$. On this event, the variable \eqref{eq:rde-D-rep} equals $E+Y_1/(p|D_1|)$, and conditionally on $D_1$ it has the strictly positive density $p|D_1|e^{-p|D_1|(x-E)}$ on $(E,\infty)$. As before, we conclude that $\pi(I)>0$ for every open interval $I$ that meets $(E,\infty)$. Every open interval meets $(-\infty,E)$ or $(E,\infty)$, which proves (iv). \proofstep{4}{Fractional cancellation.} Fix $b\in\R\setminus\{0\}$. For $a>0$, we have $|b-\sigma aT_q|=a|T_q-\sigma b/a|$ since $|\sigma|=1$, and \Cref{lem:rde-translated} implies that \[ \E|b-\sigma aT_q|^{-q}=a^{-q}\,\E\Big|T_q-\frac{\sigma b}a\Big|^{-q} \le\frac{70}{1-q}\,a^{-q}\Big(1+\frac{|b|}a\Big)^{-q}=\frac{70}{1-q}(a+|b|)^{-q}. \] For $a=0$, the left side equals $|b|^{-q}=(a+|b|)^{-q}$. Since $T_q$ is independent of $B$ and $B\neq0$ almost surely, we may condition on $B$ and obtain the first inequality in (v). For the second, we have $\E(a+|B|)^{-q}\le\min\{a^{-q},\E|B|^{-q}\}$, and $\E|B|^{-q}=g^{-q}\E|D|^{-q}$ is finite by (iii). Finally, $\min\{a^{-q},M\}\le\max\{1,M\}\min\{1,a^{-q}\}$ for $M\ge0$, as one sees by treating $a\ge1$ and $a<1$ separately. This proves (v). \end{proof} The next lemma bounds the probability that the denominator is small on the scale $p^{-1/2}$, uniformly in the energy and in the real solution. \begin{lemma}\label{lem:rde-guard} Let $E>0$, let $\pi$ be a real solution at energy $E$, let $D\sim\pi$, and for $a>0$ set $q_a=\P(|D|\le a/\sqrt p)$. Then \[ q_a\le e^{-pq_a}+2a^2. \] In particular, let $\eps\in(0,1)$, and set $a(\eps)=\sqrt\eps/2$ and $p(\eps)=\eps^{-1}\log(2/\eps)$. Then $q_a<\eps$ for every $p>\max\{1,p(\eps)\}$, every $E>0$, every real solution at energy $E$, and every $a\in(0,a(\eps)]$. \end{lemma} \begin{proof} Let $N$, $(Y_j)$ and $(D_j)$ be as in \Cref{def:rde-real}, and recall that $D$ has the law of \eqref{eq:rde-D-rep}. We call an index $j\le N$ \emph{small} if $|D_j|\le a/\sqrt p$, and we let $K$ be the number of small indices. Since the $D_j$ are independent of $N$, the variable $K$ has, conditionally on $N$, the binomial law with parameters $N$ and $q_a$. Its generating function is $\E s^K=\E(1-q_a+q_as)^N=e^{-pq_a(1-s)}$, $s\in[0,1]$, which is that of $\Pois(pq_a)$; in particular $\P(K=0)=e^{-pq_a}$. On the event $\{K\ge1\}$, let $j_*$ be the smallest small index. Then $\{K\ge1\}$ and $j_*$ are measurable with respect to $\sigma(N,(D_j)_{j\ge1})$. Let $\mathcal G$ be the $\sigma$-algebra generated by $N$, $(D_j)_{j\ge1}$ and the variables $Y_j\one_{\{j\neq j_*\}}$, $j\ge1$. For each $k\ge1$, the variable $Y_k$ is independent of $N$, $(D_j)_{j\ge1}$ and $(Y_j)_{j\neq k}$, and the event $\{j_*=k\}$ is determined by $N$ and $(D_j)$. Then on $\{K\ge1\}$, conditionally on $\mathcal G$, the variable $Y_{j_*}$ has the $\Exp(1)$ law. Further, \[ E-R=c-\frac{Y_{j_*}}{pD_{j_*}},\qquad c=E-\frac1p\sum_{j\le N,\,j\neq j_*}\frac{Y_j}{D_j}, \] where $c$ is $\mathcal G$-measurable. For a real constant $c$ and $\kappa\neq0$, the variable $c+\kappa Y_{j_*}$ has a density bounded by $|\kappa|^{-1}$. Here $|\kappa|^{-1}=p|D_{j_*}|\le a\sqrt p$, and on $\{K\ge1\}$ we have \[ \P\big(|E-R|\le a/\sqrt p\,\big|\,\mathcal G\big)\le\frac{2a}{\sqrt p}\cdot a\sqrt p=2a^2. \] Since $E-R$ has the law of $D$, we conclude that \[ q_a\le\P(K=0)+\E\big[\P(|E-R|\le a/\sqrt p\mid\mathcal G)\one_{\{K\ge1\}}\big]\le e^{-pq_a}+2a^2. \] For the second assertion, let $a\le a(\eps)$ and $p>p(\eps)$; then $2a^2\le\eps/2$ and $e^{-p\eps}<\eps/2$. If $q_a\ge\eps$, then the first assertion gives $q_a\le e^{-p\eps}+\eps/2<\eps$, which is impossible. This proves the second assertion. \end{proof} \section{Regular energies and boundary values}\label{sec:rde-regular} In this section, we define the regular energies and study the boundary values of resolvent entries at a regular energy. The main result is \Cref{prop:rde-boundary}. \subsection{Notation}\label{sec:rde-ulam} We use the realization of the marked tree on the configuration space $\Omega$ of \Cref{def:palm-config}. Thus $\omega=((N_v)_{v\in\mathbb U},(\zeta_v)_{v\in\mathbb U},(w_v)_{v\in\mathbb U_*})$ has law $\P$, the children of $v\in V=V(\omega)$ are $v1,\dots,vN_v$, the edge $\{v^-,v\}$ carries the mark $w_v$, and $Y_v=|w_v|^2=p|h_{v^-v}|^2$. We write $\theta_u$ for the shifts, $\mathbb U_{\le m}=\bigcup_{n\le m}\mathbb U_n$, $u\mathbb U=\{uv:v\in\mathbb U\}$, and $\mathcal V_n=V\cap\mathbb U_n$ for the set of vertices of generation $n$. For $u\in\mathbb U$, we set $\mathcal S_u=\{\theta_u\omega\in\mathcal S\}$, and for $u\in V$ we set $\varsigma_u=\one_{\mathcal S_u}$, so that $\varsigma_u=1$ if and only if the forward subtree $\mathcal T_u$ is infinite. We recall from \Cref{lem:palm-branching}(i) that for every $n\ge0$, the configurations $\theta_u\omega$, $u\in\mathbb U_n$, are independent with law $\P$ and independent of the $\sigma$-field \[ \mathscr F_n=\sigma\bigl(N_u,\zeta_u:\ |u|0$, we have $\Im(-1/D)=\Im D/|D|^2$, and \[ \Im G^C_{xx}(z)=|G^C_{xx}(z)|^2\Big(\eta+\sum_{y\sim_Cx}|h_{xy}|^2\,\Im\Gamma^C_{y\to x}(z)\Big). \] Letting $\eta\downarrow0$ gives \eqref{eq:rde-im-diag}. The assertions about $\Gamma^C_{x\to w}$ follow in the same way from \Cref{prop:tree-schur}(ii). Since $\mathcal K$ is countable, all these statements hold simultaneously on one almost sure event. \end{proof} All imaginary parts in \eqref{eq:rde-im-diag} and \eqref{eq:rde-im-cavity} are nonnegative, since they are limits of imaginary parts of Borel transforms of positive measures. The next lemma shows that a positive imaginary part at the real axis can only be created by an infinite forward subtree. \begin{lemma}\label{lem:rde-propagation} Let $E\in\mathcal E_p$. Almost surely, the following holds for every removed set $\Delta$, every component $C$ of $\mathcal T^\Delta$, every $x\in C$, and every neighbor $w$ of $x$ in $C$. \begin{enumerate}[label=\textup{(\roman*)}] \item If $\Im G^C_{xx}(E+\iu0)>0$ or $\Im\Gamma^C_{x\to w}(E+\iu0)>0$, then there exists $y\in V$ with $\Im\Gamma_y(E+\iu0)>0$. \item On the event that $\Im\Gamma_y(E+\iu0)>0$ for every $y\in V$ such that $\mathcal T_y$ is infinite, we have $\Im G^C_{xx}(E+\iu0)>0$ if and only if $C$ is infinite, and $\Im\Gamma^C_{x\to w}(E+\iu0)>0$ if and only if $C_{x\to w}$ is infinite. \end{enumerate} \end{lemma} \begin{proof} We work on the intersection of the almost sure events of \Cref{def:rde-regular} and \Cref{lem:rde-boundary-schur}, and we use two facts about paths in the tree. First, a \emph{non-backtracking path} $x_0,x_1,x_2,\dots$, that is, a path with $x_{k+1}\sim x_k$ and $x_{k+1}\neq x_{k-1}$, visits distinct vertices. If it is infinite, then there exists $k_0\le|x_0|$ such that $x_{k+1}$ is a child of $x_k$ for all $k\ge k_0$. Indeed, a step to the parent cannot follow a step to a child without returning to the previous vertex; the path takes at most $|x_0|$ steps toward the root, and then it moves away from the root forever. Second, if $|y|>d(\Delta)$, then the forward subtree $\mathcal T_y$ contains no vertex of $\Delta_V$ and no edge of $\Delta_E$, since all its vertices and all lower endpoints of its edges have length at least $|y|$. If moreover $y\in C$ and $y^-\in C$, then by \Cref{lem:tree-branch}(ii) the branch $C_{y\to y^-}$ is the vertex set of $\mathcal T_y$, and $\Gamma^C_{y\to y^-}=\Gamma_y$. \proofpart{Proof of (i).} Suppose that $\Im\Gamma^C_{x\to w}(E+\iu0)>0$, and set $x_0=x$ and $x_{-1}=w$. By \eqref{eq:rde-im-cavity}, there exists a neighbor $x_1\neq x_{-1}$ of $x_0$ in $C$ with $\Im\Gamma^C_{x_1\to x_0}(E+\iu0)>0$. Applying \eqref{eq:rde-im-cavity} again to the oriented edge $x_1\to x_0$, and repeating, we obtain an infinite non-backtracking path $(x_k)_{k\ge0}$ in $C$ with $\Im\Gamma^C_{x_k\to x_{k-1}}(E+\iu0)>0$ for all $k\ge0$. Let $k>k_0$ satisfy $|x_k|>d(\Delta)$. Then $x_k$ is a child of $x_{k-1}$, and by the second fact, $\Gamma^C_{x_k\to x_{k-1}}=\Gamma_{x_k}$. The vertex $y=x_k$ satisfies $\Im\Gamma_y(E+\iu0)>0$. If instead $\Im G^C_{xx}(E+\iu0)>0$, then \eqref{eq:rde-im-diag} provides a neighbor $w'$ of $x$ in $C$ with $\Im\Gamma^C_{w'\to x}(E+\iu0)>0$, and we apply the previous argument to the oriented edge $w'\to x$. \proofpart{Proof of (ii).} If $C_{x\to w}$ is finite, then condition (b) of \Cref{def:rde-regular} and the discussion after it show that $\Gamma^C_{x\to w}(E+\iu0)$ is real. Suppose that $C_{x\to w}$ is infinite. It induces an infinite, connected, locally finite subtree, and König's lemma provides an infinite non-backtracking path $x=x_0,x_1,x_2,\dots$ in $C_{x\to w}$; its first step is not $w$, since $w\notin C_{x\to w}$. Let $k>k_0$ satisfy $|x_k|>\max\{d(\Delta),|x|,|w|\}$. Then $x_k$ is a child of $x_{k-1}$, and by the second fact, $\Gamma^C_{x_k\to x_{k-1}}=\Gamma_{x_k}$. The forward subtree $\mathcal T_{x_k}$ is infinite, since it contains $x_k,x_{k+1},\dots$, and by assumption $\Im\Gamma_{x_k}(E+\iu0)>0$. For $0\le i\le k-1$, we keep only the term $y=x_{i+1}$ in \eqref{eq:rde-im-cavity} for the oriented edge $x_i\to x_{i-1}$, where $x_{-1}=w$, and obtain \[ \Im\Gamma^C_{x_i\to x_{i-1}}(E+\iu0)\ge|\Gamma^C_{x_i\to x_{i-1}}(E+\iu0)|^2\,|h_{x_ix_{i+1}}|^2\, \Im\Gamma^C_{x_{i+1}\to x_i}(E+\iu0). \] The first two factors on the right are positive by \Cref{lem:rde-boundary-schur}. By induction from $i=k-1$ down to $i=0$, we obtain $\Im\Gamma^C_{x\to w}(E+\iu0)>0$. For the diagonal entries, if $C$ is finite, then $G^C_{xx}(E+\iu0)$ is real. If $C$ is infinite, then one of the finitely many branches $C_{y\to x}$, $y\sim_Cx$, is infinite, since by \Cref{lem:tree-branch}(ii) these branches together with $\{x\}$ make up $C$. For this $y$, the previous paragraph gives $\Im\Gamma^C_{y\to x}(E+\iu0)>0$, and \eqref{eq:rde-im-diag} gives $\Im G^C_{xx}(E+\iu0)>0$. \end{proof} \subsection{The zero--one law and the real boundary law}\label{sec:rde-zero-one} Recall from \Cref{def:tree-pgw} that, for $p>1$, the survival probability $\varrho_p$ is the unique solution in $(0,1)$ of $1-\varrho=e^{-p\varrho}$. \begin{proposition}\label{prop:rde-boundary} Let $E\in\mathcal E_p$ with $E>0$, and set $u=\P(\Im\Gamma_o(E+\iu0)>0)$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $u=1-e^{-pu}$ and $u\in\{0,\varrho_p\}$, and \[ \P^\infty(\Im\Gamma_o(E+\iu0)=0)=1-\frac u{\varrho_p}\in\{0,1\}. \] If $u=\varrho_p$, then almost surely (under $\P$, and hence under $\P^\infty$), the following holds for every vertex $v\in V$, every removed set $\Delta$, every component $C$ of $\mathcal T^\Delta$ and every $x\in C$: $\Im\Gamma_v(E+\iu0)>0$ if and only if $\mathcal T_v$ is infinite, and $\Im G^C_{xx}(E+\iu0)>0$ if and only if $C$ is infinite. In particular, $\P^\infty$-almost surely, $\Im G_{vv}(E+\iu0)>0$ for every $v\in V$. \item Suppose that $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Then $u=0$, and almost surely (under $\P$, and hence under $\P^\infty$) the following hold. For every $v\in V$, the boundary value $\Gamma_v(E+\iu0)$ is finite, real and nonzero, and the denominators $D_v=-1/\Gamma_v(E+\iu0)$ satisfy \begin{equation}\label{eq:rde-boundary-recursion} D_v=E-\frac1p\sum_{k=1}^{N_v}\frac{Y_{vk}}{D_{vk}}. \end{equation} For every removed set $\Delta$, every component $C$ of $\mathcal T^\Delta$, every $x\in C$ and every neighbor $w$ of $x$ in $C$, the boundary values $G^C_{xx}(E+\iu0)$ and $\Gamma^C_{x\to w}(E+\iu0)$ are real and nonzero. Finally, the law of $D_o$ under $\P$ is a real solution at energy $E$. \end{enumerate} \end{proposition} \begin{proof} We work on the almost sure events of \Cref{def:rde-regular} and \Cref{lem:rde-boundary-schur,lem:rde-propagation}, and on the event where $\Gamma_v(z)=G_{oo}(z)\circ\theta_v$ for all $v\in V\setminus\{o\}$. Let $\mathfrak A$ be the event that $\lim_{k\to\infty}G_{oo}(E+\iu/k)$ exists and has positive imaginary part. For $v\in V$, the event $A_v=\{\Im\Gamma_v(E+\iu0)>0\}$ then coincides with $\theta_v^{-1}\mathfrak A$ up to a null set. \proofstep{1}{Finite trees.} On the event $\mathcal S^c$, the tree $\mathcal T$ is finite, and condition (b) of \Cref{def:rde-regular} with $\Lambda=V$ shows that $\Gamma_o(E+\iu0)=G_{oo}(E)$ is real. In other words, $A_o\subset\mathcal S$ up to a null set. Since $\theta_v$ preserves $\P$, we also have $\theta_v^{-1}\mathfrak A\subset\mathcal S_v$ up to a null set, for every $v\in\mathbb U$. \proofstep{2}{The fixed point equation.} By \eqref{eq:rde-im-cavity}, applied with $C=V$ and the oriented edge $v\to v^-$, and by \eqref{eq:rde-im-diag} at the root, we have \[ \Im\Gamma_v(E+\iu0)=|\Gamma_v(E+\iu0)|^2\sum_{k=1}^{N_v}|h_{v,vk}|^2\,\Im\Gamma_{vk}(E+\iu0). \] Since $\Gamma_v(E+\iu0)\neq0$ and the edge weights are nonzero, we obtain $A_v=\bigcup_{k\le N_v}A_{vk}$ almost surely. At the root, $A_o=\bigcup_{k\le N_o}\theta_k^{-1}\mathfrak A$ up to a null set. By \Cref{lem:palm-branching}(ii), the configurations $\theta_k\omega$, $k\ge1$, are independent with law $\P$ and independent of $N_o$, and the events $\theta_k^{-1}\mathfrak A$ are independent of each other and of $N_o$, with probability $u$ each. This gives \[ 1-u=\P(A_o^c)=\E(1-u)^{N_o}=e^{-pu}. \] The function $u\mapsto1-e^{-pu}-u$ is strictly concave on $[0,1]$, vanishes at $0$, has derivative $p-1>0$ at $0$, and equals $-e^{-p}<0$ at $1$; it has exactly two zeros in $[0,1]$, namely $0$ and $\varrho_p$. By Step 1, we have $\P^\infty(A_o)=\P(A_o\cap\mathcal S)/\varrho_p=u/\varrho_p$, which is $0$ or $1$. \proofstep{3}{Propagation when $u=\varrho_p$.} Suppose $u=\varrho_p$. By Step 1, $\P(\mathfrak A\,\triangle\,\mathcal S)=\P(\mathcal S)-\P(\mathfrak A)=0$. Since $\theta_v$ preserves $\P$, the event $\theta_v^{-1}\mathfrak A\,\triangle\,\mathcal S_v$ is null for every $v\in\mathbb U$. Taking the union over the countably many $v$, we find that almost surely $A_v=\mathcal S_v$ for every $v\in V$. This is the first assertion about $\Gamma_v$, and on this event \Cref{lem:rde-propagation}(ii) gives the assertion about $G^C_{xx}$. Under $\P^\infty$, the tree is infinite, and taking $\Delta$ empty and $C=V$, we find that $\Im G_{vv}(E+\iu0)>0$ for all $v\in V$. This proves (i). \proofstep{4}{The case $\P^\infty(A_o)=0$.} Suppose $\P^\infty(A_o)=0$. Decomposing according to survival and extinction, and using that $\P(A_o\cap\mathcal S^c)=0$ by Step 1, we have \[ u=\P(A_o)=\varrho_p\,\P^\infty(A_o)+\P(A_o\cap\mathcal S^c)=0. \] Since $\theta_v$ preserves $\P$, we have $\P(\theta_v^{-1}\mathfrak A)=u=0$ for every $v\in\mathbb U$, and almost surely $\Gamma_v(E+\iu0)$ is real for every $v\in V$. It is finite by condition (a) and nonzero by \Cref{lem:rde-boundary-schur}. By \Cref{lem:rde-propagation}(i), the boundary values $G^C_{xx}(E+\iu0)$ and $\Gamma^C_{x\to w}(E+\iu0)$ are real for every removed set, component and vertex, and they are nonzero by \Cref{lem:rde-boundary-schur}. Further, \Cref{lem:rde-boundary-schur} with $C=V$, applied to the oriented edge $v\to v^-$ if $v\neq o$ and to the diagonal entry $G_{oo}=\Gamma_o$ if $v=o$, gives \[ D_v=-\frac1{\Gamma_v(E+\iu0)}=E+\sum_{k=1}^{N_v}|h_{v,vk}|^2\,\Gamma_{vk}(E+\iu0) =E-\frac1p\sum_{k=1}^{N_v}\frac{Y_{vk}}{D_{vk}}, \] which is \eqref{eq:rde-boundary-recursion}. \proofstep{5}{The law of $D_o$.} Let $\mathfrak D\colon\Omega\to\R$ be the measurable map defined by $\mathfrak D=-1/\lim_{k\to\infty}G_{oo}(E+\iu/k)$ if this limit exists and is real and nonzero, and $\mathfrak D=E$ otherwise. By Step 4 and the identity $\Gamma_v=G_{oo}\circ\theta_v$, almost surely $D_v=\mathfrak D(\theta_v\omega)$ for every $v\in V$. By \eqref{eq:rde-boundary-recursion} at the root, \[ D_o=E-\frac1p\sum_{k=1}^{N_o}\frac{Y_k}{\mathfrak D(\theta_k\omega)}. \] By \Cref{lem:palm-branching}(ii), the variables $N_o$, $(Y_k)_{k\ge1}$ and $(\theta_k\omega)_{k\ge1}$ are independent, and the $\theta_k\omega$ have law $\P$. It follows that the variables $\mathfrak D(\theta_k\omega)$, $k\ge1$, are independent with the law of $D_o$, and independent of $N_o$ and of the squared moduli of the root edge marks. Since $D_o\in\R\setminus\{0\}$ almost surely, the law of $D_o$ is a real solution at energy $E$. \end{proof} The nonvanishing of the boundary denominators in \Cref{prop:rde-boundary}(ii) is a consequence of the finiteness of the boundary values, which is part of the definition of $\mathcal E_p$. We also record the structure of the law of $D_o$ given the children. Under the no-ac hypothesis, by Step 5 of the proof of \Cref{prop:rde-boundary}, $D_o$ is either the constant $E$ (if $N_o=0$) or an affine function of $Y_1,\dots,Y_{N_o}$ with the nonzero real coefficients $-1/(p\mathfrak D(\theta_k\omega))$ (if $N_o\ge1$). These coefficients are measurable with respect to the $\sigma$-field generated by $N_o$ and $(\theta_k\omega)_{k\ge1}$, and the squared moduli $Y_1,\dots,Y_{N_o}$ are fresh given this $\sigma$-field, by the example after \Cref{lem:palm-fresh-edges} and consequence (a) after \Cref{def:palm-fresh}. By \Cref{lem:palm-affine}(iii) and \eqref{eq:palm-freeze}, the conditional law of $D_o$ on the event $\{N_o\ge1\}$ has no atoms. The same argument is used in \Cref{sec:rde-fixed} for the real recursive tree process. \begin{definition}\label{def:rde-boundary-law} Let $E\in\mathcal E_p$ with $E>0$. Following \cite{AizenmanWarzel2013resonant}, we say that the \emph{no-ac hypothesis} holds at $E$ if $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. In this case, with $D_o=-1/\Gamma_o(E+\iu0)$, the \emph{real boundary law} at $E$ is $\pi_E=\Law_\P(D_o)$, and the \emph{conditioned laws} are \[ \pi^\infty_E=\Law_{\P^\infty}(D_o),\qquad\pi^{\mathrm f}_E=\Law_\P(D_o\mid\mathcal S^c). \] In the $B$-normalization ($E>2$), the real boundary law is the law of $g(E)D_o$ under $\P$. \end{definition} By \Cref{prop:rde-boundary}(ii), $\pi_E$ is a real solution at energy $E$, and all the conclusions of \Cref{prop:rde-properties} apply to it. Further, \[ \pi_E=\varrho_p\,\pi^\infty_E+(1-\varrho_p)\,\pi^{\mathrm f}_E,\qquad \pi^\infty_E\le\varrho_p^{-1}\pi_E\quad\text{as measures,} \] and $\pi^\infty_E$ has no atoms. To see the last claim, note that on $\mathcal S$ the root has at least one child, and by the previous paragraph the law of $D_o$ restricted to the event $\{N_o\ge1\}$ has no atoms. Then $\P^\infty(D_o=c)\le\varrho_p^{-1}\P(D_o=c,\,N_o\ge1)=0$ for every $c\in\R$. In \Cref{rem:rde-endogeny} below, we note that the boundary denominators form a recursive tree process with marginal $\pi_E$ which is a function of the tree. \section{Recursive tree processes}\label{sec:rde-rtp} For an arbitrary real solution, a family of denominators satisfying the recursion need not be a function of the tree. The following lemma constructs such a family on an extended probability space. It is used in \Cref{sec:loc}, where the real solution need not be the real boundary law. \begin{lemma}\label{lem:rde-rtp} Let $E>0$ and let $\pi$ be a real solution at energy $E$. There exist a probability space and, on it, a random configuration $\omega$ with law $\P$ (\Cref{def:palm-config}) and real random variables $(D_v)_{v\in\mathbb U}$ with the following properties. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $v\in\mathbb U$, we have $D_v\sim\pi$, and almost surely $D_v\in\R\setminus\{0\}$ and \[ D_v=E-\frac1p\sum_{k=1}^{N_v}\frac{Y_{vk}}{D_{vk}}. \] \item For every $n\ge0$, the variables $D_v$, $v\in\mathbb U_n$, are independent with law $\pi$, and they are independent of the $\sigma$-field $\mathscr F_n$ of \Cref{sec:rde-ulam}. \item Let $v_1,\dots,v_k\in\mathbb U$ be distinct words none of which is a prefix of another, and let $\mathscr F(v_1,\dots,v_k)$ be the $\sigma$-field generated by the coordinates $N_u,\zeta_u$ with $u\notin\bigcup_iv_i\mathbb U$ and $w_u$ with $u\notin\bigcup_iv_i\mathbb U_*$, where $v\mathbb U_*=v\mathbb U\setminus\{v\}$. Then the pairs $\bigl(\theta_{v_i}\omega,(D_{v_iu})_{u\in\mathbb U}\bigr)$, $1\le i\le k$, are independent of each other and of $\mathscr F(v_1,\dots,v_k)$, and each has the law of $\bigl(\omega,(D_u)_{u\in\mathbb U}\bigr)$. In particular, for every $n\ge0$ this holds for the pairs indexed by $v\in\mathbb U_n$, with $\mathscr F_n$ in place of $\mathscr F(v_1,\dots,v_k)$. \end{enumerate} \end{lemma} We call $(\omega,(D_v)_{v\in\mathbb U})$ a \emph{recursive tree process} with marginal $\pi$, in the sense of \cite{AldousBandyopadhyay2005survey}. By (i), the restriction of $(D_v)$ to $V$ is a family of real denominators on the tree that satisfies the real recursion at every vertex. \begin{remark}\label{rem:rde-rtp-structure} We comment on the measurability structure of the process. \begin{enumerate}[label=\textup{(\roman*)}] \item For $|v|=n$, the variable $D_v$ is not independent of $\theta_v\omega$; for instance, $D_v=E$ on the event $\{N_v=0\}$. Property (ii) asserts independence from $\mathscr F_n$ only, and $\mathscr F_n$ contains the offspring numbers $N_u$ with $|u|0$, suppose that the no-ac hypothesis holds at $E$, let $\mathfrak D$ be the map of Step 5 of the proof of \Cref{prop:rde-boundary}, and set $D_v=\mathfrak D(\theta_v\omega)$ for $v\in\mathbb U$, where $\omega$ has law $\P$. Then $(\omega,(D_v)_{v\in\mathbb U})$ satisfies the conclusions of \Cref{lem:rde-rtp} with $\pi=\pi_E$, every $D_v$ is a measurable function of $\omega$, and almost surely $D_v=-1/\Gamma_v(E+\iu0)$ for every $v\in V$. In the terminology of \cite{AldousBandyopadhyay2005survey}, this is an endogenous recursive tree process with marginal $\pi_E$. In particular, when the real solution in \Cref{lem:rde-rtp} is the real boundary law, the process may be taken to be this one. To verify the claim, note that $\mathfrak D$ takes values in $\R\setminus\{0\}$ by its definition, and that the last assertion is Step 5 of the proof of \Cref{prop:rde-boundary}. Let $A$ be the event that $\mathfrak D(\omega)=E-p^{-1}\sum_{k\le N_o}Y_k/\mathfrak D(\theta_k\omega)$; it has probability one by Step 5 of that proof. Since $\theta_{vk}=\theta_k\circ\theta_v$, and since the offspring number and the root edge marks of $\theta_v\omega$ are $N_v$ and $w_{vk}$, $k\ge1$, the recursion of \Cref{lem:rde-rtp}(i) at the word $v$ is the event $\theta_v^{-1}A$. This event has probability one because $\theta_v$ preserves $\P$, and $D_v$ has the law of $D_o$, which is $\pi_E$. This proves (i). For (ii) and (iii), we note that the pair $(\theta_{v_i}\omega,(D_{v_iu})_{u\in\mathbb U})$ is a measurable function of $\theta_{v_i}\omega$, since $D_{v_iu}=\mathfrak D(\theta_u(\theta_{v_i}\omega))$. The configurations $\theta_{v_1}\omega,\dots,\theta_{v_k}\omega$ and the generators of $\mathscr F(v_1,\dots,v_k)$ are functions of pairwise disjoint sets of the independent coordinates of $\omega$, and each $\theta_{v_i}\omega$ has law $\P$; this is the argument of the proof of \Cref{lem:palm-branching}(i). This gives (iii), and (ii) follows in the same way from \Cref{lem:palm-branching}(i). \end{remark} \section{Fractional moments at positive imaginary part}\label{sec:rde-moments} The following lemma integrates one step of the cavity recursion, at a single vertex. It extends \Cref{lem:palm-affine} to products of several negative powers sharing the same fresh variables, with additional weights given by powers of these variables. We write $C_{r,d}=(1-r)^{-1}(2/d)^r$ for the constant of \Cref{lem:palm-affine}. \begin{lemma}\label{lem:rde-row} Let $r\in(0,1)$, $d>0$, $b\ge0$ and $k\in\N_0$, and set $c_b=\max\{1,(2b/e)^b\}$. Let $n\ge0$, let $Y_1,\dots,Y_n$ be independent $\Exp(1)$ variables, let $c\in\C$ with $|c|\ge d$ and $\gamma_1,\dots,\gamma_n\in\C$, let $J_1,\dots,J_l\subset\{1,\dots,n\}$, let $a_1,\dots,a_l\ge0$ with $a_1+\dots+a_l\le r$, and let $K\subset\{1,\dots,n\}$ with $|K|\le k$ and $b_j\in[0,b]$ for $j\in K$. Then \[ \E\Big[\prod_{i=1}^l\Big|c+\sum_{j\in J_i}\gamma_jY_j\Big|^{-a_i}\prod_{j\in K}Y_j^{b_j}\Big] \le(2c_b)^k\big(1+C_{r,d}(n+1)^{r/2}\big). \] The same bound holds for the conditional expectation given a $\sigma$-field $\mathscr G$ if $Y_1,\dots,Y_n$ are fresh given $\mathscr G$ (\Cref{def:palm-fresh}), and $n$, $c$, the $\gamma_j$, the $J_i$, the $a_i$, $K$ and the $b_j$ are $\mathscr G$-measurable, with $|c|\ge d$ almost surely. \end{lemma} \begin{proof} For $j\in K$ and $y>0$, we have $y^{b_j}e^{-y}\le c_be^{-y/2}$, since the maximum of $y^{b_j}e^{-y/2}$ is $(2b_j/e)^{b_j}\le c_b$. For every measurable $\Phi\ge0$, this gives \[ \E\big[Y_j^{b_j}\Phi(Y_j)\big]\le c_b\int_0^\infty e^{-y/2}\Phi(y)\dd y=2c_b\,\E\Phi(2Y_j). \] Applying this to each $j\in K$ in turn, using the independence of the $Y_j$ and Tonelli's theorem, we find that the left side of the claimed bound is at most $(2c_b)^{|K|}$ times $\E\prod_i|X_i|^{-a_i}$, where $X_i=c+\sum_{j\in J_i}\gamma'_jY_j$ with $\gamma'_j=2\gamma_j$ for $j\in K$ and $\gamma'_j=\gamma_j$ otherwise. Let $A=a_1+\dots+a_l$. If $A=0$, there is nothing more to prove. Otherwise, Hölder's inequality with the exponents $A/a_i$ gives \[ \E\prod_{i=1}^l|X_i|^{-a_i}\le\prod_{i=1}^l\big(\E|X_i|^{-A}\big)^{a_i/A} \le\max_i\E|X_i|^{-A}. \] Since $|x|^{-A}\le1+|x|^{-r}$ for $00$. Then the following hold for every $z\in\C\setminus\R$ with $|z|\ge d$. \begin{enumerate}[label=\textup{(\roman*)}] \item $\E|\Gamma_o(z)|^r\le C_{r,d}(p+1)^{r/2}$ and $\E^\infty|\Gamma_o(z)|^r\le\varrho_p^{-1}C_{r,d}(p+1)^{r/2}$. \item Let $J$ be a random set of children of the root that is a function of the offspring numbers and the auxiliary marks, and let \[ \Gamma^J_o(z)=-\Big(z+\frac1p\sum_{k\in J}Y_k\,\Gamma_k(z)\Big)^{-1} \] be the diagonal entry at the root of the resolvent after the edges from the root to the children outside $J$ are deleted. Then $\E|\Gamma^J_o(z)|^r\le C_{r,d}(p+1)^{r/2}$. If $\mathbb Q$ is a probability measure on $\Omega$ with $\mathbb Q\le K\,\P$ for a constant $K$, then all bounds of (i) and (ii) hold under $\mathbb Q$ with an extra factor $K$. \item For $q\in(0,1)$ and $g>0$, \[ \E|D_o(z)|^q\le|z|^q+p^{1-q}\Gamma(1+q)\,C_{q,d}(p+1)^{q/2},\qquad \E|gD_o(z)|^{-q}\le g^{-q}C_{q,d}(p+1)^{q/2}. \] \item Let $E>2$ and $X_\eta=-\Gamma_o(E+\iu\eta)/g(E)$. Then $\sup_{\eta>0}\E|X_\eta|^r\le C_{r,1}(p+1)^{r/2}$. \end{enumerate} In particular, for every compact interval $I\subset(0,\infty)$ and $r\in(0,1)$, there exists $C=C(p,I,r)$ such that \[ \sup_{E\in I,\,0<\eta\le1}\Big(\E|\Gamma_o(E+\iu\eta)|^r+\E^\infty|\Gamma_o(E+\iu\eta)|^r +\E|D_o(E+\iu\eta)|^r+\E|D_o(E+\iu\eta)|^{-r}\Big)\le C. \] \end{lemma} \begin{proof} Part (i) is \Cref{cor:palm-root-moment}, since $\Gamma_o=G_{oo}$ and $|z|\ge d$. For (ii), let $\mathscr G$ be the $\sigma$-field generated by $N_o$, the auxiliary marks, the configurations $\theta_k\omega$, $k\ge1$, and the phases of the root edge marks. As shown after \Cref{lem:palm-fresh-edges}, the squared moduli $Y_1,\dots,Y_{N_o}$ are fresh given $\mathscr G$, and the set $J$ and the coefficients $\one_{\{k\in J\}}\Gamma_k(z)/p$ are $\mathscr G$-measurable. \Cref{lem:palm-affine}(iv), with constant $z$ and these coefficients, gives $\E[|\Gamma^J_o(z)|^r\mid\mathscr G]\le C_{r,d}(N_o+1)^{r/2}$, and taking expectations and using Jensen's inequality, we find the bound. For $\mathbb Q\le K\P$, we have $\E_{\mathbb Q}Z\le K\,\E Z$ for every $Z\ge0$. For (iii), we use \eqref{eq:rde-complex-recursion} and $|a+b|^q\le|a|^q+|b|^q$ to obtain \[ \E|D_o(z)|^q\le|z|^q+p^{-q}\,\E\sum_{k=1}^{N_o}Y_k^q|\Gamma_k(z)|^q =|z|^q+p^{1-q}\Gamma(1+q)\,\E|\Gamma_o(z)|^q, \] where we used that $N_o$, the $Y_k$ and the $\Gamma_k(z)=G_{oo}(z)\circ\theta_k$ are independent and that the $\Gamma_k(z)$ have the law of $\Gamma_o(z)$, by \Cref{lem:palm-branching}(ii). Combined with (i), this gives the first bound in (iii). The second bound is (i) with $r=q$, since $|gD_o(z)|^{-q}=g^{-q}|\Gamma_o(z)|^q$. For (iv), let $z=E+\iu\eta$ and $X_k=-\Gamma_k(z)/g(E)$; then $X_\eta=1/(g(E)D_o(z))$. We multiply \eqref{eq:rde-complex-recursion} at the root by $g(E)$ and use $g(E)E=1+\theta$ from \Cref{lem:rde-normalizations} and $g(E)\Gamma_k(z)=-\theta X_k$. We obtain \[ X_\eta=\Big(1+\theta+\iu g(E)\eta-\frac\theta p\sum_{k=1}^{N_o}Y_kX_k\Big)^{-1}. \] The constant term has modulus at least $1+\theta\ge1$, and the argument of (ii) with $d=1$ applies. The final assertion follows from (i) and (iii), with $d=\min I$ and $|z|\le\max I+1$. \end{proof} The bounds of \Cref{lem:rde-moments}(ii) also apply when structural decorations are prescribed at the root. For example, under the law conditioned on the event that the root has exactly one child, which has $\P$-probability $pe^{-p}$, all bounds hold with the factor $K=e^p/p$. If a vertex is required to have no children other than a selected one, then its denominator without the selected child is the deterministic number $z$. \section{Fixed-depth convergence}\label{sec:rde-fixed} In \Cref{sec:res,sec:deloc}, the resonance argument compares, at a fixed depth, expectations of functionals of the cavity variables at $E+\iu\eta$, $\eta>0$, with the corresponding expectations for real denominators, and then lets $\eta$ tend to zero with the depth fixed. This section provides the convergence statement for this step. The functionals are sums over one or two vertices of a fixed generation, which the many-to-one and many-to-two formulas of \Cref{prop:palm-spine} express through the spine laws. We first record the properties of the spine laws that we use, then define the depth-$m$ data and the real recursive tree process, and finally prove the convergence. \subsection{Spine laws}\label{sec:rde-spine} We use the spine laws of \Cref{def:palm-spine-laws}. These are the one-spine laws $\mathbb Q_L$ and $\mathbb Q_L^\infty$, with canonical random elements $(\omega,\mathsf v)$, and the two-spine laws $\mathbb Q_{L,j}$ and $\mathbb Q_{L,j}^\infty$, with canonical random elements $(\omega,\mathsf u,\mathsf v)$; here $L\ge0$ and $0\le j0$, suppose that the no-ac hypothesis holds at $E$, let $L\ge0$, and let $\pi'\in\{\pi_E,\pi^\infty_E\}$. \begin{enumerate}[label=\textup{(\roman*)}] \item The \emph{one-spine real recursive tree process} of length $L$ with endpoint law $\pi'$ consists of independent random variables $D^{\mathrm{side}}_l$, $0\le l0$, suppose that the no-ac hypothesis holds at $E$, and let $\mathbb Q$ be a spine law of length $L$. \begin{enumerate}[label=\textup{(\roman*)}] \item $\mathbb Q$-almost surely, all conclusions of \Cref{def:rde-regular}, \Cref{lem:rde-boundary-schur}, \Cref{lem:rde-propagation} and \Cref{prop:rde-boundary}(ii) hold. In particular, for every $v\in V$, we have $D_v(E+\iu\eta)\to D_v=\mathfrak D(\theta_v\omega)\in\R\setminus\{0\}$, and for every spine vertex $x$ which is not an endpoint, $D^{\mathrm{side}}_x(E+\iu\eta)\to D^{\mathrm{side}}_x(E)\in\R\setminus\{0\}$, where \[ D^{\mathrm{side}}_x(E)=E-\frac1p\sum_{y\text{ side child of }x}\frac{Y_y}{D_y}. \] \item Under $\mathbb Q_L$, respectively $\mathbb Q_L^\infty$, the variables $D^{\mathrm{side}}_{\mathsf v_l}(E)$, $0\le l0$. Then for every $z\in\C\setminus\R$ with $|z|\ge d$, \[ \E_{\mathbb Q}W(\omega,\mathbf v;z)\le\varrho_p^{-2}\Big((2c_b)^k\big(1+C_{r,d}(p+3)^{r/2}\big)\Big)^{n}, \] where $\mathbf v=\mathsf v$ or $(\mathsf u,\mathsf v)$, $n$ is the number of spine vertices, and $c_b$ and $C_{r,d}$ are as in \Cref{lem:rde-row}. \end{lemma} \begin{proof} We argue by induction on $L$, integrating the factor of the root first. Let $R_o(z)$ be the factor of $W$ indexed by $x=o$, and let $W'(z)$ be the product of the other factors; then $W=R_oW'$. Let $\mathcal A$ be the set of children of the root, and let $\mathscr G$ be the $\sigma$-field of \Cref{lem:rde-spine}(ii) for this set. Then the squared moduli $Y_1,\dots,Y_{N_o}$ of the root edge marks are fresh given $\mathscr G$. The forward cavities $\Gamma_k(z)=G_{oo}(z)\circ\theta_k$, the sets $J_{o,i}$ and $K_o$, the numbers $b_{o,y}$, and the factor $W'(z)$ are $\mathscr G$-measurable, since they depend only on the offspring numbers, the auxiliary marks, the distinguished vertices, and the configurations $\theta_k\omega$, which do not contain the marks $w_k$. The conditional version of \Cref{lem:rde-row}, with $n=N_o$, $c=z$ and $\gamma_k=\Gamma_k(z)/p$, gives \[ \E_{\mathbb Q}\big[R_o(z)\,\big|\,\mathscr G\big]\le(2c_b)^k\big(1+C_{r,d}(N_o+1)^{r/2}\big). \] Taking expectations, we find \begin{equation}\label{eq:rde-products-step} \E_{\mathbb Q}W(z)\le(2c_b)^k\,\E_{\mathbb Q}\Big[\big(1+C_{r,d}(N_o+1)^{r/2}\big)W'(z)\Big]. \end{equation} If $L=0$, then $W'=1$, the law of $N_o$ is $\Pois(p)$ under $\P$, and $\E^\infty Z\le\varrho_p^{-1}\E Z$ for $Z\ge0$; Jensen's inequality and \eqref{eq:rde-products-step} give the claim. Let $L\ge1$, and consider a one-spine law. By \Cref{lem:rde-spine}(iii), $N_o-1$ has law $\Pois(p)$ and is independent of the pair $(\theta_{I_1}\omega,\mathsf v')$, which has law $\mathbb Q_{L-1}$ or $\mathbb Q_{L-1}^\infty$. The factor $W'$ is a function of this pair, and in the coordinates of $\theta_{I_1}\omega$ it is a fractional product of depth $L-1$ with the same parameters, by \Cref{def:rde-products}(i) and (iii) and since $m-1\ge L$. Using this independence in \eqref{eq:rde-products-step} and Jensen's inequality $\E(N_o+1)^{r/2}\le(p+2)^{r/2}$, we find \[ \E_{\mathbb Q}W(z)\le(2c_b)^k\big(1+C_{r,d}(p+2)^{r/2}\big)\,\E_{\mathbb Q'}W'(z), \] where $\mathbb Q'$ is the law of the pair, and the induction hypothesis applies to the last factor. For two-spine laws with $j\ge1$, the same argument applies with the triple of \Cref{lem:rde-spine}(iii). For $j=0$, the factor $W'$ is the product of two fractional products of depth $L-1$, one for each of the two independent pairs of \Cref{lem:rde-spine}(iii), and $N_o-2$ has law $\Pois(p)$; the expectation factorizes, and $\E(N_o+1)^{r/2}\le(p+3)^{r/2}$. In each case, the factor $\varrho_p^{-1}$ appears once for each endpoint with endpoint law $\P^\infty$, and there are at most two endpoints. \end{proof} \subsection{Convergence at fixed depth}\label{sec:rde-convergence} \begin{proposition}\label{prop:rde-fixed-depth} Let $E\in\mathcal E_p$ with $E>0$, and suppose that the no-ac hypothesis holds at $E$, that is, $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Let $\mathbb Q$ be a spine law of length $L$, write $\mathbf v=\mathsf v$ or $\mathbf v=(\mathsf u,\mathsf v)$ for its distinguished vertices, and let $m\ge L+1$. \begin{enumerate}[label=\textup{(\roman*)}] \item $\mathbb Q$-almost surely, $\mathcal L_m(E+\iu\eta)\to\mathcal L_m(E)$ in $\mathcal X_m$ as $\eta\downarrow0$. The side denominators and the endpoint denominators converge almost surely, and their limits, together with the squared moduli of the spine edge marks, have the joint law of the real recursive tree process of \Cref{lem:rde-comparison-law}(ii); the denominators at the spine vertices converge to the corresponding denominators of this process. In particular, the joint law of these variables at $E+\iu\eta$ converges weakly to that of the real recursive tree process. \item Let $\Phi$ be a bounded measurable function of $\mathcal L_m$ and $\mathbf v$, and suppose that $\mathbb Q$-almost surely $\Phi(\cdot,\mathbf v)$ is continuous at $\mathcal L_m(E)$. Then \[ \lim_{\eta\downarrow0}\E_{\mathbb Q}\Phi(\mathcal L_m(E+\iu\eta),\mathbf v)=\E_{\mathbb Q}\Phi(\mathcal L_m(E),\mathbf v). \] \item Let $00$ there exists $\eta_0>0$ such that, for all $0<\eta\le\eta_0$ and all $k$, \[ \Big|\E_{\mathbb Q_k}\big[W_k(\omega,\mathbf v;E+\iu\eta)\Phi_k(\mathcal L_m(E+\iu\eta),\mathbf v)\big] -\E_{\mathbb Q_k}\big[W_k(\omega,\mathbf v;E)\Phi_k(\mathcal L_m(E),\mathbf v)\big]\Big|\le\eps. \] The same holds for finitely many sums as in \textup{(iv)}. \end{enumerate} \end{proposition} \begin{proof} \proofstep{1}{Almost sure convergence.} By \Cref{def:rde-regular}(a) and \Cref{lem:rde-spine}(i), $\mathbb Q$-almost surely $F_\kappa(E+\iu\eta)\to F_\kappa(E+\iu0)$ for every $\kappa\in\mathcal K_m$. The other components of $\mathcal L_m(z)$ do not depend on $z$. Since convergence in the product topology is coordinatewise convergence, $\mathcal L_m(E+\iu\eta)\to\mathcal L_m(E)$ almost surely. The convergence of the side denominators and the law of the limit are \Cref{lem:rde-comparison-law}(i) and (ii), and almost sure convergence implies weak convergence of the joint law. This proves (i). \proofstep{2}{Bounded functionals.} By Step 1 and the continuity assumption, $\Phi(\mathcal L_m(E+\iu\eta),\mathbf v)\to\Phi(\mathcal L_m(E),\mathbf v)$ almost surely. Since $\Phi$ is bounded, the dominated convergence theorem implies (ii). \proofstep{3}{Fractional products.} Let $0<\eta\le1$ and $z=E+\iu\eta$; note that $|z|\ge E$. Raising $W$ to the power $r/s$ multiplies the exponents $a_{x,i}$ and $b_{x,y}$ by $r/s$, which shows that $W^{r/s}$ is a fractional product with cavity exponents at most $r$ and exponents of the squared moduli at most $br/s$. By \Cref{lem:rde-products} with $d=E$, we have a bound on $\E_{\mathbb Q}W(\omega,\mathbf v;z)^{r/s}$ that does not depend on $\eta$. Since $r/s>1$, the family $\{W(\omega,\mathbf v;E+\iu\eta)\}_{0<\eta\le1}$ is uniformly integrable. Since each $\Gamma^J_x(z)$ appearing in $W$ has $|x|\le L0$. We take $\eta_0=\min_k\eta_k$, and we treat finitely many sums as in (iv) in the same way. \end{proof} \begin{remark}\label{rem:rde-fixed-depth-use} We comment on the use of \Cref{prop:rde-fixed-depth}. \begin{enumerate}[label=\textup{(\roman*)}] \item The order of choices is: the energy $E$ and the depth $m$ are fixed first, then the finite family of spine laws and functionals, then the tolerance $\eps$, and finally the imaginary part $\eta_0$. The proposition does not interchange the limits $m\to\infty$ and $\eta\downarrow0$. \item In the backbone version, by \Cref{lem:rde-comparison-law}(ii), the side denominators of the limiting real recursive tree process have the unconditioned real boundary law $\pi_E$, and only the endpoint denominators have the conditioned law $\pi^\infty_E$. Functionals that look beyond the endpoints are described by \Cref{lem:rde-comparison-law}(iii). \item Conditioning on a structural event of positive probability, such as a selection of decorations at an endpoint as in \Cref{lem:palm-decorations}, amounts to multiplying $\Phi$ by the indicator of an event depending only on the genealogy, the survival indicators and the auxiliary marks in the ball, and dividing by its probability. If the event depends on the auxiliary ranks only through their relative order, then its indicator is continuous at every point of $\mathcal X_m$ at which the ranks in the ball are distinct, since the genealogy and survival coordinates are discrete; this holds almost surely. Indicators of events defined by finitely many inequalities on the real parts of side denominators, spine denominators and endpoint denominators, and on squared moduli, satisfy the hypothesis of (ii) when the levels are those allowed by \Cref{lem:rde-atoms}. \end{enumerate} \end{remark} \chapter{The transfer operator and the free energy}\label{sec:press} In this section, we study the positive operator that governs the growth of fractional moments of products of transfer matrices, and we define the free energy. We work with a general law of the rescaled denominator. Throughout this section, we fix $p>1$ and $K>0$, and we let $\nu$ be a Borel probability measure on $\R$ such that \begin{equation}\label{eq:press-hyp} \nu(\{0\})=0,\qquad \supp\nu=\R,\qquad \int_\R|b|^q\,\nu(\dd b)<\infty\quad\text{for every }q\in(0,1). \end{equation} We let $B$ be a random variable with law $\nu$, and we let $\beta$ be a random variable independent of $B$ with the exponential law of rate $K$, that is, with density $Ke^{-Kt}$ on $(0,\infty)$. If $\pi$ is a real solution at energy $E>0$ (\Cref{def:rde-real}), $D$ has law $\pi$, and $g>0$, then the law of $B=gD$ satisfies \eqref{eq:press-hyp} by \Cref{prop:rde-properties}, and $\beta=g^2Y/p$ with $Y\sim\Exp(1)$ has the exponential law of rate $K=p/g^2$. This is the situation in which the results of this section are applied. We now describe the contents of this section. In \Cref{sec:press-projective}, we define the transfer matrices, their action on the projective line, and the transfer operator $\mathcal Q_s$, and we record some elementary facts about angular integrals of matrices. In \Cref{sec:press-perron}, we prove \Cref{thm:press-perron}, which states that $\mathcal Q_s$ has a simple dominant eigenvalue $\lambda(s)$ with a strictly positive eigenfunction, that $\lambda(s)$ is real-analytic in $s$, and that $\lambda(s)$ determines the growth of angular moments of products of independent transfer matrices. In \Cref{sec:press-pressure}, we define the free energy $\varphi(s)=\log\lambda(s)$ and establish its basic properties: independence of the scaling, convexity, the symmetry $\varphi(s)=\varphi(2-s)$, monotonicity, and the strict inequality $\varphi(1)<\log p$. In \Cref{sec:press-rows}, we prove inverse moment bounds for products of three independent transfer matrices, and we use them to compare the free energy with moments of norms of products. Finally, in \Cref{sec:press-paths}, we identify the free energy with the exponential growth rate of the path moments $\Phi_L(s)$, which are the quantities controlled in \Cref{sec:loc}. \section{Transfer matrices and projective coordinates}\label{sec:press-projective} We identify the real projective line $\RP$ with $\R\cup\{\infty\}$, where $x\in\R$ corresponds to the line through $(x,1)$ and $\infty$ corresponds to the line through $(1,0)$; this is a homeomorphism when $\R\cup\{\infty\}$ carries the topology of the one-point compactification. We write $C(\RP)$ for the Banach space of continuous functions $F\colon\RP\to\C$ with the norm $\|F\|_\infty=\max_{x\in\RP}|F(x)|$. We say that a real-valued function $F$ is nonnegative, and write $F\ge0$, if $F(x)\ge0$ for all $x$. Vectors in $\C^2$ carry the Hermitian norm $\|\cdot\|$, and matrices carry the corresponding operator norm. \begin{definition}\label{def:press-operators} \begin{enumerate}[label=\textup{(\roman*)}] \item For $b\in\R$ and $\beta>0$, define \begin{equation}\label{eq:press-M} M(b,\beta)=\beta^{-1/2}\begin{pmatrix}0&1\\-\beta&b\end{pmatrix}. \end{equation} We have $\det M(b,\beta)=1$. \item For $x\in\R$, set $v_x=(x,1)/\sqrt{1+x^2}$, and set $v_\infty=(1,0)$. For an invertible real $2\times2$ matrix $A$ and $x\in\RP$, let $A\cdot x\in\RP$ denote the point corresponding to the line through $Av_x$. The number $\|Av_x\|$ does not depend on the choice of sign of the unit vector representing $x$. \item For $b\in\R$ and $\beta>0$, define the projective map $f_{b,\beta}(x)=M(b,\beta)\cdot x$. In coordinates, we have \[ f_{b,\beta}(x)=(b-\beta x)^{-1}\quad (x\in\R,\ x\neq b/\beta),\qquad f_{b,\beta}(b/\beta)=\infty,\qquad f_{b,\beta}(\infty)=0. \] \item Let $m$ be the Borel probability measure on $\RP$ given by $m(\dd x)=\pi^{-1}(1+x^2)^{-1}\dd x$ on $\R$ and $m(\{\infty\})=0$, where $\pi=3.14\ldots$ (elsewhere, $\pi$ denotes a real solution); it is the image of the uniform probability measure on the unit circle under the map sending a unit vector to the line it spans. For an invertible real $2\times2$ matrix $A$ and $s\in[0,2]$, define \[ J_s(A)=\int_{\RP}\|Av_x\|^{-s}\,m(\dd x). \] \item For $s\in(0,2)$, define $e_s(x)=|x|^s$ for $x\in\R$, and define $h_s\colon\RP\to[0,1]$ by \[ h_s(x)=\frac{|x|^s}{(1+x^2)^{s/2}}\quad(x\in\R),\qquad h_s(\infty)=1. \] The function $h_s$ is continuous on $\RP$, vanishes only at $0$, and $e_s=h_s(1+x^2)^{s/2}$ on $\R$. \item For $s\in(0,2)$ and a Borel function $F\colon\R\to[0,\infty]$, define the transfer operator in the $x$-coordinate \[ (\mathcal A_sF)(x)=p\,\E\Big[\beta^{s/2}|x|^sF\big(f_{B,\beta}(x)\big)\Big],\qquad x\in\R. \] This is well defined, because $f_{B,\beta}(x)\in\R$ almost surely for every $x\in\R$: for $x=0$ we have $f_{B,\beta}(0)=1/B$ and $\P(B=0)=0$, and for $x\neq0$ we have $f_{B,\beta}(x)=\infty$ only if $\beta=B/x$, an event of probability zero since $\beta$ has a density and is independent of $B$. \item For $s\in(0,2)$, a bounded Borel function $F\colon\RP\to\C$, and $x\in\RP$, define the transfer operator \[ (\mathcal Q_sF)(x)=p\,\E\Big[\big\|M(B,\beta)v_x\big\|^{-s}F\big(f_{B,\beta}(x)\big)\Big]. \] The expectation converges absolutely by \Cref{lem:press-row-moment} below. \end{enumerate} \end{definition} For $x\in\R$, we have $M(b,\beta)(x,1)^{\mathsf T}=\beta^{-1/2}(1,b-\beta x)^{\mathsf T}$ and $M(b,\beta)(1,0)^{\mathsf T}=\beta^{-1/2}(0,-\beta)^{\mathsf T}$. Taking norms, we find \begin{equation}\label{eq:press-norm} \big\|M(b,\beta)v_x\big\|^{-s}=\frac{\beta^{s/2}(1+x^2)^{s/2}}{\big(1+(b-\beta x)^2\big)^{s/2}}\quad(x\in\R), \qquad \big\|M(b,\beta)v_\infty\big\|^{-s}=\beta^{-s/2}. \end{equation} For fixed $(b,\beta)$, the functions $x\mapsto\|M(b,\beta)v_x\|^{-s}$ and $x\mapsto f_{b,\beta}(x)$ are continuous on $\RP$. The next lemma collects two elementary facts about the projective action. Part (i) is the identity that makes the norm multipliers in $\mathcal Q_s$ telescope along products, and part (ii) computes the projective Jacobian. \begin{lemma}\label{lem:press-projective} Let $A$ and $A'$ be real $2\times 2$ matrices with determinant one. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $x\in\RP$, we have $(A'A)\cdot x=A'\cdot(A\cdot x)$ and \begin{equation}\label{eq:press-cocycle} \|A'Av_x\|=\|A'v_{A\cdot x}\|\,\|Av_x\|. \end{equation} \item For every Borel function $F\colon\RP\to[0,\infty]$, we have \begin{equation}\label{eq:press-jacobian} \int_{\RP}F(A\cdot x)\,\|Av_x\|^{-2}\,m(\dd x)=\int_{\RP}F(y)\,m(\dd y). \end{equation} \end{enumerate} \end{lemma} \begin{proof} For (i), since the vector $Av_x$ spans the line $A\cdot x$, we have $Av_x=\pm\|Av_x\|v_{A\cdot x}$. Applying $A'$, we obtain $A'Av_x=\pm\|Av_x\|A'v_{A\cdot x}$. This vector spans the line $A'\cdot(A\cdot x)$, and taking norms gives \eqref{eq:press-cocycle}. For (ii), we parametrize $\RP$ by angles. For $\theta\in\R$, let $u_\theta=(\cos\theta,\sin\theta)$. The map $\theta\mapsto[u_\theta]$ from $[0,\pi)$ to $\RP$ is a bijection, and the image of $\pi^{-1}\dd\theta$ under it is $m$; indeed, $u_\theta$ spans the line of $x=\cot\theta$, and $\dd x=-(1+x^2)\,\dd\theta$. The map $u\mapsto Au/\|Au\|$ is a homeomorphism of the unit circle, with inverse $u\mapsto A^{-1}u/\|A^{-1}u\|$. Let $\Theta\colon\R\to\R$ be a continuous lift, so that $Au_\theta=\|Au_\theta\|u_{\Theta(\theta)}$. Writing $w(\theta)=Au_\theta$, we have \[ \Theta'(\theta)=\frac{\det\big(w(\theta),w'(\theta)\big)}{\|w(\theta)\|^2} =\frac{\det(A)\det(u_\theta,u_\theta')}{\|Au_\theta\|^2}=\|Au_\theta\|^{-2}>0, \] where we used that $w'(\theta)=Au_\theta'$ and $\det(u_\theta,u'_\theta)=1$. Since a homeomorphism of the circle has degree $\pm1$ and $\Theta$ is increasing, we have $\Theta(\theta+2\pi)=\Theta(\theta)+2\pi$. Further, since $Au_{\theta+\pi}=-Au_\theta$, the continuous function $\theta\mapsto\Theta(\theta+\pi)-\Theta(\theta)$ takes values in $\pi+2\pi\Z$, and it is therefore constant; since twice this constant equals $\Theta(\theta+2\pi)-\Theta(\theta)=2\pi$, it equals $\pi$. We conclude that $\Theta$ is an increasing $C^1$ bijection of $\R$ with $\Theta(\theta+\pi)=\Theta(\theta)+\pi$, and it induces the projective action of $A$ on $\R/\pi\Z\simeq\RP$. For $G(\theta)=F([u_\theta])$, which is $\pi$-periodic, the change of variables $\vartheta=\Theta(\theta)$ gives \[ \int_0^\pi G(\Theta(\theta))\,\|Au_\theta\|^{-2}\,\dd\theta =\int_{\Theta(0)}^{\Theta(0)+\pi}G(\vartheta)\,\dd\vartheta=\int_0^\pi G(\vartheta)\,\dd\vartheta. \] Dividing by $\pi$ gives \eqref{eq:press-jacobian}. \end{proof} \begin{lemma}\label{lem:press-angular} Let $A$ be a real $2\times2$ matrix with determinant one. Then $J_0(A)=J_2(A)=1$, and for every $s\in[0,2]$, we have \begin{equation}\label{eq:press-symmetry} J_s(A)=J_{2-s}(A^{-1})=J_{2-s}(A). \end{equation} Further, $s\mapsto\log J_s(A)$ is convex on $[0,2]$, and $J_s(A)\le1$ for $s\in[0,2]$. \end{lemma} \begin{proof} We have $J_0(A)=m(\RP)=1$, and $J_2(A)=1$ by \eqref{eq:press-jacobian} with $F=1$. Fix $s\in[0,2]$ and $x\in\RP$, and set $y=A\cdot x$. By \eqref{eq:press-cocycle} applied to $A^{-1}A=I$, we have $1=\|A^{-1}v_y\|\,\|Av_x\|$, and we can write \[ \|Av_x\|^{-s}=\|Av_x\|^{-2}\,\|Av_x\|^{2-s}=\|Av_x\|^{-2}\,\|A^{-1}v_{A\cdot x}\|^{-(2-s)}. \] Integrating against $m$ and using \eqref{eq:press-jacobian} with $F(y)=\|A^{-1}v_y\|^{-(2-s)}$, we find $J_s(A)=J_{2-s}(A^{-1})$. To prove that $J_t(A^{-1})=J_t(A)$ for $t\in[0,2]$, we use the singular value decomposition. There exist orthogonal matrices $U,V$ and $\sigma\ge1$ such that $A=U\operatorname{diag}(\sigma,\sigma^{-1})V^{\mathsf T}$; the two singular values are $\sigma$ and $\sigma^{-1}$ because $\det A=1$. Then $A^{-1}=V\operatorname{diag}(\sigma^{-1},\sigma)U^{\mathsf T}$. The quantity $J_t$ is invariant under multiplication by orthogonal matrices on either side: on the left because $\|UAv\|=\|Av\|$, and on the right because $\|AVv_x\|=\|Av_{V\cdot x}\|$ for orthogonal $V$ and the projective action of an orthogonal matrix preserves $m$ (it is induced by a rotation or a reflection of the unit circle). It follows that $J_t(A)=J_t(D_\sigma)$ and $J_t(A^{-1})=J_t(D_\sigma^{-1})$, where $D_\sigma=\operatorname{diag}(\sigma,\sigma^{-1})$. Let $R$ be the rotation by $\pi/2$. Then $D_\sigma^{-1}=RD_\sigma R^{-1}$, and the same invariance implies $J_t(D_\sigma^{-1})=J_t(D_\sigma)$. This proves \eqref{eq:press-symmetry}. For $s_0,s_1\in[0,2]$ and $\tau\in(0,1)$, H\"older's inequality with exponents $1/(1-\tau)$ and $1/\tau$ gives \[ J_{(1-\tau)s_0+\tau s_1}(A)=\int\big(\|Av_x\|^{-s_0}\big)^{1-\tau}\big(\|Av_x\|^{-s_1}\big)^{\tau}\,m(\dd x)\le J_{s_0}(A)^{1-\tau}J_{s_1}(A)^{\tau}, \] which is the convexity of $\log J_s(A)$. Taking $s_0=0$ and $s_1=2$, we find $J_s(A)\le J_0(A)^{1-s/2}J_2(A)^{s/2}=1$. \end{proof} For exponents below one, the angular integral $J_s(A)$ is comparable to a power of the norm of $A$. \begin{lemma}\label{lem:press-norm-angular} For every $s\in(0,1)$, there exists $C_s<\infty$ such that for every real $2\times2$ matrix $A$ with determinant one, setting $t=\|A\|$, we have \[ t^{-s}\le J_s(A)\le C_st^{-s}. \] One can take $C_s=\pi^{-1}\int_0^\pi|\cos\theta|^{-s}\,\dd\theta$. \end{lemma} \begin{proof} The lower bound follows from $\|Av_x\|\le t$ for every $x$. For the upper bound, write $A=U\operatorname{diag}(t,t^{-1})V^{\mathsf T}$ with $U,V$ orthogonal. As in the proof of \Cref{lem:press-angular}, we have $J_s(A)=J_s(\operatorname{diag}(t,t^{-1}))$. In the angular parametrization $u_\theta=(\cos\theta,\sin\theta)$, we have $\|\operatorname{diag}(t,t^{-1})u_\theta\|\ge t|\cos\theta|$, and therefore \[ J_s(A)\le\frac{t^{-s}}{\pi}\int_0^\pi|\cos\theta|^{-s}\,\dd\theta=C_st^{-s}. \] The constant $C_s$ is finite because $s<1$. \end{proof} \section{The Perron--Frobenius eigenvalue of the transfer operator}\label{sec:press-perron} \subsection{Boundedness and holomorphy} We begin with a bound on the expected norm multiplier of one factor $M(b,\beta)$, uniform over the input direction. \begin{lemma}\label{lem:press-row-moment} For $s\in(0,2)$ and $K>0$, set \[ C_{s,K}=2^{s/2}K^{-s/2}\,\Gamma(1+s/2)+2^{s}K^{s/2}\,\Gamma(1-s/2). \] Then for every $b\in\R$, we have \begin{equation}\label{eq:press-row-moment} \sup_{x\in\RP}\int_0^\infty Ke^{-Kt}\,\big\|M(b,t)v_x\big\|^{-s}\,\dd t\le C_{s,K}\big(1+|b|^{s/2}\big). \end{equation} The constant $C_{s,K}$ is continuous in $(s,K)\in(0,2)\times(0,\infty)$. \end{lemma} \begin{proof} Let $r=s/2\in(0,1)$, and let $I(x)$ denote the integral on the left side of \eqref{eq:press-row-moment}. By \eqref{eq:press-norm}, we have $I(\infty)=K\int_0^\infty t^{-r}e^{-Kt}\,\dd t=K^r\Gamma(1-r)$. For $x\in\R$ with $|x|\le1$, we discard the denominator in \eqref{eq:press-norm} and obtain $I(x)\le2^r\int_0^\infty Kt^re^{-Kt}\,\dd t=2^rK^{-r}\Gamma(1+r)$. Now let $|x|=a\ge1$ and $\sigma=\sgn x$. Substituting $u=at$ in the integral defining $I(x)$ and using \eqref{eq:press-norm}, we have \begin{equation}\label{eq:press-row-sub} I(x)=K\,\frac{(1+a^2)^r}{a^{1+r}}\int_0^\infty e^{-Ku/a}\,\frac{u^r}{\big(1+(b-\sigma u)^2\big)^r}\,\dd u. \end{equation} Since $u=|\sigma u|\le|b|+|b-\sigma u|$ and $r<1$, we have $u^r\le|b|^r+|b-\sigma u|^r$. Using $|b-\sigma u|^r\le(1+(b-\sigma u)^2)^{r/2}$ and $(1+(b-\sigma u)^2)^{-r}\le (1+(b-\sigma u)^2)^{-r/2}$, we have \[ \frac{u^r}{\big(1+(b-\sigma u)^2\big)^r}\le(1+|b|^r)\,H(b-\sigma u),\qquad H(v)=(1+v^2)^{-r/2}. \] The function $H$ is even and nonincreasing on $[0,\infty)$. Hence $\int_{J'}H\le\int_{-\ell/2}^{\ell/2}H$ for every interval $J'\subset\R$ of length $\ell$. (Indeed, if $J$ denotes the centered interval of length $\ell$, then $J'\setminus J$ and $J\setminus J'$ have the same length, and $H\le H(\ell/2)$ on $J'\setminus J$ while $H\ge H(\ell/2)$ on $J\setminus J'$.) We write $e^{-Ku/a}=\int_u^\infty (K/a)e^{-Kt/a}\,\dd t$ and use Tonelli's theorem. Since $\{b-\sigma u:0\le u\le t\}$ is an interval of length $t$, this gives \begin{align*} \int_0^\infty e^{-Ku/a}H(b-\sigma u)\,\dd u &=\int_0^\infty\frac Ka e^{-Kt/a}\int_0^tH(b-\sigma u)\,\dd u\,\dd t\\ &\le\int_0^\infty\frac Ka e^{-Kt/a}\,2\int_0^{t/2}H(v)\,\dd v\,\dd t\\ &=2\int_0^\infty H(v)e^{-2Kv/a}\,\dd v\\ &\le2\int_0^\infty v^{-r}e^{-2Kv/a}\,\dd v=2\,\Gamma(1-r)\Big(\frac{a}{2K}\Big)^{1-r}. \end{align*} Inserting this bound into \eqref{eq:press-row-sub} and using $(1+a^2)^r\le2^ra^{2r}$, we find \[ I(x)\le K\,2^ra^{r-1}\cdot2\,\Gamma(1-r)\Big(\frac{a}{2K}\Big)^{1-r}(1+|b|^r)=4^rK^r\,\Gamma(1-r)\,(1+|b|^r). \] Combining the three cases gives \eqref{eq:press-row-moment}. The continuity of $C_{s,K}$ is evident from its formula. \end{proof} To prove analyticity in $s$, we extend $\mathcal Q_s$ to complex exponents. Let $\Sigma=\{z\in\C:0<\Re z<2\}$. For $z\in\Sigma$, $x\in\RP$, $b\in\R$, and $\beta>0$, set \[ R_z(x;b,\beta)=\exp\big(-z\log\|M(b,\beta)v_x\|\big), \] so that $|R_z|=R_{\Re z}$ and $R_s(x;b,\beta)=\|M(b,\beta)v_x\|^{-s}$ for real $s$. For a bounded Borel function $F\colon\RP\to\C$, define \[ (\mathcal Q_zF)(x)=p\,\E\Big[R_z(x;B,\beta)\,F\big(f_{B,\beta}(x)\big)\Big],\qquad x\in\RP. \] For real $z=s\in(0,2)$, this is the operator $\mathcal Q_s$ of \Cref{def:press-operators}. \begin{lemma}\label{lem:press-bounded} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $z\in\Sigma$, the operator $\mathcal Q_z$ maps $C(\RP)$ into $C(\RP)$, and \[ \|\mathcal Q_z\|\le p\,C_{\Re z,K}\big(1+\E|B|^{\Re z/2}\big). \] The right side is bounded on every compact subset of $\Sigma$. \item For real $s\in(0,2)$, the operator $\mathcal Q_s$ is positive, that is, $\mathcal Q_sF\ge0$ for $F\ge0$, and $\|\mathcal Q_s\|=\|\mathcal Q_s\one\|_\infty$. Further, $\mathcal Q_s\one(x)>0$ for every $x\in\RP$. \item The map $z\mapsto\mathcal Q_z$ from $\Sigma$ to the bounded operators on $C(\RP)$ is holomorphic in the operator norm. \end{enumerate} \end{lemma} \begin{proof} For $\sigma\in(0,2)$, conditioning on $B$ and applying \Cref{lem:press-row-moment}, we have \begin{equation}\label{eq:press-row-uniform} \sup_{x\in\RP}\E\big[R_\sigma(x;B,\beta)\big]\le C_{\sigma,K}\big(1+\E|B|^{\sigma/2}\big)<\infty, \end{equation} where the finiteness follows from \eqref{eq:press-hyp} because $\sigma/2<1$. Since $\E|B|^{\sigma/2}$ is a finite continuous function of $\sigma\in(0,2)$ (by dominated convergence, with dominating function $1+|B|^{\sigma_+/2}$ on $\sigma\le\sigma_+<2$), the right side of \eqref{eq:press-row-uniform} is bounded on compact subsets of $(0,2)$. We now prove (i). Fix $z\in\Sigma$ and $F\in C(\RP)$, and write $\sigma=\Re z$. Since $|R_z|=R_\sigma$, \eqref{eq:press-row-uniform} implies that $|\mathcal Q_zF(x)|\le p\,C_{\sigma,K}(1+\E|B|^{\sigma/2})\|F\|_\infty$ for every $x$. It remains to show that $\mathcal Q_zF$ is continuous. Let $x_n\to x$ in $\RP$. For each fixed $(b,\beta)$, the maps $y\mapsto R_z(y;b,\beta)$ and $y\mapsto f_{b,\beta}(y)$ are continuous on $\RP$ by \eqref{eq:press-norm}. Then the random variables $Z_n=R_z(x_n;B,\beta)F(f_{B,\beta}(x_n))$ converge almost surely to $Z=R_z(x;B,\beta)F(f_{B,\beta}(x))$. Fix $t\in(\sigma,2)$. Then $|Z_n|^{t/\sigma}\le\|F\|_\infty^{t/\sigma}R_t(x_n;B,\beta)$, and \eqref{eq:press-row-uniform} with $\sigma$ replaced by $t$ shows that $\sup_n\E|Z_n|^{t/\sigma}<\infty$. Since $t/\sigma>1$, the sequence $(Z_n)$ is uniformly integrable, and by Vitali's convergence theorem, we have $\E Z_n\to\E Z$. This proves (i). For (ii), positivity is immediate from the definition, and for real $F$ we have $|\mathcal Q_sF|\le\mathcal Q_s|F|\le\|F\|_\infty\mathcal Q_s\one$, which gives $\|\mathcal Q_s\|=\|\mathcal Q_s\one\|_\infty$; the same bound holds for complex $F$ since $|\mathcal Q_sF|\le\mathcal Q_s|F|$. Since the integrand $R_s(x;B,\beta)$ in $\mathcal Q_s\one(x)$ is strictly positive, we have $\mathcal Q_s\one(x)>0$. For (iii), fix $z\in\Sigma$, let $\sigma=\Re z$, and choose $\eps>0$ with $[\sigma-2\eps,\sigma+2\eps]\subset(0,2)$. We write $\ell=\log\|M(B,\beta)v_x\|$, and we define the operator \[ (\mathcal D_zF)(x)=-p\,\E\Big[\ell\,R_z(x;B,\beta)\,F\big(f_{B,\beta}(x)\big)\Big]. \] For every $w\in\C$, we have $|e^w-1-w|\le|w|^2e^{|w|}/2$. Applying this with $w=-h\ell$ for $h\in\C$ with $|h|\le\eps/2$, we obtain \begin{align*} \big|R_{z+h}-R_z+h\ell R_z\big|&=|R_z|\,\big|e^{-h\ell}-1+h\ell\big|\le\frac{|h|^2}2\,\ell^2e^{\eps|\ell|/2}R_\sigma\\ &\le\frac{8|h|^2}{\eps^2}\,e^{\eps|\ell|}R_\sigma\le\frac{8|h|^2}{\eps^2}\big(R_{\sigma-\eps}+R_{\sigma+\eps}\big). \end{align*} In the second inequality, we used $\ell^2e^{-\eps|\ell|/2}\le16\eps^{-2}e^{-2}\le16\eps^{-2}$, and in the third, we used $e^{\eps|\ell|}e^{-\sigma\ell}\le e^{-(\sigma-\eps)\ell}+e^{-(\sigma+\eps)\ell}$. By \eqref{eq:press-row-uniform}, we conclude that \[ \big\|\mathcal Q_{z+h}-\mathcal Q_z-h\mathcal D_z\big\|\le\frac{8p|h|^2}{\eps^2}\sup_{x\in\RP}\E\big[R_{\sigma-\eps}(x;B,\beta)+R_{\sigma+\eps}(x;B,\beta)\big]\le C|h|^2, \] with $C$ independent of $h$. Since $|\ell|R_\sigma\le\eps^{-1}(R_{\sigma-\eps}+R_{\sigma+\eps})$, the same reasoning as in (i) shows that $\mathcal D_z$ is a bounded operator on $C(\RP)$. We conclude that $z\mapsto\mathcal Q_z$ is complex differentiable in operator norm at every point of $\Sigma$, with derivative $\mathcal D_z$. \end{proof} \subsection{Kernel measures and total variation} For $s\in(0,2)$ and $x\in\RP$, we let $\kappa_s(x,\cdot)$ be the finite Borel measure on $\RP$ defined by \[ \kappa_s(x,A)=p\,\E\Big[\big\|M(B,\beta)v_x\big\|^{-s}\one_A\big(f_{B,\beta}(x)\big)\Big], \] so that $\mathcal Q_sF(x)=\int F(y)\,\kappa_s(x,\dd y)$ for bounded Borel $F$. We call $\kappa_s(x,\cdot)$ the kernel measure of $\mathcal Q_s$ at $x$. By Tonelli's theorem, $x\mapsto\kappa_s(x,A)$ is Borel for each Borel set $A$; that is, $\kappa_s$ is a finite kernel. We define its iterates by $\kappa_s^{(1)}=\kappa_s$ and $\kappa_s^{(n+1)}(x,A)=\int\kappa_s(x,\dd y)\,\kappa_s^{(n)}(y,A)$. By induction and Tonelli's theorem, we have $\mathcal Q_s^nF(x)=\int F\,\dd\kappa_s^{(n)}(x,\cdot)$ for bounded Borel $F$, the Chapman--Kolmogorov relations $\kappa_s^{(m+n)}(x,\cdot)=\int\kappa_s^{(m)}(x,\dd y)\,\kappa_s^{(n)}(y,\cdot)$ hold, and \begin{equation}\label{eq:press-row-mass} \kappa_s^{(n)}(x,\RP)=\mathcal Q_s^n\one(x)\le\|\mathcal Q_s\|^n . \end{equation} Directly from the definition, we also have \begin{equation}\label{eq:press-row-first-step} \kappa_s^{(n+1)}(x,\cdot)=p\,\E\Big[\big\|M(B,\beta)v_x\big\|^{-s}\,\kappa_s^{(n)}\big(f_{B,\beta}(x),\cdot\big)\Big]. \end{equation} For a finite signed Borel measure $\mu$ on $\RP$, we write $\|\mu\|_{\TV}=|\mu|(\RP)$ for its total variation norm. We say that the $n$-step kernel measure is continuous in total variation at $x_0\in\RP$ if $\|\kappa_s^{(n)}(x,\cdot)-\kappa_s^{(n)}(x_0,\cdot)\|_{\TV}\to0$ as $x\to x_0$. The kernel measures at the two special points are explicit. Since $f_{b,\beta}(\infty)=0$ and $f_{b,\beta}(0)=1/b$, \eqref{eq:press-norm} yields \begin{equation}\label{eq:press-special-rows} \kappa_s(\infty,\cdot)=p\,\E\big[\beta^{-s/2}\big]\,\delta_0,\qquad \kappa_s(0,A)=p\,\E\big[\beta^{s/2}\big]\,\E\Big[(1+B^2)^{-s/2}\one_A(1/B)\Big]. \end{equation} At a finite nonzero input, the kernel measure has a density. To describe it, let $\iota\colon\R\to\RP$ be the map $\iota(w)=1/w$, with $\iota(0)=\infty$; it is a Borel bijection onto $\RP\setminus\{0\}$ with Borel inverse. Set $\phi(t)=t^{s/2}e^{-Kt}\one_{\{t>0\}}$, which is a bounded continuous function on $\R$ because $s>0$. For $x\in\R\setminus\{0\}$ and $w\in\R$, define \begin{equation}\label{eq:press-row-density} k_x(w)=\frac{pK}{|x|}\,(1+x^2)^{s/2}(1+w^2)^{-s/2}\int\phi\Big(\frac{b-w}{x}\Big)\,\nu(\dd b). \end{equation} \begin{lemma}\label{lem:press-tv} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $s\in(0,2)$. \begin{enumerate}[label=\textup{(\roman*)}] \item For $x\in\R\setminus\{0\}$, the kernel measure $\kappa_s(x,\cdot)$ is the image of the measure $k_x(w)\,\dd w$ under $\iota$. Further, $k_x(w)>0$ for every $w\in\R$. \item The one-step kernel measure is continuous in total variation at every $x\in\R\setminus\{0\}$. \item The two-step kernel measure is continuous in total variation at every $x\in\R$. \item The three-step kernel measure is continuous in total variation at every $x\in\RP$. \end{enumerate} Further, $\mathcal Q_s^3$ is a compact operator on $C(\RP)$. \end{lemma} \begin{proof} We begin with (i). Fix $x\in\R\setminus\{0\}$ and a Borel set $A\subset\RP$. For fixed $b$, we substitute $w=b-\beta x$ in the integral over $\beta$, so that $\beta=(b-w)/x$, $\dd\beta=\dd w/|x|$, and the constraint $\beta>0$ becomes $(b-w)/x>0$. By \eqref{eq:press-norm}, the integrand $\|M(b,\beta)v_x\|^{-s}$ equals $\beta^{s/2}(1+x^2)^{s/2}(1+w^2)^{-s/2}$, and $f_{b,\beta}(x)=\iota(w)$. Using Tonelli's theorem to integrate over $b$ last, we obtain \[ \kappa_s(x,A)=\int_\R k_x(w)\,\one_A\big(\iota(w)\big)\,\dd w, \] which is the first claim. For the second claim, note that $\phi(t)>0$ for $t>0$. If $x>0$, the integral in \eqref{eq:press-row-density} is positive because $\nu((w,\infty))>0$, and if $x<0$, it is positive because $\nu((-\infty,w))>0$; both follow from $\supp\nu=\R$. For (ii), let $x_n\to x$ in $\R\setminus\{0\}$. Since $\phi$ is bounded and continuous, dominated convergence gives $\int\phi((b-w)/x_n)\,\nu(\dd b)\to\int\phi((b-w)/x)\,\nu(\dd b)$ for every $w$. The prefactors in \eqref{eq:press-row-density} are continuous in $x$, so $k_{x_n}(w)\to k_x(w)$ for every $w$. Further, $\int k_{x_n}=\mathcal Q_s\one(x_n)\to\mathcal Q_s\one(x)=\int k_x$ by \Cref{lem:press-bounded}(i). We now use Scheff\'e's lemma: since $(k_x-k_{x_n})^+\le k_x$ and $(k_x-k_{x_n})^+\to0$ pointwise, dominated convergence gives $\int(k_x-k_{x_n})^+\to0$, and then \[ \int|k_x-k_{x_n}|=2\int(k_x-k_{x_n})^+-\int(k_x-k_{x_n})\to0 . \] Since $\iota$ is a Borel bijection with Borel inverse, the total variation norm of the difference of the image measures equals $\int|k_x-k_{x_n}|$. This proves (ii). The remaining parts rest on the following claim. \emph{Claim.} Let $n\ge1$ and $x_0\in\RP$, and suppose that for $\nu\otimes\Exp(K)$-almost every $(b,\beta)$, the $n$-step kernel measure is continuous in total variation at $f_{b,\beta}(x_0)$. Then the $(n+1)$-step kernel measure is continuous in total variation at $x_0$. To prove the claim, let $x_k\to x_0$, and write $R^{(k)}=\|M(B,\beta)v_{x_k}\|^{-s}$, $R=\|M(B,\beta)v_{x_0}\|^{-s}$, $y_k=f_{B,\beta}(x_k)$, and $y=f_{B,\beta}(x_0)$. By \eqref{eq:press-row-first-step} and \eqref{eq:press-row-mass}, we have \begin{multline*} \big\|\kappa_s^{(n+1)}(x_k,\cdot)-\kappa_s^{(n+1)}(x_0,\cdot)\big\|_{\TV}\\ \le p\,\|\mathcal Q_s\|^n\,\E\big|R^{(k)}-R\big|+p\,\E\Big[R\,\big\|\kappa_s^{(n)}(y_k,\cdot)-\kappa_s^{(n)}(y,\cdot)\big\|_{\TV}\Big]. \end{multline*} (The total variation distance is a Borel function of $(y_k,y)$, because it equals the supremum of $|\mathcal Q_s^nF(y_k)-\mathcal Q_s^nF(y)|$ over a countable dense subset of the unit ball of $C(\RP;\R)$, and each $\mathcal Q_s^nF$ is continuous.) By continuity of the projective action, $R^{(k)}\to R$ and $y_k\to y$ almost surely. The family $(R^{(k)})$ is uniformly integrable by the argument in the proof of \Cref{lem:press-bounded}(i), and the first term tends to zero by Vitali's convergence theorem. In the second term, the integrand tends to zero almost surely by hypothesis, and it is bounded by $2\|\mathcal Q_s\|^nR$, which is integrable. By dominated convergence, the second term tends to zero. This proves the claim. We now prove (iii). Let $x_0\in\R\setminus\{0\}$. Almost surely, $f_{B,\beta}(x_0)$ is finite (as noted in \Cref{def:press-operators}(vi)) and nonzero (since $f_{b,\beta}(x_0)=0$ only if $x_0=\infty$). By (ii) and the claim with $n=1$, the two-step kernel measure is continuous at $x_0$. For $x_0=0$, we have $f_{B,\beta}(0)=1/B$, which lies in $\R\setminus\{0\}$ almost surely because $\P(B=0)=0$. Again (ii) and the claim imply continuity at $0$. For (iv), let $x_0\in\R$. By (iii), the two-step kernel measure is continuous at every point of $\R$, and $f_{B,\beta}(x_0)\in\R$ almost surely. The claim with $n=2$ implies continuity of the three-step kernel measure at $x_0$. For $x_0=\infty$, we have $f_{b,\beta}(\infty)=0$ for all $(b,\beta)$, and the two-step kernel measure is continuous at $0$ by (iii). The claim with $n=2$ implies continuity at $\infty$. Finally, we deduce compactness. By (iv) and the compactness of $\RP$, the map $x\mapsto\kappa_s^{(3)}(x,\cdot)$ is uniformly continuous in total variation. For $F\in C(\RP)$ with $\|F\|_\infty\le1$ and $x,x'\in\RP$, we have \[ \big|\mathcal Q_s^3F(x)-\mathcal Q_s^3F(x')\big|\le\big\|\kappa_s^{(3)}(x,\cdot)-\kappa_s^{(3)}(x',\cdot)\big\|_{\TV}. \] The image of the unit ball under $\mathcal Q_s^3$ is therefore equicontinuous, and it is bounded by $\|\mathcal Q_s\|^3$. By the Arzel\`a--Ascoli theorem, it is relatively compact in $C(\RP)$. \end{proof} \begin{remark}\label{rem:press-third-power} The argument above does not apply to $\mathcal Q_s$ or $\mathcal Q_s^2$. By \eqref{eq:press-special-rows}, the kernel measure at $\infty$ is a multiple of the point mass $\delta_0$, while by \Cref{lem:press-tv}(i) the kernel measures at finite nonzero inputs are absolutely continuous with respect to $m$; since such measures form a closed set in total variation, the one-step kernel measure is not continuous in total variation at $\infty$. For the second power, \eqref{eq:press-special-rows} gives $\kappa_s^{(2)}(\infty,\cdot)=p\,\E[\beta^{-s/2}]\,\kappa_s(0,\cdot)$, and $\kappa_s(0,\cdot)$ has an atom at $1/b_0$ whenever $\nu(\{b_0\})>0$. A real solution always has such an atom: by \Cref{prop:rde-properties}, the law of $D$ has an atom of mass $e^{-p}$ at $E$, and the law of $B=gD$ has an atom at $gE$. On the other hand, $\kappa_s^{(2)}(x,\cdot)$ is absolutely continuous for $x\in\R\setminus\{0\}$, since it is a mixture of the absolutely continuous kernel measures $\kappa_s(y,\cdot)$, $y\in\R\setminus\{0\}$. It follows that the two-step kernel measure is not continuous in total variation at $\infty$ in this case. The third step from $\infty$ passes through $0$ and then through the almost surely finite nonzero point $1/B$, where the kernel measures are absolutely continuous; this is why the third power is the first one to which the argument applies. \end{remark} \begin{lemma}\label{lem:press-positive} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $s\in(0,2)$. For every $n\ge2$ and $x\in\RP$, the measure $\kappa_s^{(n)}(x,\cdot)$ has full support, that is, $\kappa_s^{(n)}(x,O)>0$ for every nonempty open set $O\subset\RP$. In particular, for every $n\ge2$ and every real-valued $F\in C(\RP)$ with $F\ge0$ and $F\neq0$, we have $\min_{x\in\RP}\mathcal Q_s^nF(x)>0$. \end{lemma} \begin{proof} Let $O\subset\RP$ be open and nonempty. We first show that $\kappa_s(y,O)>0$ for every $y\in\R$. For $y\in\R\setminus\{0\}$, this follows from \Cref{lem:press-tv}(i), since $\iota^{-1}(O\setminus\{0\})$ is a nonempty open subset of $\R$ and $k_y>0$ everywhere. For $y=0$, the set $\{b\in\R\setminus\{0\}:1/b\in O\}$ is open and nonempty (because $O\setminus\{0,\infty\}$ is open and nonempty), and it has positive $\nu$-measure since $\supp\nu=\R$. By \eqref{eq:press-special-rows}, we conclude that $\kappa_s(0,O)>0$. Next, let $x\in\RP$. If $x\in\R$, then $\kappa_s(x,\cdot)$ has positive total mass and gives no mass to $\{\infty\}$ (by \Cref{lem:press-tv}(i) for $x\neq0$, and by $\P(B=0)=0$ for $x=0$), which implies that $\kappa_s^{(2)}(x,O)=\int\kappa_s(x,\dd y)\,\kappa_s(y,O)>0$. If $x=\infty$, then $\kappa_s^{(2)}(\infty,O)=p\,\E[\beta^{-s/2}]\,\kappa_s(0,O)>0$ by \eqref{eq:press-special-rows}. This proves the claim for $n=2$. For $n\ge3$, we have $\kappa_s^{(n)}(x,O)=\int\kappa_s^{(n-2)}(x,\dd y)\,\kappa_s^{(2)}(y,O)>0$, because $\kappa_s^{(n-2)}(x,\RP)=\mathcal Q_s^{n-2}\one(x)>0$ by \Cref{lem:press-bounded}(ii) and induction. Finally, if $F\ge0$ is continuous and not identically zero, then $\{F>0\}$ is open and nonempty, and we have $\mathcal Q_s^nF(x)=\int F\,\dd\kappa_s^{(n)}(x,\cdot)>0$ for every $x$. Since $\mathcal Q_s^nF$ is continuous and $\RP$ is compact, its minimum is positive. \end{proof} \subsection{The Perron--Frobenius eigenvalue} We use the following form of the stability of an isolated simple eigenvalue under small perturbations, both for analyticity in $s$ and for continuity in the law of $B$. For a bounded operator $T$ on a complex Banach space, we write $\sigma(T)$ for its spectrum. \begin{lemma}\label{lem:press-perturbation} Let $X$ be a complex Banach space and let $T_0$ be a bounded operator on $X$. Suppose that $\mu_0\in\sigma(T_0)$ is an isolated point of $\sigma(T_0)$ whose Riesz projection has rank one, and that there exists $r_0<|\mu_0|$ such that $|w|\le r_0$ for every $w\in\sigma(T_0)\setminus\{\mu_0\}$. Fix $\delta\in(0,(|\mu_0|-r_0)/3)$, let $\Gamma$ be the circle $\{|w-\mu_0|=\delta\}$, and let $\psi\in X$ and $\ell\in X^*$ satisfy $\ell(\Pi_0\psi)\neq0$, where $\Pi_0$ is the Riesz projection of $T_0$ at $\mu_0$. Then there exists $\eps>0$ such that every bounded operator $T$ with $\|T-T_0\|<\eps$ has the following properties. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\sigma(T)\subset\{|w|\|T_0\|$, we have $M=\sup_{w\in\Omega}\|(w-T_0)^{-1}\|<\infty$. Suppose that $\|T-T_0\|<1/(2M)$. For $w\in\Omega$, we write $w-T=(w-T_0)\big(I-(w-T_0)^{-1}(T-T_0)\big)$; the second factor is invertible by the Neumann series, with inverse of norm at most $2$. Then $\Omega$ lies in the resolvent set of $T$, $\|(w-T)^{-1}\|\le2M$, and the resolvent identity gives \[ \big\|(w-T)^{-1}-(w-T_0)^{-1}\big\|=\big\|(w-T)^{-1}(T-T_0)(w-T_0)^{-1}\big\|\le2M^2\|T-T_0\| \] for $w\in\Omega$. This proves the first inclusion in (i). Since $|\mu_0|-\delta>r_0+2\delta$, the circle $\Gamma$ lies in $\Omega$, and integrating the last bound over $\Gamma$ gives $\|\Pi_T-\Pi_0\|\le2\delta M^2\|T-T_0\|$. We now choose $\eps\le1/(2M)$ so small that $2\delta M^2\eps<1$ and $|\ell(\Pi_T\psi)-\ell(\Pi_0\psi)|<|\ell(\Pi_0\psi)|$ whenever $\|T-T_0\|<\eps$. Since $\|\Pi_T-\Pi_0\|<1$, the projection $\Pi_T$ has rank one. Indeed, if $f$ lies in the range of $\Pi_T$ and $\Pi_0f=0$, then $\|f\|=\|(\Pi_T-\Pi_0)f\|<\|f\|$ unless $f=0$, so $\Pi_0$ is injective on the range of $\Pi_T$; in the same way, $\Pi_T$ is injective on the range of $\Pi_0$, which is one-dimensional (see also \cite[Section I.4.6]{Kato1995perturbation}). By the properties of Riesz projections \cite[Section III.6.4]{Kato1995perturbation}, the range of $\Pi_T$ is invariant under $T$, and the spectrum of the restriction of $T$ to it equals $\sigma(T)\cap\{|w-\mu_0|<\delta\}$. Since the range is one-dimensional, $T$ acts on it as multiplication by a scalar $\mu(T)$, and this scalar is the only point of $\sigma(T)$ in the disc. Every other point of $\sigma(T)$ has modulus less than $r_0+\delta<|\mu_0|-\delta<|\mu(T)|$. This proves (i). For (ii), we have $T\Pi_T=\mu(T)\Pi_T$, since $T$ acts on the range of $\Pi_T$ as multiplication by $\mu(T)$. Applying $\ell$ to $T\Pi_T\psi=\mu(T)\Pi_T\psi$ gives the formula for $\mu(T)$, and its denominator is nonzero by the choice of $\eps$. For $T$ and $T_1$ in the ball $\{\|T-T_0\|<\eps\}$ and $w\in\Gamma$, both resolvents are bounded by $2M$, and the resolvent identity gives $\|(w-T)^{-1}-(w-T_1)^{-1}\|\le4M^2\|T-T_1\|$. Integrating over $\Gamma$, we obtain $\|\Pi_T-\Pi_{T_1}\|\le4\delta M^2\|T-T_1\|$. This proves continuity of $\Pi_T$, and continuity of $\mu(T)$ follows from the formula. For (iii), fix $w\in\Gamma$. The map $z\mapsto(w-T(z))^{-1}$ is holomorphic on $U$, being the composition of a holomorphic map with inversion, which is holomorphic on the invertible operators; its derivative $(w-T(z))^{-1}T'(z)(w-T(z))^{-1}$ is jointly continuous in $(w,z)\in\Gamma\times U$. It follows that $z\mapsto \Pi_{T(z)}$ is holomorphic in operator norm (see also \cite[Section VII.1.3]{Kato1995perturbation}), and $z\mapsto\mu(T(z))=\ell(T(z)\Pi_{T(z)}\psi)/\ell(\Pi_{T(z)}\psi)$ is a quotient of holomorphic functions with nonvanishing denominator. \end{proof} We now state and prove the main result of this section. \begin{theorem}\label{thm:press-perron} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $s\in(0,2)$. Let $\lambda(s)$ be the spectral radius of $\mathcal Q_s$ on $C(\RP)$. \begin{enumerate}[label=\textup{(\roman*)}] \item The operator $\mathcal Q_s$ is bounded and positive on $C(\RP)$. For every $n\ge2$, the operator $\mathcal Q_s^n$ is strongly positive: $\min_{x\in\RP}\mathcal Q_s^nF(x)>0$ for every real-valued $F\in C(\RP)$ with $F\ge0$ and $F\neq0$. Further, $\mathcal Q_s^3$ is compact. \item We have $\lambda(s)>0$, and $\lambda(s)$ is an eigenvalue of $\mathcal Q_s$ whose Riesz projection has rank one. There exists a real-valued $\psi_s\in C(\RP)$ with $\min_{x}\psi_s(x)>0$ and $\mathcal Q_s\psi_s=\lambda(s)\psi_s$, and there exists $r(s)<\lambda(s)$ such that $|z|\le r(s)$ for every $z\in\sigma(\mathcal Q_s)\setminus\{\lambda(s)\}$. \item There exists a finite positive Borel measure $\ell_s$ on $\RP$ with full support such that $\int\psi_s\,\dd\ell_s=1$, $\int\mathcal Q_sF\,\dd\ell_s=\lambda(s)\int F\,\dd\ell_s$ for $F\in C(\RP)$, and \[ \lim_{L\to\infty}\sup_{\|F\|_\infty\le1}\Big\|\lambda(s)^{-L}\mathcal Q_s^LF-\Big(\int F\,\dd\ell_s\Big)\psi_s\Big\|_\infty=0 . \] \item The function $s\mapsto\lambda(s)$ is real-analytic on $(0,2)$. \item Let $M_1,M_2,\dots$ be independent copies of $M(B,\beta)$. For every $L\ge1$, every bounded Borel function $F$, and every $x\in\RP$, we have \begin{equation}\label{eq:press-product-formula} \mathcal Q_s^LF(x)=p^L\,\E\Big[\big\|M_L\cdots M_1v_x\big\|^{-s}F\big((M_L\cdots M_1)\cdot x\big)\Big]. \end{equation} Further, there exists $C\ge1$, depending on $s,p,K$, and $\nu$, such that for every $L\ge1$, \begin{equation}\label{eq:press-angular-growth} C^{-1}\lambda(s)^L\le p^L\,\E J_s(M_L\cdots M_1)\le C\lambda(s)^L. \end{equation} In particular, the limit below exists and \[ \log\lambda(s)=\log p+\lim_{L\to\infty}\frac1L\log\E J_s(M_L\cdots M_1). \] \end{enumerate} \end{theorem} \begin{proof} The proof proceeds as follows. Part (i) collects \Cref{lem:press-bounded,lem:press-tv,lem:press-positive}. We apply the Krein--Rutman theorem to the compact strongly positive operator $T=\mathcal Q_s^3$, and we transfer the conclusions to $\mathcal Q_s$ using the Riesz projection of $T$, which commutes with $\mathcal Q_s$. Analyticity then follows from \Cref{lem:press-perturbation} and the holomorphy of $z\mapsto\mathcal Q_z$, and \eqref{eq:press-angular-growth} follows from two-sided bounds on $\psi_s$. \proofstep{1}{Part (i).} Boundedness and positivity are \Cref{lem:press-bounded}(i) and (ii), strong positivity is \Cref{lem:press-positive}, and compactness of $\mathcal Q_s^3$ is \Cref{lem:press-tv}. \proofstep{2}{The Perron--Frobenius eigenvector of $T=\mathcal Q_s^3$.} Let $\mathcal K=\{F\in C(\RP;\R):F\ge0\}$. This is a closed cone in the real Banach space $C(\RP;\R)$, and its interior $\{F:\min F>0\}$ is nonempty. By Step 1, the operator $T$ is compact, and it maps $\mathcal K\setminus\{0\}$ into the interior of $\mathcal K$. By the Krein--Rutman theorem \cite{KreinRutman1948linear}, in the form for compact strongly positive operators \cite[Theorem 19.3]{Deimling1985nonlinear}, the spectral radius $r_T$ of $T$ is positive, and there exists $\psi$ in the interior of $\mathcal K$ with $T\psi=r_T\psi$. Since $|T^nF|\le T^n|F|$ for complex $F$, we have $\|T^n\|=\|T^n\one\|_\infty$ both on $C(\RP;\R)$ and on $C(\RP)$, and by Gelfand's formula, $r_T$ is also the spectral radius of $T$ on the complex space $C(\RP)$. The cited theorem also asserts that $r_T$ is simple and is the only eigenvalue of modulus $r_T$. Since we work on the complex space $C(\RP)$ and use algebraic simplicity below, we verify these two assertions directly. First, let $w\in C(\RP)$ be nonzero with $Tw=\mu w$ and $|\mu|=r_T$. Since $|Tw|\le T|w|$ pointwise, we have $r_T|w|\le T|w|$. The function $\psi/|w|$ on $\{w\neq0\}$ tends to infinity at the boundary of this set, because $\min\psi>0$, and it attains its infimum $t_*\in(0,\infty)$ at some $x_0$. The function $\chi=\psi-t_*|w|$ is nonnegative and vanishes at $x_0$, and \[ T\chi=r_T\psi-t_*T|w|\le r_T\psi-t_*r_T|w|=r_T\chi. \] If $\chi\neq0$, then $T\chi(x_0)>0$ by strong positivity, contradicting $T\chi(x_0)\le r_T\chi(x_0)=0$. We conclude that $|w|=\psi/t_*$, and then $T|w|=r_T|w|=|Tw|$. Fix $x\in\RP$. Writing $T|w|(x)=\int|w|\,\dd\kappa_s^{(3)}(x,\cdot)$ and $|Tw(x)|=|\int w\,\dd\kappa_s^{(3)}(x,\cdot)|$, equality in the triangle inequality for integrals shows that $w=c|w|$ holds $\kappa_s^{(3)}(x,\cdot)$-almost everywhere for some constant $c$ with $|c|=1$. Since both sides are continuous and $\kappa_s^{(3)}(x,\cdot)$ has full support by \Cref{lem:press-positive}, we have $w=c|w|=(c/t_*)\psi$ everywhere. We conclude that every eigenvector for an eigenvalue of modulus $r_T$ is a multiple of $\psi$, and that $r_T$ is the only eigenvalue of modulus $r_T$. Second, we show that $r_T$ is algebraically simple. Since $T$ is compact, $r_T$ is an isolated point of $\sigma(T)$ and its Riesz projection $\Pi$ has finite rank, equal to the dimension of the generalized eigenspace $\bigcup_k\ker(T-r_T)^k$ \cite[Theorem III.6.26]{Kato1995perturbation}. If this dimension were larger than one, then, since $\ker(T-r_T)$ is spanned by $\psi$, there would exist $w$ with $(T-r_T)w=\psi$, and taking real parts we could assume that $w$ is real. By induction, $T^Lw=r_T^Lw+Lr_T^{L-1}\psi$ for $L\ge1$. Let $a=\|w\|_\infty/\min\psi$, so that $|w|\le a\psi$. Positivity gives $|T^Lw|\le T^L|w|\le aT^L\psi=ar_T^L\psi$, and therefore \[ Lr_T^{L-1}\psi\le|T^Lw|+r_T^L|w|\le2ar_T^L\psi. \] This gives $L\le2ar_T$ for every $L$, which is impossible. This contradiction shows that $\Pi$ has rank one, and $\Pi F=\ell'(F)\psi$ for some $\ell'\in C(\RP)^*$ with $\ell'(\psi)=1$. Finally, by \cite[Theorem III.6.26]{Kato1995perturbation}, the nonzero points of $\sigma(T)$ are eigenvalues and can accumulate only at $0$. By the first assertion above, every point of $\sigma(T)\setminus\{r_T\}$ has modulus less than $r_T$, and only finitely many of these points have modulus at least $r_T/2$. Then there exists $r'0$, we have $\lambda>0$, and $\lambda^3=r_T$. Let $X_1=\ker\Pi$. The decomposition $C(\RP)=\Ran\Pi\oplus X_1$ is invariant under $\mathcal Q_s$, and $\mathcal Q_s$ acts on $\Ran\Pi=\spn\{\psi\}$ as multiplication by $\lambda$. By the properties of Riesz projections \cite[Section III.6.4]{Kato1995perturbation}, we have $\sigma(T|_{X_1})=\sigma(T)\setminus\{r_T\}\subset\{|z|\le r'\}$. By the spectral mapping theorem for polynomials, $\sigma(T|_{X_1})=\{z^3:z\in\sigma(\mathcal Q_s|_{X_1})\}$, and we obtain $\sigma(\mathcal Q_s|_{X_1})\subset\{|z|\le(r')^{1/3}\}$. We set $r(s)=(r')^{1/3}<\lambda$. Then $\sigma(\mathcal Q_s)=\{\lambda\}\cup\sigma(\mathcal Q_s|_{X_1})$, the point $\lambda$ is isolated in $\sigma(\mathcal Q_s)$, and its Riesz projection is the projection onto $\Ran\Pi$ along $X_1$, which is $\Pi$; this has rank one. The spectral radius of $\mathcal Q_s$ is therefore $\lambda$, that is, $\lambda=\lambda(s)$. Setting $\psi_s=\psi$ proves (ii). \proofstep{4}{Part (iii).} Let $N=\mathcal Q_s(I-\Pi)$. Then $\mathcal Q_s=\lambda(s)\Pi+N$ and $\Pi N=N\Pi=0$, which give $\mathcal Q_s^L=\lambda(s)^L\Pi+N^L$ for $L\ge1$. The spectral radius of $N$ equals that of $\mathcal Q_s|_{X_1}$, which is at most $r(s)$. By Gelfand's formula, we have $\|N^L\|^{1/L}\to r(N)\le r(s)<\lambda(s)$. This gives $\|\lambda(s)^{-L}\mathcal Q_s^L-\Pi\|=\|(N/\lambda(s))^L\|\to0$. In particular, $\Pi$ is positive, being a norm limit of positive operators. For real $F\ge0$, we have $\ell'(F)\psi=\Pi F\ge0$. Then $\ell'$ is a positive linear functional, and by the Riesz--Markov theorem it is given by integration against a finite positive Borel measure $\ell_s$. We have $\int\psi_s\,\dd\ell_s=\ell'(\psi)=1$, and $\Pi\mathcal Q_s=\lambda(s)\Pi$ gives $\int\mathcal Q_sF\,\dd\ell_s=\lambda(s)\int F\,\dd\ell_s$. For full support, let $F\ge0$ be real, continuous, and nonzero. By Step 1, there exists $c>0$ with $\mathcal Q_s^3F\ge c\psi_s$, and positivity of $\Pi$ gives \[ \lambda(s)^3\Big(\int F\,\dd\ell_s\Big)\psi_s=\Pi\mathcal Q_s^3F\ge c\,\Pi\psi_s=c\,\psi_s. \] Since $\psi_s>0$, we have $\int F\,\dd\ell_s>0$. Applying this to a nonzero continuous $F\ge0$ supported in a given nonempty open set shows that $\ell_s$ has full support. \proofstep{5}{Part (iv).} Fix $s_0\in(0,2)$. By Step 3 and Step 4, the operator $T_0=\mathcal Q_{s_0}$ satisfies the hypotheses of \Cref{lem:press-perturbation} with $\mu_0=\lambda(s_0)$, $r_0=r(s_0)$, $\psi=\psi_{s_0}$, and $\ell=\ell_{s_0}$; here $\ell(\Pi_0\psi)=\int\psi_{s_0}\,\dd\ell_{s_0}=1$, since the Riesz projection $\Pi_0$ of $\mathcal Q_{s_0}$ at $\lambda(s_0)$ is $F\mapsto(\int F\,\dd\ell_{s_0})\psi_{s_0}$ by Steps 3 and 4. Let $\eps>0$ be as in that lemma. By \Cref{lem:press-bounded}(iii), $z\mapsto\mathcal Q_z$ is holomorphic, in particular continuous, and there exists an open disc $U\subset\Sigma$ centered at $s_0$ with $\|\mathcal Q_z-\mathcal Q_{s_0}\|<\eps$ for $z\in U$. By \Cref{lem:press-perturbation}(iii), $z\mapsto\mu(\mathcal Q_z)$ is holomorphic on $U$. For real $s\in U$, the spectral radius $\lambda(s)$ belongs to $\sigma(\mathcal Q_s)$ by (ii) and has maximal modulus in $\sigma(\mathcal Q_s)$, and \Cref{lem:press-perturbation}(i) implies that $\lambda(s)=\mu(\mathcal Q_s)$. Then $\lambda$ coincides on $U\cap\R$ with a holomorphic function, which proves that $\lambda$ is real-analytic near $s_0$. \proofstep{6}{Part (v).} We prove \eqref{eq:press-product-formula} by induction on $L$; the case $L=1$ is the definition of $\mathcal Q_s$. Suppose \eqref{eq:press-product-formula} holds for $L$, and let $M_1,\dots,M_{L+1}$ be independent copies of $M(B,\beta)$. Then $\mathcal Q_s^{L+1}F(x)=p\,\E\big[\|M_1v_x\|^{-s}(\mathcal Q_s^LF)(M_1\cdot x)\big]$. We apply the induction hypothesis to $\mathcal Q_s^LF$ with the matrices $M_2,\dots,M_{L+1}$, which are independent of $M_1$, and use Fubini's theorem. Fubini's theorem applies because, by Tonelli's theorem and the same computation with $|F|$ in place of $F$, the absolute value of the integrand has expectation at most $p^{-L-1}\mathcal Q_s^{L+1}|F|(x)\le p^{-L-1}\|\mathcal Q_s\|^{L+1}\|F\|_\infty$. We obtain \[ \mathcal Q_s^{L+1}F(x)=p^{L+1}\,\E\Big[\|M_1v_x\|^{-s}\,\big\|M_{L+1}\cdots M_2v_{M_1\cdot x}\big\|^{-s}F\big((M_{L+1}\cdots M_2)\cdot(M_1\cdot x)\big)\Big]. \] By \Cref{lem:press-projective}(i), we have $\|M_{L+1}\cdots M_2v_{M_1\cdot x}\|\,\|M_1v_x\|=\|M_{L+1}\cdots M_1v_x\|$ and $(M_{L+1}\cdots M_2)\cdot(M_1\cdot x)=(M_{L+1}\cdots M_1)\cdot x$. This completes the induction. Taking $F=\one$ in \eqref{eq:press-product-formula}, integrating against $m$, and using Tonelli's theorem, we obtain $\int\mathcal Q_s^L\one\,\dd m=p^L\,\E J_s(M_L\cdots M_1)$. Let $a=\min\psi_s>0$ and $A=\max\psi_s$. Since $\mathcal Q_s^L$ is positive and $a\le\psi_s\le A$, we have \[ \frac aA\,\lambda(s)^L\le\frac1A\,\mathcal Q_s^L\psi_s\le\mathcal Q_s^L\one\le\frac1a\,\mathcal Q_s^L\psi_s\le\frac Aa\,\lambda(s)^L \] pointwise on $\RP$. Integrating against the probability measure $m$ gives \eqref{eq:press-angular-growth} with $C=A/a$. Taking logarithms, dividing by $L$, and letting $L\to\infty$ gives the last statement. \end{proof} \subsection{Continuity in the law of the denominator} We now show that the Perron--Frobenius eigenvalue depends continuously on the law of $B$ and on the rate of $\beta$, under weak convergence of the laws and a uniform moment bound. Since atoms of $\nu$ produce atoms in the kernel measures of $\mathcal Q_s$ (see \Cref{rem:press-third-power}), the operator $\mathcal Q_s$ itself is not norm-continuous in this sense in general, and we work with the third power. \begin{proposition}\label{prop:press-continuity} Let $I\subset\R$ be an interval. For each $E\in I$, let $\nu_E$ be a Borel probability measure on $\R$ satisfying \eqref{eq:press-hyp}, and let $K(E)>0$. Suppose that $E\mapsto K(E)$ is continuous, that $E\mapsto\nu_E$ is continuous for the topology of weak convergence, and that there exists $q_0\in(0,1)$ such that \[ \sup_{E\in I'}\int|b|^{q_0}\,\nu_E(\dd b)<\infty\qquad\text{for every compact }I'\subset I . \] For $E\in I$ and $s\in(0,2)$, let $\mathcal Q_{E,s}$ and $\lambda_E(s)$ be the transfer operator and its Perron--Frobenius eigenvalue from \Cref{thm:press-perron}, defined with the law $\nu_E$ and the rate $K(E)$. Then for every $s\in(0,2q_0)$, the map $E\mapsto\mathcal Q_{E,s}^3$ is continuous from $I$ to the bounded operators on $C(\RP)$ in operator norm, and $E\mapsto\lambda_E(s)$ is continuous on $I$. In particular, if the moment bound holds for some $q_0>1/2$, then $E\mapsto\lambda_E(1)$ is continuous on $I$, and if it holds for every $q_0\in(0,1)$, then $E\mapsto\lambda_E(s)$ is continuous on $I$ for every $s\in(0,2)$. \end{proposition} \begin{proof} The proof repeats the total variation argument of \Cref{lem:press-tv}, with the law and the input varying together, on a probability space on which the denominators converge almost surely. Fix $s\in(0,2q_0)$, $E_0\in I$, and a sequence $E_n\to E_0$ in $I$. The set $I'=\{E_0\}\cup\{E_n:n\ge1\}$ is a compact subset of $I$. We write $K_n=K(E_n)$ for $n\ge0$, and let $k_-=\inf_nK_n>0$ and $k_+=\sup_nK_n<\infty$. Fix $t\in(s,2q_0)$. In the following, $n$ ranges over $\{0,1,2,\dots\}$, and the index $0$ refers to $E_0$. \proofstep{1}{Coupling and uniform bounds.} By the Skorokhod representation theorem \cite[Theorem 6.7]{Billingsley1999convergence}, there exist random variables $B_n$ with law $\nu_{E_n}$, defined on one probability space, such that $B_n\to B_0$ almost surely. Enlarging the space if necessary, let $Y$ be an $\Exp(1)$ random variable independent of $(B_n)_{n\ge0}$, and set $\beta_n=Y/K_n$. Then $(B_n,\beta_n)$ has law $\nu_{E_n}\otimes\Exp(K_n)$ for each $n$, and $\beta_n\to\beta_0$ almost surely. For $\sigma\in\{s,t\}$, by \Cref{lem:press-row-moment}, we have \begin{equation}\label{eq:press-cont-uniform} m_\sigma=\sup_{n\ge0}\sup_{x\in\RP}\E\big[R_\sigma(x;B_n,\beta_n)\big]\le\sup_{n\ge0}C_{\sigma,K_n}\big(2+\E|B_n|^{q_0}\big)<\infty, \end{equation} where we used $|b|^{\sigma/2}\le1+|b|^{q_0}$ (since $\sigma/21$, every family of random variables of the form $R_s(x_n;B_n,\beta_n)$ with $x_n\in\RP$ is bounded in $L^{t/s}$, and in particular uniformly integrable. Let $\kappa_n^{(k)}(x,\cdot)$ denote the $k$-step kernel measure of $\mathcal Q_{E_n,s}$ at $x$. By \eqref{eq:press-row-mass}, we have $\kappa_n^{(k)}(x,\RP)\le(p\,m_s)^k$ for all $n,k,x$. We say that the $k$-step kernel measures are jointly continuous at $y\in\RP$ if $\|\kappa_n^{(k)}(y_n,\cdot)-\kappa_0^{(k)}(y,\cdot)\|_{\TV}\to0$ for every sequence $y_n\to y$ in $\RP$. The maps $(b,\beta,x)\mapsto\|M(b,\beta)v_x\|^{-s}$ and $(b,\beta,x)\mapsto f_{b,\beta}(x)$ are continuous on $\R\times(0,\infty)\times\RP$, because $M(b,\beta)$ depends continuously on $(b,\beta)$ and $x\mapsto v_x$ is continuous up to sign. \proofstep{2}{The one-step kernel measures are jointly continuous at every $y\in\R\setminus\{0\}$.} For $K>0$, let $\phi_K(u)=u^{s/2}e^{-Ku}\one_{\{u>0\}}$. The map $(K,u)\mapsto\phi_K(u)$ is continuous on $(0,\infty)\times\R$, and $\phi_K\le\sup_{u>0}u^{s/2}e^{-k_-u}<\infty$ for $K\ge k_-$. Let $y_n\to y\in\R\setminus\{0\}$; then $y_n\in\R\setminus\{0\}$ for large $n$. By \Cref{lem:press-tv}(i), applied to $\nu_{E_n}$ and $K_n$, the kernel measure $\kappa_n^{(1)}(y_n,\cdot)$ is the image under $\iota$ of the measure with density \[ k_n(w)=\frac{pK_n}{|y_n|}\,(1+y_n^2)^{s/2}(1+w^2)^{-s/2}\,\E\Big[\phi_{K_n}\Big(\frac{B_n-w}{y_n}\Big)\Big]. \] For each $w$, the random variables $\phi_{K_n}((B_n-w)/y_n)$ converge almost surely to $\phi_{K_0}((B_0-w)/y)$ and are uniformly bounded. By dominated convergence, $k_n(w)$ converges to the density $k_\infty(w)$ of $\kappa_0^{(1)}(y,\cdot)$. Further, $\int k_n=p\,\E R_s(y_n;B_n,\beta_n)\to p\,\E R_s(y;B_0,\beta_0)=\int k_\infty$, by almost sure convergence and uniform integrability (Step 1). Scheff\'e's lemma, in the form proved in \Cref{lem:press-tv}(ii), gives $\int|k_n-k_\infty|\to0$, which is the claim. \proofstep{3}{Propagation.} We claim that if $k\ge1$, $x\in\RP$, and the $k$-step kernel measures are jointly continuous at $f_{B_0,\beta_0}(x)$ almost surely, then the $(k+1)$-step kernel measures are jointly continuous at $x$. Let $x_n\to x$, and write $R^{(n)}=R_s(x_n;B_n,\beta_n)$ and $y_n=f_{B_n,\beta_n}(x_n)$ for $n\ge1$, and $R=R_s(x;B_0,\beta_0)$ and $y=f_{B_0,\beta_0}(x)$. Since \eqref{eq:press-row-first-step} is an identity between expectations over the law of $(B_n,\beta_n)$, it holds on the coupled space, and we obtain \begin{multline*} \big\|\kappa_n^{(k+1)}(x_n,\cdot)-\kappa_0^{(k+1)}(x,\cdot)\big\|_{\TV}\\ \le p\,(p\,m_s)^k\,\E\big|R^{(n)}-R\big|+p\,\E\Big[R\,\big\|\kappa_n^{(k)}(y_n,\cdot)-\kappa_0^{(k)}(y,\cdot)\big\|_{\TV}\Big]. \end{multline*} By Step 1, $R^{(n)}\to R$ and $y_n\to y$ almost surely. The first term tends to zero by uniform integrability. The integrand in the second term is measurable, by the countable supremum used in the proof of \Cref{lem:press-tv}. For almost every outcome, the hypothesis applies to the sequence $y_n\to y$, and this integrand tends to zero; it is bounded by $2(p\,m_s)^kR$, which is integrable. Dominated convergence completes the proof of the claim. \proofstep{4}{The three-step kernel measures are jointly continuous at every $x\in\RP$.} As in the proof of \Cref{lem:press-tv}, for $x\in\R$ the point $f_{B_0,\beta_0}(x)$ lies in $\R\setminus\{0\}$ almost surely. By Step 2 and Step 3 with $k=1$, the two-step kernel measures are jointly continuous at every $x\in\R$. For $x\in\R$, the point $f_{B_0,\beta_0}(x)$ lies in $\R$ almost surely, and $f_{B_0,\beta_0}(\infty)=0$. Step 3 with $k=2$ implies the claim. \proofstep{5}{Norm continuity of the third power.} We show that $\sup_{x\in\RP}\|\kappa_n^{(3)}(x,\cdot)-\kappa_0^{(3)}(x,\cdot)\|_{\TV}\to0$. If not, then there exist $\eta>0$, a subsequence (which we do not relabel), and points $x_n$ with $\|\kappa_n^{(3)}(x_n,\cdot)-\kappa_0^{(3)}(x_n,\cdot)\|_{\TV}\ge\eta$. By compactness of $\RP$, we may pass to a further subsequence with $x_n\to x$. Steps 1--4 apply to every sequence converging to $E_0$, in particular to this subsequence, and they give $\|\kappa_n^{(3)}(x_n,\cdot)-\kappa_0^{(3)}(x,\cdot)\|_{\TV}\to0$. By \Cref{lem:press-tv}(iv) for $\nu_{E_0}$, we also have $\|\kappa_0^{(3)}(x_n,\cdot)-\kappa_0^{(3)}(x,\cdot)\|_{\TV}\to0$. This contradicts the choice of $x_n$. Since $|\mathcal Q_{E_n,s}^3F(x)-\mathcal Q_{E_0,s}^3F(x)|\le\|F\|_\infty\|\kappa_n^{(3)}(x,\cdot)-\kappa_0^{(3)}(x,\cdot)\|_{\TV}$, we conclude that $\|\mathcal Q_{E_n,s}^3-\mathcal Q_{E_0,s}^3\|\to0$. \proofstep{6}{Continuity of the eigenvalue.} Let $T_n=\mathcal Q_{E_n,s}^3$. By Step 2 of the proof of \Cref{thm:press-perron}, applied to $\nu_{E_0}$ and $K(E_0)$, the point $\mu_0=\lambda_{E_0}(s)^3$ is isolated in $\sigma(T_0)$, its Riesz projection $\Pi$ has rank one, there exists $r'<\mu_0$ with $|z|\le r'$ for every $z\in\sigma(T_0)\setminus\{\mu_0\}$, and $\Pi\psi=\psi$ for $\psi=\psi_{E_0,s}$. Let $\ell\in C(\RP)^*$ be the functional with $\Pi F=\ell(F)\psi$, so that $\ell(\Pi\psi)=1$. Fix $\delta\in(0,(\mu_0-r')/3)$ and let $\eps$ be as in \Cref{lem:press-perturbation}. By Step 5, $\|T_n-T_0\|<\eps$ for all large $n$. For such $n$, \Cref{lem:press-perturbation}(i) shows that $\sigma(T_n)$ has a unique point of maximal modulus, and that this point lies within distance $\delta$ of $\mu_0$. By \Cref{thm:press-perron}(ii) for $\nu_{E_n}$ and the spectral mapping theorem, the spectral radius of $T_n$ is $\lambda_{E_n}(s)^3$ and belongs to $\sigma(T_n)$; it is therefore this point, and $|\lambda_{E_n}(s)^3-\lambda_{E_0}(s)^3|<\delta$ for all large $n$. Since $\delta$ can be taken arbitrarily small, $\lambda_{E_n}(s)\to\lambda_{E_0}(s)$. This completes the proof. \end{proof} \section{The free energy}\label{sec:press-pressure} \subsection{Independence of the scaling and the definition of the free energy} The transfer matrices of a real solution depend on the choice of scaling $g>0$. The next lemma shows that different scalings give similar transfer operators. \begin{lemma}[Change of scaling]\label{lem:press-gauge} Let $E>0$ and let $\pi$ be a real solution at energy $E$. For $g>0$, let $\nu_g$ be the law of $gD$, where $D$ has law $\pi$, and let $K_g=p/g^2$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $g>0$, the law $\nu_g$ satisfies \eqref{eq:press-hyp}, and $\int|b|^{-q}\,\nu_g(\dd b)<\infty$ for every $q\in(0,1)$. \item Let $s\in(0,2)$, and for $g>0$ let $\mathcal Q_s^{(g)}$ be the transfer operator defined with $\nu_g$ and $K_g$. For $g,g'>0$, set $S=\operatorname{diag}(c^{-1/2},c^{1/2})$ with $c=g'/g$, and define $(\Theta F)(y)=\|Sv_y\|^{-s}F(S\cdot y)$. Then $\Theta$ is a bounded invertible operator on $C(\RP)$, both $\Theta$ and $\Theta^{-1}$ are positive, and \[ \mathcal Q_s^{(g')}=\Theta^{-1}\mathcal Q_s^{(g)}\Theta . \] In particular, the spectral radius of $\mathcal Q_s^{(g)}$ does not depend on $g$. \end{enumerate} \end{lemma} \begin{proof} For (i), \Cref{prop:rde-properties} states that $\pi$ has full support, that its only atom is at $E\neq0$, and that $\E|D|^{\pm q}<\infty$ for $q\in(0,1)$; these properties are preserved by the scaling $D\mapsto gD$. For (ii), let $Y\sim\Exp(1)$ be independent of $D$ and set $\alpha=Y/p$. Then $(gD,g^2\alpha)$ has law $\nu_g\otimes\Exp(K_g)$, and $\mathcal Q_s^{(g)}$ is defined by the random matrix $M=M(gD,g^2\alpha)$, and similarly $\mathcal Q_s^{(g')}$ is defined by $\widetilde M=M(g'D,g'^2\alpha)$. A direct computation from \eqref{eq:press-M} shows that for every $d\in\R$ and $a,\gamma>0$, we have \begin{equation}\label{eq:press-gauge-conj} M(\gamma d,\gamma^2a)=S_\gamma M(d,a)S_\gamma^{-1},\qquad S_\gamma=\operatorname{diag}(\gamma^{-1/2},\gamma^{1/2}). \end{equation} Indeed, the $(1,2)$, $(2,1)$, and $(2,2)$ entries of both sides are $\gamma^{-1}a^{-1/2}$, $-\gamma a^{1/2}$, and $a^{-1/2}d$, respectively, and the $(1,1)$ entries vanish. Since $S_{g'}=S_cS_g$, \eqref{eq:press-gauge-conj} gives $\widetilde M=SMS^{-1}$ with $S=S_c$. The map $y\mapsto S\cdot y$ is a homeomorphism of $\RP$ and $y\mapsto\|Sv_y\|^{-s}$ is continuous and positive. Then $\Theta$ is bounded and positive, with positive inverse $(\Theta^{-1}G)(x)=\|Sv_{S^{-1}\cdot x}\|^{s}G(S^{-1}\cdot x)$. Fix $y\in\RP$ and a realization of $M$. By \Cref{lem:press-projective}(i), we have $\widetilde M\cdot(S\cdot y)=S\cdot(M\cdot y)$ and \[ \|\widetilde Mv_{S\cdot y}\|=\|SMv_y\|\,\|S^{-1}v_{S\cdot y}\|=\|Sv_{M\cdot y}\|\,\|Mv_y\|\,\|Sv_y\|^{-1}, \] where in the last step we used $\|S^{-1}v_{S\cdot y}\|\,\|Sv_y\|=\|S^{-1}Sv_y\|=1$. For $F\in C(\RP)$, we obtain \[ \big(\mathcal Q_s^{(g')}F\big)(S\cdot y)=\|Sv_y\|^{s}\,p\,\E\Big[\|Mv_y\|^{-s}\,\|Sv_{M\cdot y}\|^{-s}F\big(S\cdot(M\cdot y)\big)\Big]=\|Sv_y\|^{s}\,\big(\mathcal Q_s^{(g)}\Theta F\big)(y). \] Multiplying by $\|Sv_y\|^{-s}$, we find $\Theta\mathcal Q_s^{(g')}F=\mathcal Q_s^{(g)}\Theta F$, which is the claimed identity. Since similar operators have the same spectrum, this implies the last statement. \end{proof} \begin{definition}\label{def:press-pressure} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $\lambda(s)$ be the Perron--Frobenius eigenvalue of $\mathcal Q_s$ from \Cref{thm:press-perron}. \begin{enumerate}[label=\textup{(\roman*)}] \item The free energy is $\varphi(s)=\log\lambda(s)$ for $s\in(0,2)$. The name follows \cite{AizenmanWarzel2013resonant,AggarwalLopatto2025mobility}, where the same quantity, up to normalization, is called the free energy. \item Let $M_1,M_2,\dots$ be independent copies of $M(B,\beta)$. For $s\in(0,2)$ and $L\ge0$, we set $f_L(s)=\E\|M_L\cdots M_1\|^{-s}$, with $f_0(s)=1$. \item Let $\pi$ be a real solution at energy $E>0$. The free energy of $\pi$ is $\varphi_\pi(s)=\log\lambda(s)$, where $\lambda(s)$ is computed with the law $\nu_g$ and the rate $K_g=p/g^2$ of \Cref{lem:press-gauge} for some $g>0$; by \Cref{lem:press-gauge}(ii), it does not depend on $g$. When the real solution is determined by the energy, we also write $\varphi_E(s)$. We define $f_L(s)$ for $\pi$ as in (ii), with a fixed scaling. \end{enumerate} \end{definition} \subsection{Properties of the free energy} \begin{proposition}\label{prop:press-properties} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $\varphi$ be its free energy. \begin{enumerate}[label=\textup{(\roman*)}] \item The function $\varphi$ is real-analytic and convex on $(0,2)$, and $\varphi(s)=\varphi(2-s)$ for $s\in(0,2)$. \item We have $\varphi'(1)=0$, and $\varphi$ is nonincreasing on $(0,1]$ and nondecreasing on $[1,2)$. \item We have $0\le\E\log\|M(B,\beta)\|<\infty$, and for every $s\in(0,2)$, \[ \log p-s\,\E\log\|M(B,\beta)\|\le \varphi(s)\le\log p . \] In particular, $\lim_{s\downarrow0}\varphi(s)=\log p$. \item We have $\varphi(1)<\log p$. \item We have $\varphi'(s)<0$ for every $s\in(0,1)$. In particular, $\varphi$ is strictly decreasing on $(0,1]$, and $\varphi(s)<\log p$ for every $s\in(0,2)$. \end{enumerate} If $\pi$ is a real solution at energy $E>0$, then $\varphi_\pi$ is independent of the scaling, and \textup{(i)--(v)} hold for $\varphi_\pi$. \end{proposition} \begin{proof} Throughout the proof, $M_1,M_2,\dots$ are independent copies of $M=M(B,\beta)$, and we abbreviate $W_L=M_L\cdots M_1$ and $j_L(s)=\E J_s(W_L)$. By \Cref{thm:press-perron}(v), we have \begin{equation}\label{eq:press-P-limit} \varphi(s)=\log p+\lim_{L\to\infty}\frac1L\log j_L(s),\qquad s\in(0,2). \end{equation} \proofpart{Proof of (i).} Real-analyticity is \Cref{thm:press-perron}(iv). By \Cref{lem:press-angular}, we have $J_s(W_L)=J_{2-s}(W_L)$ for every realization. Then $j_L(s)=j_L(2-s)$, and \eqref{eq:press-P-limit} implies that $\varphi(s)=\varphi(2-s)$. For convexity, fix $L$, $s_0,s_1\in(0,2)$, and $\tau\in(0,1)$. Applying H\"older's inequality with exponents $1/(1-\tau)$ and $1/\tau$ to the integral over $m\otimes\P$ of $\|W_Lv_x\|^{-(1-\tau)s_0}\|W_Lv_x\|^{-\tau s_1}$, we obtain $j_L((1-\tau)s_0+\tau s_1)\le j_L(s_0)^{1-\tau}j_L(s_1)^\tau$. Then $s\mapsto L^{-1}\log j_L(s)$ is convex, and so is its pointwise limit $\varphi(s)-\log p$. \proofpart{Proof of (ii).} Differentiating $\varphi(s)=\varphi(2-s)$ gives $\varphi'(s)=-\varphi'(2-s)$, and taking $s=1$ gives $\varphi'(1)=0$. Since $\varphi$ is convex and differentiable, $\varphi'$ is nondecreasing. Then $\varphi'\le0$ on $(0,1]$ and $\varphi'\ge0$ on $[1,2)$. \proofpart{Proof of (iii).} Since $\det M=1$, the product of the two singular values of $M$ is one; in particular, $\|M\|\ge1$ and $\log\|M\|\ge0$. Bounding the operator norm by the Hilbert--Schmidt norm, we have $\|M(b,\beta)\|^2\le\beta^{-1}(1+\beta^2+b^2)$, and therefore \[ \log\|M(b,\beta)\|\le\frac12|\log\beta|+\frac12\log(1+\beta^2)+\frac12\log(1+b^2). \] The first two terms have finite expectation since $\beta$ is exponential. For the third, fix $q\in(0,1)$; since $\log y\le(2/q)y^{q/2}$ for $y\ge1$, we have $\log(1+B^2)\le(2/q)(1+B^2)^{q/2}\le(2/q)(1+|B|^q)$, which is integrable by \eqref{eq:press-hyp}. This proves $\E\log\|M\|<\infty$. By \Cref{lem:press-angular}, we have $J_s(W_L)\le1$. Then $j_L(s)\le1$, which implies $\varphi(s)\le\log p$ by \eqref{eq:press-P-limit}. For the lower bound, by Jensen's inequality for the probability measure $m$, we have \[ J_s(A)=\int\exp\big(-s\log\|Av_x\|\big)\,m(\dd x)\ge\exp\Big(-s\int\log\|Av_x\|\,m(\dd x)\Big)\ge\|A\|^{-s} \] for every $A$ with determinant one. Using Jensen's inequality again, and then $\log\|W_L\|\le\sum_{j=1}^L\log\|M_j\|$, we obtain \[ j_L(s)\ge\E\exp\big(-s\log\|W_L\|\big)\ge\exp\big(-s\,\E\log\|W_L\|\big)\ge\exp\big(-sL\,\E\log\|M\|\big). \] Inserting this into \eqref{eq:press-P-limit} gives the lower bound. Letting $s\downarrow0$ gives $\lim_{s\downarrow0}\varphi(s)=\log p$. \proofpart{Proof of (iv).} We work in the real Hilbert space $L^2(m)$ of real-valued functions, with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|_2$. For a real $2\times2$ matrix $A$ with determinant one, define $(U_AF)(x)=\|Av_x\|^{-1}F(A\cdot x)$. By \eqref{eq:press-jacobian} applied to $|F|^2$, $U_A$ is an isometry of $L^2(m)$. By \eqref{eq:press-cocycle}, we have $U_AU_{A'}=U_{A'A}$; indeed, \[ (U_AU_{A'}F)(x)=\|Av_x\|^{-1}\,\|A'v_{A\cdot x}\|^{-1}F\big(A'\cdot(A\cdot x)\big)=\|A'Av_x\|^{-1}F\big((A'A)\cdot x\big). \] Taking $A'=A^{-1}$ and then $A=A'^{-1}$, we have $U_{A^{-1}}U_A=U_AU_{A^{-1}}=U_I=I$, and $U_A$ is unitary with $U_A^{-1}=U_{A^{-1}}$. Let $\psi=\psi_1$ be the eigenfunction from \Cref{thm:press-perron}(ii) with $s=1$; it is continuous, real-valued, and bounded below by a positive constant. By definition, $\mathcal Q_1\psi(x)=p\,\E[(U_M\psi)(x)]$. By Tonelli's theorem (all integrands are nonnegative), the Cauchy--Schwarz inequality, and unitarity, we have \[ \lambda(1)\|\psi\|_2^2=\langle\psi,\mathcal Q_1\psi\rangle=p\,\E\langle\psi,U_M\psi\rangle\le p\,\E\big[\|\psi\|_2\|U_M\psi\|_2\big]=p\,\|\psi\|_2^2 . \] We have shown that $\lambda(1)\le p$. Suppose, for the sake of contradiction, that $\lambda(1)=p$. Then the display implies $\E\big[\|\psi\|_2^2-\langle\psi,U_M\psi\rangle\big]=0$. Since $\|U_M\psi\|_2=\|\psi\|_2$, we have $\|\psi\|_2^2-\langle\psi,U_M\psi\rangle=\|U_M\psi-\psi\|_2^2/2\ge0$, and therefore $\E\|U_M\psi-\psi\|_2^2=0$. The function $(b,\beta)\mapsto\|U_{M(b,\beta)}\psi-\psi\|_2^2$ is continuous on $\R\times(0,\infty)$, by dominated convergence: the integrand is continuous in $(b,\beta)$ for each $x$, and $\|M(b,\beta)v_x\|^{-1}\le\|M(b,\beta)^{-1}\|$ is bounded uniformly in $x$ and in $(b,\beta)$ in compact sets. This function vanishes $\nu\otimes\Exp(K)$-almost everywhere, and this measure charges every nonempty open subset of $\R\times(0,\infty)$ because $\supp\nu=\R$. Since the function is continuous, it vanishes identically. In particular, $U_{M(0,1)}\psi=\psi$ and $U_{M(0,4)}\psi=\psi$. A direct computation from \eqref{eq:press-M} gives \[ M(0,4)^{-1}M(0,1)=\begin{pmatrix}0&-1/2\\2&0\end{pmatrix}\begin{pmatrix}0&1\\-1&0\end{pmatrix}=\begin{pmatrix}1/2&0\\0&2\end{pmatrix}=\Delta . \] Then $U_\Delta=U_{M(0,1)}U_{M(0,4)^{-1}}=U_{M(0,1)}U_{M(0,4)}^{-1}$, and $U_\Delta\psi=\psi$ in $L^2(m)$. Since both sides are continuous and $m$ has full support, we have $\psi(x)=\|\Delta v_x\|^{-1}\psi(\Delta\cdot x)$ for every $x$. Let $\mu$ be the finite measure $\psi^2m$. For every Borel $F\ge0$, \eqref{eq:press-jacobian} with $A=\Delta$ gives \[ \int F(\Delta\cdot x)\,\mu(\dd x)=\int F(\Delta\cdot x)\,\psi(\Delta\cdot x)^2\,\|\Delta v_x\|^{-2}\,m(\dd x)=\int F(y)\,\mu(\dd y), \] so $\mu$ is invariant under the map $x\mapsto\Delta\cdot x$, which is $x\mapsto x/4$ in coordinates. For $k\in\Z$, let $I_k=[4^k,4^{k+1})$. The preimage of $I_k$ under $x\mapsto x/4$ is $I_{k+1}$, and invariance gives $\mu(I_{k+1})=\mu(I_k)$ for every $k$. The intervals $I_k$ are disjoint with union $(0,\infty)$, and $\mu((0,\infty))\le\mu(\RP)<\infty$, which forces $\mu(I_k)=0$ for every $k$ and $\mu((0,\infty))=0$. This contradicts $\mu((0,\infty))=\int_{(0,\infty)}\psi^2\,\dd m>0$, which holds because $\psi>0$ and $m((0,\infty))>0$. We conclude that $\lambda(1)0$ by \Cref{lem:press-gauge}(i), and $\varphi_\pi$ does not depend on $g$ by \Cref{lem:press-gauge}(ii). Applying (i)--(v) to $\nu_g$, we obtain the last statement. \end{proof} The symmetry $\varphi(s)=\varphi(2-s)$ is inherited from the identity $J_s(A)=J_{2-s}(A)$ of \Cref{lem:press-angular}. A symmetry of this kind for the linearized cavity equation goes back to Abou-Chacra, Anderson, and Thouless \cite{AbouChacraAndersonThouless1973}; see \cite{ParisiPascazioPietracaprinaRosScardicchio2020anderson} for a discussion. The proof of \Cref{prop:press-properties}(iv) uses only \eqref{eq:press-hyp}, and in particular it does not use atoms of $\nu$. The next lemma selects an exponent below one at which the free energy has a useful tangent line. \begin{lemma}\label{lem:press-gap} Suppose that $\nu$ satisfies \eqref{eq:press-hyp} and that $\varphi(1)>0$. Then there exists $s\in(1/2,1)$ such that \[ \gamma=-\varphi'(s)>0,\qquad F=\varphi(s)-(1-s)\gamma>0,\qquad \varphi(s)<\log p . \] In particular, this holds for the free energy $\varphi_\pi$ of a real solution $\pi$ with $0<\varphi_\pi(1)$. \end{lemma} \begin{proof} By \Cref{prop:press-properties}(v), we have $\varphi'(s)<0$ and $\varphi(s)<\log p$ for every $s\in(0,1)$. Since $\varphi$ is continuously differentiable on $(0,2)$ and $\varphi'(1)=0$, we have \[ \lim_{s\uparrow1}\big(\varphi(s)-(1-s)(-\varphi'(s))\big)=\lim_{s\uparrow1}\big(\varphi(s)+(1-s)\varphi'(s)\big)=\varphi(1)>0 . \] Choosing $s\in(1/2,1)$ sufficiently close to one gives all three inequalities. \end{proof} \section{Products of three independent factors}\label{sec:press-rows} In this section, we show that the product of three independent transfer matrices has bounded inverse fractional moments of its bilinear forms, uniformly in the vectors. Here and below, a factor is a random matrix $M(B,\beta)$ in which $\beta$ has an exponential law and is independent of $B$, as in \Cref{sec:press-perron}. The smoothing comes from the exponential variables $\beta_i$ alone; the variables $B_i$ enter only through their moments. We then use this bound to compare the free energy with moments of norms of products. \subsection{Inverse moments of multiaffine polynomials} A multiaffine polynomial in $n$ variables is a polynomial of the form $\mathfrak P(t)=\sum_{\alpha\subset\{1,\dots,n\}}c_\alpha\prod_{i\in\alpha}t_i$ with coefficients $c_\alpha\in\C$, that is, a polynomial of degree at most one in each variable. For $s\in(0,1)$ and $K>0$, let $\omega_{s,K}$ be the measure on $\R$ given by $\omega_{s,K}(\dd t)=Kt^{s/2}e^{-Kt}\one_{\{t>0\}}\,\dd t$. Its total mass is $\omega_{s,K}(\R)=K^{-s/2}\Gamma(1+s/2)$. \begin{lemma}\label{lem:press-multiaffine} Let $s\in(0,1)$, $K>0$, and $n\ge1$. There exists a constant $C_{s,K,n}<\infty$ such that for every nonzero multiaffine polynomial $\mathfrak P$ in $n$ variables with coefficients $c_\alpha$, \[ \int_{(0,\infty)^n}|\mathfrak P(t)|^{-s}\,\omega_{s,K}(\dd t_1)\cdots\omega_{s,K}(\dd t_n)\le C_{s,K,n}\Big(\max_\alpha|c_\alpha|\Big)^{-s}. \] One can take $C_{s,K,n}=C_{s,K,1}^n$, and $C_{s,K,1}$ is bounded for $(s,K)$ in compact subsets of $(0,1)\times(0,\infty)$. \end{lemma} \begin{proof} We argue by induction on $n$. \proofstep{1}{A translated singularity.} Let \[ A_{s,K}=\sup_{x_0\in\R}\int_0^\infty|t-x_0|^{-s}\,\omega_{s,K}(\dd t). \] Let $w_*=\sup_{t>0}Kt^{s/2}e^{-Kt}<\infty$. On $\{|t-x_0|<1\}$, we bound the density of $\omega_{s,K}$ by $w_*$, and $\int_{|t-x_0|<1}|t-x_0|^{-s}\,\dd t=2/(1-s)$. On $\{|t-x_0|\ge1\}$, we bound $|t-x_0|^{-s}$ by one. This gives $A_{s,K}\le2w_*/(1-s)+K^{-s/2}\Gamma(1+s/2)<\infty$. \proofstep{2}{The case $n=1$.} Let $\mathfrak P(t)=at+b$ with $(a,b)\in\C^2\setminus\{0\}$. Replacing $\mathfrak P$ by $\mathfrak P/\max(|a|,|b|)$ multiplies both sides of the claimed inequality by $\max(|a|,|b|)^s$, so we may assume that $\max(|a|,|b|)=1$. Suppose first that $|a|\ge1/2$. Then $|at+b|=|a|\,|t-z|$ with $z=-b/a$, and $|t-z|\ge|t-\Re z|$ for real $t$, and integrating gives $\int|\mathfrak P|^{-s}\,\dd\omega_{s,K}\le|a|^{-s}A_{s,K}\le2^sA_{s,K}$. Suppose next that $|a|<1/2$, so that $|b|=1$. Let $R=1/(2|a|)$, with $R=\infty$ if $a=0$. For $0R$, we write $|at+b|^{-s}=|a|^{-s}|t-z|^{-s}\le|a|^{-s}|t-\Re z|^{-s}$ and $e^{-Kt}\le e^{-KR/2}e^{-Kt/2}$. Since $Kt^{s/2}e^{-Kt/2}\,\dd t=2\,\omega_{s,K/2}(\dd t)$, the range $t>R$ contributes at most \[ |a|^{-s}e^{-KR/2}\cdot2A_{s,K/2}=2A_{s,K/2}\,|a|^{-s}e^{-K/(4|a|)}\le2A_{s,K/2}\Big(\frac{4s}{eK}\Big)^s, \] where we used $\sup_{y>0}y^{-s}e^{-K/(4y)}=(4s/(eK))^s$. We conclude that the lemma holds for $n=1$ with \[ C_{s,K,1}=\max\Big\{2^sA_{s,K},\ 2^sK^{-s/2}\Gamma(1+s/2)+2A_{s,K/2}\Big(\frac{4s}{eK}\Big)^s\Big\}. \] The bounds on $A_{s,K}$ in Step 1 are bounded for $(s,K)$ in compact subsets of $(0,1)\times(0,\infty)$, and hence so is $C_{s,K,1}$. \proofstep{3}{The induction step.} Let $n\ge2$ and suppose that the lemma holds for $n-1$ variables with the constant $C_{s,K,1}^{n-1}$. We write $t=(t',t_n)$ with $t'\in\R^{n-1}$. Every multiaffine polynomial $\mathfrak P$ in $n$ variables can be written uniquely as $\mathfrak P(t)=t_n\mathfrak P_1(t')+\mathfrak P_0(t')$, where $\mathfrak P_1$ and $\mathfrak P_0$ are multiaffine in $n-1$ variables; the coefficients of $\mathfrak P_1$ are the $c_\alpha$ with $n\in\alpha$, and those of $\mathfrak P_0$ are the $c_\alpha$ with $n\notin\alpha$. Let $\mathfrak Q\in\{\mathfrak P_1,\mathfrak P_0\}$ be a polynomial that has a coefficient of maximal modulus among all coefficients of $\mathfrak P$, so that $\mathfrak Q\neq0$ and $\max_\alpha|c_\alpha(\mathfrak Q)|=\max_\alpha|c_\alpha(\mathfrak P)|$. The zero set of the nonzero polynomial $\mathfrak Q$ in $\R^{n-1}$ has Lebesgue measure zero (its real or imaginary part is a nonzero real polynomial); in particular, it is null for $\omega_{s,K}^{\otimes(n-1)}$. For $t'$ with $\mathfrak Q(t')\neq0$, we have $(\mathfrak P_1(t'),\mathfrak P_0(t'))\neq(0,0)$, and the case $n=1$ applied to $t_n\mapsto t_n\mathfrak P_1(t')+\mathfrak P_0(t')$ gives \[ \int|\mathfrak P(t',t_n)|^{-s}\,\omega_{s,K}(\dd t_n)\le C_{s,K,1}\max\big(|\mathfrak P_1(t')|,|\mathfrak P_0(t')|\big)^{-s}\le C_{s,K,1}|\mathfrak Q(t')|^{-s}. \] Integrating over $t'$, and using Tonelli's theorem and the induction hypothesis for $\mathfrak Q$, we obtain \[ \int|\mathfrak P|^{-s}\,\dd\omega_{s,K}^{\otimes n}\le C_{s,K,1}\int|\mathfrak Q|^{-s}\,\dd\omega_{s,K}^{\otimes(n-1)}\le C_{s,K,1}^n\Big(\max_\alpha|c_\alpha(\mathfrak P)|\Big)^{-s}. \] This completes the induction. \end{proof} \subsection{Three independent factors} For $b\in\C$ and $\beta>0$, we define $M(b,\beta)$ by the formula \eqref{eq:press-M}; it is a complex $2\times2$ matrix with determinant one. For vectors $u,v\in\C^2$ and a matrix $G$, we write $u^{\mathsf T}Gv=\sum_{i,j}u_iG_{ij}v_j$ for the bilinear (not sesquilinear) pairing. \begin{proposition}\label{prop:press-three-rows} Let $s\in(0,1)$, $K>0$, and $a_+,a_-\ge0$. Let $B_1,B_2,B_3$ be independent real random variables such that \[ \E|B_i|^s\le a_+,\qquad\E|B_i|^{-s}\le a_-,\qquad i\in\{1,2,3\}, \] and let $\beta_1,\beta_2,\beta_3$ be independent random variables with the exponential law of rate $K$, independent of $(B_1,B_2,B_3)$. Set $M_i=M(B_i,\beta_i)$ and $G=M_3M_2M_1$, and let \[ C_*=2^sC_{s,K,3}\,(1+a_+)(1+2a_-), \] where $C_{s,K,3}$ is the constant of \Cref{lem:press-multiaffine}. Then the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For all nonzero $u,v\in\R^2$, we have $\E|u^{\mathsf T}Gv|^{-s}\le C_*\|u\|^{-s}\|v\|^{-s}$. \item For all nonzero real $2\times2$ matrices $A$ and $C$, we have $\E\|CGA\|^{-s}\le C_*\|C\|^{-s}\|A\|^{-s}$. \end{enumerate} The laws of $B_1,B_2,B_3$ need not coincide, and each may be any law with the stated moment bounds, including a law with atoms or a point mass at a nonzero constant. The same bounds hold for complex factors, in the sense of \Cref{prop:press-three-rows-complex}. \end{proposition} \begin{proof} Both sides of (i) are homogeneous of degree $-s$ in $u$ and in $v$, so we may assume that $\|u\|=\|v\|=1$. We write $u=(u_1,u_2)$ and $v=(v_1,v_2)$, and we let $N_i=\beta_i^{1/2}M_i=\begin{pmatrix}0&1\\-\beta_i&B_i\end{pmatrix}$. Multiplying out, we have \begin{align*} N_2N_1&=\begin{pmatrix}-\beta_1&B_1\\-B_2\beta_1&B_1B_2-\beta_2\end{pmatrix},\\ N_3N_2N_1&=\begin{pmatrix}-B_2\beta_1&B_1B_2-\beta_2\\\beta_1\beta_3-B_2B_3\beta_1&B_1B_2B_3-B_1\beta_3-B_3\beta_2\end{pmatrix}. \end{align*} We see that $\mathfrak P=u^{\mathsf T}N_3N_2N_1v$ is a multiaffine polynomial in $(\beta_1,\beta_2,\beta_3)$, with coefficients depending on $B_1,B_2,B_3,u,v$. Set \[ a=u_1+B_3u_2,\qquad b=u_2 . \] Collecting monomials, we find that the coefficients of $\beta_1\beta_3$, $\beta_1$, $\beta_2$, $\beta_3$, and $1$ are \[ c_{13}=bv_1,\qquad c_1=-B_2av_1,\qquad c_2=-av_2,\qquad c_3=-B_1bv_2,\qquad c_\emptyset=B_1B_2av_2, \] and the coefficients of $\beta_1\beta_2$, $\beta_2\beta_3$, and $\beta_1\beta_2\beta_3$ vanish. Let $d=\min(1,|B_1|,|B_2|)$. The four coefficients $c_{13},c_1,c_2,c_3$ are the products $bv_1,av_1,av_2,bv_2$, multiplied by $1,-B_2,-1,-B_1$, respectively. Since $d\le1$, $d\le|B_1|$, and $d\le|B_2|$, we find \begin{align*} \max_\alpha|c_\alpha|&\ge d\max\big(|bv_1|,|av_1|,|av_2|,|bv_2|\big)\\ &\ge\frac d2\Big(|bv_1|^2+|av_1|^2+|av_2|^2+|bv_2|^2\Big)^{1/2}=\frac d2\big(|a|^2+|b|^2\big)^{1/2}, \end{align*} where in the second inequality we bounded the maximum of four numbers below by half of their Euclidean norm, and in the last step we used $|v_1|^2+|v_2|^2=1$. Further, $(a,b)^{\mathsf T}=Su$ with $S=\begin{pmatrix}1&B_3\\0&1\end{pmatrix}$, and $S^{-1}=\begin{pmatrix}1&-B_3\\0&1\end{pmatrix}$ has norm at most $1+|B_3|$. Since $1=\|u\|\le\|S^{-1}\|\,\|Su\|\le(1+|B_3|)\|Su\|$, we have $(|a|^2+|b|^2)^{1/2}\ge(1+|B_3|)^{-1}$. We conclude that \begin{equation}\label{eq:press-coefficient-bound} \max_\alpha|c_\alpha|\ge\frac{d}{2(1+|B_3|)}. \end{equation} Since $u^{\mathsf T}Gv=(\beta_1\beta_2\beta_3)^{-1/2}\mathfrak P$, we have $|u^{\mathsf T}Gv|^{-s}=(\beta_1\beta_2\beta_3)^{s/2}|\mathfrak P|^{-s}$. The variables $\beta_1,\beta_2,\beta_3$ are independent of $(B_1,B_2,B_3)$ and have joint density $\prod_iKe^{-K\beta_i}$. By Fubini's theorem, the conditional expectation given $(B_1,B_2,B_3)$ is \begin{align*} \E\Big[|u^{\mathsf T}Gv|^{-s}\,\Big|\,B_1,B_2,B_3\Big]&=\int_{(0,\infty)^3}|\mathfrak P(\beta)|^{-s}\,\omega_{s,K}(\dd\beta_1)\,\omega_{s,K}(\dd\beta_2)\,\omega_{s,K}(\dd\beta_3)\\ &\le2^sC_{s,K,3}(1+|B_3|)^sd^{-s}, \end{align*} where we used \Cref{lem:press-multiaffine} and \eqref{eq:press-coefficient-bound}. We have $(1+|B_3|)^s\le1+|B_3|^s$ and $d^{-s}\le1+|B_1|^{-s}+|B_2|^{-s}$. Taking expectations and using the independence of $B_3$ and $(B_1,B_2)$, we obtain \[ \E|u^{\mathsf T}Gv|^{-s}\le2^sC_{s,K,3}\big(1+\E|B_3|^s\big)\big(1+\E|B_1|^{-s}+\E|B_2|^{-s}\big)\le C_* . \] This proves (i). For (ii), choose a unit vector $v_0$ with $\|Av_0\|=\|A\|$ and a unit vector $w_0$ with $\|C^{\mathsf T}w_0\|=\|C^{\mathsf T}\|=\|C\|$, and set $v=Av_0/\|A\|$ and $u=C^{\mathsf T}w_0/\|C\|$. These are unit vectors depending only on $A$ and $C$, and \[ \|CGA\|\ge\big|w_0^{\mathsf T}CGAv_0\big|=\big|(C^{\mathsf T}w_0)^{\mathsf T}G(Av_0)\big|=\|C\|\,\|A\|\,|u^{\mathsf T}Gv| . \] Raising to the power $-s$, taking expectations, and applying (i), we obtain (ii). \end{proof} The same argument applies to complex denominators, which arise when the spectral parameter has positive imaginary part. \begin{proposition}\label{prop:press-three-rows-complex} Let $s\in(0,1)$, $K>0$, and $a_+,a_-\ge0$. Let $B_1,B_2,B_3$ be independent complex random variables with $\E|B_i|^s\le a_+$ and $\E|B_i|^{-s}\le a_-$ for $i\in\{1,2,3\}$, let $\beta_1,\beta_2,\beta_3$ be as in \Cref{prop:press-three-rows}, and set $M_i=M(B_i,\beta_i)$ and $G=M_3M_2M_1$. Let $C_*$ be the constant of \Cref{prop:press-three-rows}. Then the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For all nonzero $u,v\in\C^2$, we have $\E|u^{\mathsf T}Gv|^{-s}\le C_*\|u\|^{-s}\|v\|^{-s}$. \item For all nonzero complex $2\times2$ matrices $A$ and $C$, we have $\E\|CGA\|^{-s}\le C_*\|C\|^{-s}\|A\|^{-s}$. \end{enumerate} In particular, the bounds hold with the same constant $C_*$ for every member of a family of triples $(B_1,B_2,B_3)$ of independent complex random variables satisfying the moment bounds with the same $a_+$ and $a_-$. \end{proposition} \begin{proof} For (i), the proof of \Cref{prop:press-three-rows}(i) applies with the following changes. The expansion of $u^{\mathsf T}N_3N_2N_1v$ is a polynomial identity and is unchanged, and $\mathfrak P$ is a multiaffine polynomial in $(\beta_1,\beta_2,\beta_3)$ with the complex coefficients displayed there. The identity $|bv_1|^2+|av_1|^2+|av_2|^2+|bv_2|^2=(|a|^2+|b|^2)(|v_1|^2+|v_2|^2)$ holds for complex numbers, the bound $\|S^{-1}\|\le1+|B_3|$ holds for the Hermitian operator norm, and \Cref{lem:press-multiaffine} allows complex coefficients. Then \eqref{eq:press-coefficient-bound} and the rest of the argument hold verbatim. For (ii), choose unit vectors $v_0,w_0\in\C^2$ with $\|Av_0\|=\|A\|$ and $\|C^*w_0\|=\|C\|$, where $C^*$ is the conjugate transpose. Set $v=Av_0/\|A\|$, and let $u=\overline{C^*w_0}/\|C\|$, the entrywise complex conjugate of $C^*w_0$ divided by $\|C\|$. Then $(C^*w_0)^*=\|C\|u^{\mathsf T}$, and \[ \|CGA\|\ge\big|w_0^*CGAv_0\big|=\big|(C^*w_0)^*G(Av_0)\big|=\|C\|\,\|A\|\,|u^{\mathsf T}Gv|. \] Part (i) completes the proof. The last statement holds because $C_*$ depends only on $s$, $K$, $a_+$, and $a_-$. \end{proof} \begin{remark}\label{rem:press-regulated} \Cref{prop:press-three-rows-complex} applies to the denominators of the tree at the spectral parameter $E+\iu\eta$ with $\eta>0$. Let $E>0$, $g>0$, and $0<\eta\le1$, and let $B_\eta=gD_\eta$ with $D_\eta=-1/\Gamma_o(E+\iu\eta)$. By \Cref{lem:rde-moments}, both $\sup_{0<\eta\le1}\E|B_\eta|^{-s}=g^{-s}\sup_{0<\eta\le1}\E|\Gamma_o(E+\iu\eta)|^s$ and $\sup_{0<\eta\le1}\E|B_\eta|^s=g^s\sup_{0<\eta\le1}\E|D_\eta|^s$ are finite for $s\in(0,1)$. Then the constant $C_*$ can be chosen uniformly in $\eta\in(0,1]$. \end{remark} \begin{corollary}\label{cor:press-buffers} Let $s\in(0,1)$, $K>0$, $a_+,a_-\ge0$, and let $C_*$ be as in \Cref{prop:press-three-rows}. Let $G$ and $G'$ be independent random matrices, each a product of three factors as in \Cref{prop:press-three-rows-complex} (with possibly different laws of the $B_i$, all satisfying the moment bounds with $a_+$ and $a_-$). \begin{enumerate}[label=\textup{(\roman*)}] \item For all nonzero $u,v\in\C^2$ and every nonzero complex $2\times2$ matrix $A$, we have \[ \E\big|u^{\mathsf T}GAG'v\big|^{-s}\le C_*^2\|u\|^{-s}\|A\|^{-s}\|v\|^{-s}. \] \item For all nonzero complex $2\times2$ matrices $A$, $C$, and $F$, we have \[ \E\|CGFG'A\|^{-s}\le C_*^2\|C\|^{-s}\|F\|^{-s}\|A\|^{-s}. \] If $\|F\|\ge1$, which holds in particular if $|\det F|\ge1$, the right side is at most $C_*^2\|C\|^{-s}\|A\|^{-s}$. \item Let $\mathcal G$ be a $\sigma$-algebra independent of $(G,G')$, and let $u,v,A,C,F$ be $\mathcal G$-measurable random vectors and matrices that are almost surely nonzero. Then the bounds in \textup{(i)} and \textup{(ii)} hold almost surely with $\E$ replaced by $\E[\,\cdot\,|\,\mathcal G]$. \end{enumerate} \end{corollary} \begin{proof} For a nonzero complex matrix $A$, choose a unit vector $w_0$ with $\|A^*w_0\|=\|A\|$, and let $u_A=\overline{A^*w_0}/\|A\|$. As in the proof of \Cref{prop:press-three-rows-complex}(ii), we have \begin{equation}\label{eq:press-left-vector} \|Ax\|\ge|w_0^*Ax|=\|A\|\,|u_A^{\mathsf T}x|\qquad\text{for every }x\in\C^2 . \end{equation} For (i), we condition on $G'$; since $G$ is independent of $G'$, \Cref{prop:press-three-rows-complex}(i) applied to the vectors $u$ and $AG'v$ gives $\E[|u^{\mathsf T}GAG'v|^{-s}\,|\,G']\le C_*\|u\|^{-s}\|AG'v\|^{-s}$ (both sides are infinite if $AG'v=0$). By \eqref{eq:press-left-vector}, we have $\|AG'v\|^{-s}\le\|A\|^{-s}|u_A^{\mathsf T}G'v|^{-s}$, and \Cref{prop:press-three-rows-complex}(i) implies $\E|u_A^{\mathsf T}G'v|^{-s}\le C_*\|v\|^{-s}$. Combining these bounds proves (i). For (ii), we condition on $G'$ and apply \Cref{prop:press-three-rows-complex}(ii) to $G$ with the matrices $C$ and $FG'A$, which gives $\E[\|CGFG'A\|^{-s}\,|\,G']\le C_*\|C\|^{-s}\|FG'A\|^{-s}$. Applying \Cref{prop:press-three-rows-complex}(ii) to $G'$ with the matrices $F$ and $A$ gives $\E\|FG'A\|^{-s}\le C_*\|F\|^{-s}\|A\|^{-s}$. If $|\det F|\ge1$, then $\|F\|^2\ge|\det F|\ge1$, since $|\det F|$ is the product of the two singular values of $F$. For (iii), since $(G,G')$ is independent of $\mathcal G$, the conditional expectation given $\mathcal G$ of a nonnegative measurable function of $(G,G')$ and of $\mathcal G$-measurable variables is obtained by integrating over $(G,G')$ with the $\mathcal G$-measurable variables held fixed \cite[Chapter 6]{Kallenberg2002foundations}. Applying (i) and (ii) for fixed values of $u,v,A,C,F$ gives (iii). \end{proof} \subsection{Moments of norms of products} \begin{corollary}\label{cor:press-norm-moments} Let $s\in(0,1)$. Suppose that $\nu$ satisfies \eqref{eq:press-hyp} and $\int|b|^{-s}\,\nu(\dd b)<\infty$, and let $f_L(s)=\E\|M_L\cdots M_1\|^{-s}$ be as in \Cref{def:press-pressure}. Then the limit $\varrho_s=\lim_{L\to\infty}f_L(s)^{1/L}$ exists, and $\varrho_s=\lambda(s)/p=e^{\varphi(s)-\log p}\in(0,1)$. Further, let $C_*$ be the constant of \Cref{prop:press-three-rows} with $a_+=\E|B|^s$ and $a_-=\E|B|^{-s}$. Then for every $L\ge1$, \[ C_*^{-1}\varrho_s^{L+3}\le f_L(s)\le\varrho_s^L . \] \end{corollary} \begin{proof} We abbreviate $f_L=f_L(s)$ and $W_L=M_L\cdots M_1$. Since $\det M_j=1$, every product has norm at least one, and $00$, and that $f_L\le\varrho_s^L$. By \Cref{lem:press-norm-angular}, we have $f_L\le\E J_s(W_L)\le C_sf_L$, with $C_s$ independent of $L$. Then \Cref{thm:press-perron}(v) implies that $\log\varrho_s=\log\lambda(s)-\log p=\varphi(s)-\log p$, and $\varphi(s)<\log p$ by \Cref{prop:press-properties}(v). For the lower bound, let $L,N\ge1$, and write $W_{L+N+3}=C'GA$ with $A=W_L$, $G=M_{L+3}M_{L+2}M_{L+1}$, and $C'=M_{L+N+3}\cdots M_{L+4}$. The matrix $G$ is a product of three independent factors as in \Cref{prop:press-three-rows}, with $a_+=\E|B|^s$ and $a_-=\E|B|^{-s}$, and it is independent of $(A,C')$. Since $G$ is independent of $(A,C')$, we may apply \Cref{prop:press-three-rows}(ii) conditionally on $(A,C')$, integrating over $G$ with $A$ and $C'$ held fixed (as in the proof of \Cref{cor:press-buffers}(iii)), and we obtain $\E[\|C'GA\|^{-s}\,|\,A,C']\le C_*\|C'\|^{-s}\|A\|^{-s}$. Taking expectations and using the independence of $A$ and $C'$, where $C'$ has the law of $W_N$, we obtain \[ f_{L+N+3}\le C_*f_Nf_L . \] By induction on $k\ge1$, this gives $f_{k(L+3)-3}\le C_*^{k-1}f_L^k$. Since $n^{-1}\log f_n\to\log\varrho_s$, taking logarithms and dividing by $k$ gives \[ \frac{k(L+3)-3}{k}\cdot\frac{\log f_{k(L+3)-3}}{k(L+3)-3}\le\frac{k-1}k\log C_*+\log f_L . \] Letting $k\to\infty$, we find $(L+3)\log\varrho_s\le\log C_*+\log f_L$, which is the lower bound. \end{proof} \section{Path moments}\label{sec:press-paths} In this section, we identify the free energy with the growth rate of the path moments $\Phi_L(s)$, which are iterates of the transfer operator $\mathcal A_s$ in the $x$-coordinate. We begin by writing the iterates of $\mathcal A_s$ along a chain of independent factors. \begin{lemma}\label{lem:press-chain} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $s\in(0,2)$ and $L\ge1$. \begin{enumerate}[label=\textup{(\roman*)}] \item Let $B'_0,\dots,B'_{L-1}$ have law $\nu$ and $\beta_1,\dots,\beta_L$ have the exponential law of rate $K$, all independent. For $x\in\R$, set $X_L=x$ and $X_{j-1}=f_{B'_{j-1},\beta_j}(X_j)$ for $j=L,\dots,1$. Then almost surely $X_j\in\R$ for every $j$, and for every Borel $F\colon\R\to[0,\infty]$, \begin{equation}\label{eq:press-chain} (\mathcal A_s^LF)(x)=p^L\,\E\bigg[\prod_{j=1}^L\beta_j^{s/2}|X_j|^s\,F(X_0)\bigg]. \end{equation} \item Let $\pi$ be a real solution at energy $E>0$, and for $g>0$ let $\mathcal A_s^{(g)}$ be the transfer operator in the $x$-coordinate defined with $\nu_g$ and $K_g=p/g^2$, and let $\pi_g^x$ be the law of $1/(gD)$, where $D$ has law $\pi$. Let $D_L,D'_0,\dots,D'_{L-1}$ have law $\pi$ and $Y_1,\dots,Y_L$ have law $\Exp(1)$, all independent, and set $D_{j-1}=D'_{j-1}-Y_j/(pD_j)$ for $j=L,\dots,1$. Then almost surely $D_j\in\R\setminus\{0\}$ for every $j$, and for every $g>0$, \begin{equation}\label{eq:press-chain-D} g^s\int\mathcal A_s^{(g)L}e_s\,\dd\pi_g^x=p^L\,\E\bigg[|D_0|^{-s}\prod_{j=1}^L\Big(\frac{Y_j}p\Big)^{s/2}|D_j|^{-s}\bigg]. \end{equation} In particular, the left side does not depend on $g$. \end{enumerate} \end{lemma} \begin{proof} For (i), we first show that $X_j\in\R$ almost surely. Suppose that $X_j\in\R$ almost surely. The variables $(B'_{j-1},\beta_j)$ are independent of $X_j$, which is a function of $B'_{j},\dots,B'_{L-1},\beta_{j+1},\dots,\beta_L$. By \Cref{def:press-operators}(vi), for every fixed $y\in\R$ we have $f_{B'_{j-1},\beta_j}(y)\in\R$ almost surely, and Fubini's theorem implies $X_{j-1}\in\R$ almost surely. Formula \eqref{eq:press-chain} for $L=1$ is the definition of $\mathcal A_s$. Suppose it holds for $L$. Then \[ (\mathcal A_s^{L+1}F)(x)=p\,\E\Big[\beta_{L+1}^{s/2}|x|^s\,(\mathcal A_s^LF)\big(f_{B'_L,\beta_{L+1}}(x)\big)\Big]. \] Inserting \eqref{eq:press-chain} for $\mathcal A_s^LF$, evaluated at the point $f_{B'_L,\beta_{L+1}}(x)$ with the independent variables $B'_0,\dots,B'_{L-1},\beta_1,\dots,\beta_L$, and using Tonelli's theorem, we obtain \eqref{eq:press-chain} for $L+1$, with $X_{L+1}=x$ and $X_L=f_{B'_L,\beta_{L+1}}(x)$. For (ii), the variables $D_j$ are almost surely finite and nonzero by the same argument: given $D_j\in\R\setminus\{0\}$, the variable $D_{j-1}$ is finite, and it vanishes only if $D'_{j-1}=Y_j/(pD_j)$, which has probability zero because $Y_j$ has a density and is independent of $(D'_{j-1},D_j)$. Set $B'_j=gD'_j$, $\beta_j=g^2Y_j/p$, and $X_j=1/(gD_j)$. Then $B'_j$ has law $\nu_g$, $\beta_j$ has the exponential law of rate $K_g$, $X_L$ has law $\pi_g^x$, all are independent, and \[ f_{B'_{j-1},\beta_j}(X_j)=\Big(gD'_{j-1}-\frac{g^2Y_j}p\cdot\frac1{gD_j}\Big)^{-1}=\frac1{gD_{j-1}}=X_{j-1}. \] By (i) with $F=e_s$, applied conditionally on $X_L$ and integrated against $\pi_g^x$ using Tonelli's theorem, we obtain \begin{align*} g^s\int\mathcal A_s^{(g)L}e_s\,\dd\pi_g^x&=g^sp^L\,\E\bigg[|X_0|^s\prod_{j=1}^L\beta_j^{s/2}|X_j|^s\bigg]\\ &=g^sp^L\,\E\bigg[g^{-s}|D_0|^{-s}\prod_{j=1}^Lg^s\Big(\frac{Y_j}p\Big)^{s/2}g^{-s}|D_j|^{-s}\bigg], \end{align*} which is \eqref{eq:press-chain-D}. \end{proof} \begin{definition}\label{def:press-path-moment} Let $\pi$ be a real solution at energy $E>0$, let $s\in(0,1)$, and let $L\ge0$. The path moment is \[ \Phi_L(s)=\Phi_L(s;E)=g^s\int\mathcal A_s^{(g)L}e_s\,\dd\pi_g^x\in[0,\infty], \] where $g>0$, and $\mathcal A_s^{(g)}$ and $\pi_g^x$ are as in \Cref{lem:press-chain}(ii); for $L=0$, we set $\mathcal A_s^0e_s=e_s$. By \Cref{lem:press-chain}(ii), $\Phi_L(s)$ does not depend on $g$ and is given by the right side of \eqref{eq:press-chain-D}. \end{definition} \begin{remark}\label{rem:press-path-moment} We explain the name. Let $\pi$ be a real solution at $E$, and let $(D_v)$ be the variables of the recursive tree process of \Cref{lem:rde-rtp}, which have law $\pi$ and satisfy $D_v=E-p^{-1}\sum_{u}Y_u/D_u$, the sum running over the children $u$ of $v$. (When the no-ac hypothesis of \Cref{prop:rde-boundary} holds and $\pi$ is the real boundary law, the boundary denominators $-1/\Gamma_v(E+\iu0)$ satisfy the same recursion.) For a vertex $v$ at generation $L$ with path $o=v_0,\dots,v_L=v$, the formula of \Cref{prop:tree-path} with $\Gamma_{v_j}=-1/D_{v_j}$ and $|h_{v_{j-1}v_j}|^2=Y_{v_j}/p$ gives the formal column entry \[ \big|\Gamma_o\textstyle\prod_{j=1}^Lh_{v_{j-1}v_j}\Gamma_{v_j}\big|^s=|D_o|^{-s}\prod_{j=1}^L\Big(\frac{Y_{v_j}}p\Big)^{s/2}|D_{v_j}|^{-s}. \] Summing over $v$ and applying the many-to-one formula (\Cref{prop:palm-spine}), the sum becomes $p^L$ times an expectation along a single path, on which $D_{v_{j-1}}=D'_{j-1}-Y_{v_j}/(pD_{v_j})$, where $D'_{j-1}$ is the contribution of the other children of $v_{j-1}$; since these children form an independent Poisson family, the cavity equation shows that $D'_{j-1}$ has law $\pi$. Comparing with \eqref{eq:press-chain-D}, the expected generation-$L$ sum is $\Phi_L(s;E)$. This identity is stated and proved in \Cref{prop:loc-columns}. \end{remark} The operators $\mathcal A_s$ and $\mathcal Q_s$ are related by the weight function $h_s$ of \Cref{def:press-operators}(v). For the identification, we use the following conjugacy. Here, for a Borel function $G\colon\RP\to[0,\infty]$, we define $\mathcal Q_s^LG(x)=\int G\,\dd\kappa_s^{(L)}(x,\cdot)\in[0,\infty]$; for bounded $G$, this agrees with the operator iterate. \begin{lemma}\label{lem:press-conjugacy} Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and let $s\in(0,2)$. For $x\in\R\setminus\{0\}$, $b\in\R$, and $\beta>0$, we have \begin{equation}\label{eq:press-one-step-conj} \beta^{s/2}|x|^s\,h_s\big(f_{b,\beta}(x)\big)=h_s(x)\,\big\|M(b,\beta)v_x\big\|^{-s}. \end{equation} Further, for every Borel $G\colon\RP\to[0,\infty]$, every $L\ge1$, and every $x\in\R\setminus\{0\}$, \begin{equation}\label{eq:press-conjugacy} \mathcal A_s^L(h_sG)(x)=h_s(x)\,\mathcal Q_s^LG(x). \end{equation} \end{lemma} \begin{proof} Let $w=b-\beta x$. If $w\neq0$, then $f_{b,\beta}(x)=1/w$ and $h_s(1/w)=(1+w^2)^{-s/2}$, and \eqref{eq:press-norm} gives \[ h_s(x)\,\|M(b,\beta)v_x\|^{-s}=\frac{|x|^s}{(1+x^2)^{s/2}}\cdot\frac{\beta^{s/2}(1+x^2)^{s/2}}{(1+w^2)^{s/2}}=\beta^{s/2}|x|^s\,h_s(1/w). \] If $w=0$, then $f_{b,\beta}(x)=\infty$, $h_s(\infty)=1$, and both sides of \eqref{eq:press-one-step-conj} equal $\beta^{s/2}|x|^s$. This proves \eqref{eq:press-one-step-conj}. We prove \eqref{eq:press-conjugacy} by induction on $L$. For $x\in\R\setminus\{0\}$, the point $f_{B,\beta}(x)$ lies in $\R\setminus\{0\}$ almost surely (it equals $\infty$ only on a null event, and it equals $0$ only if $x=\infty$). For $L=1$, by \eqref{eq:press-one-step-conj}, we have \begin{align*} \mathcal A_s(h_sG)(x)&=p\,\E\Big[\beta^{s/2}|x|^sh_s\big(f_{B,\beta}(x)\big)G\big(f_{B,\beta}(x)\big)\Big]\\ &=h_s(x)\,p\,\E\Big[\|M(B,\beta)v_x\|^{-s}G\big(f_{B,\beta}(x)\big)\Big]=h_s(x)\,\mathcal Q_sG(x). \end{align*} Suppose that \eqref{eq:press-conjugacy} holds for $L$. Since $f_{B,\beta}(x)\in\R\setminus\{0\}$ almost surely, we may apply the induction hypothesis at this point, and \eqref{eq:press-one-step-conj} gives \[ \mathcal A_s^{L+1}(h_sG)(x)=p\,\E\Big[\beta^{s/2}|x|^s\,h_s\big(f_{B,\beta}(x)\big)\,(\mathcal Q_s^LG)\big(f_{B,\beta}(x)\big)\Big]=h_s(x)\,\mathcal Q_s^{L+1}G(x). \] In the last step, we used \eqref{eq:press-row-first-step} to write $\mathcal Q_s^{L+1}G(x)=p\,\E[\|M(B,\beta)v_x\|^{-s}(\mathcal Q_s^LG)(f_{B,\beta}(x))]$. \end{proof} The identification of the free energy with the path moments requires a fractional moment bound for the denominator perturbed by a tilted squared modulus. For $q\in(0,1)$, let $T_q$ be a random variable with density $t^{q/2}e^{-t}/\Gamma(1+q/2)$ on $(0,\infty)$, independent of $B$. We consider the bound \begin{equation}\label{eq:press-fc} \E|B-\sigma aT_q|^{-q}\le C_q\min\{1,a^{-q}\}\qquad\text{for all }\sigma\in\{-1,1\}\text{ and }a\ge0 . \end{equation} By \Cref{prop:rde-properties}, if $B=gD$ with $D$ distributed according to a real solution at energy $E>0$, then \eqref{eq:press-fc} holds for every $q\in(0,1)$, with a constant $C_q$ depending on $p,E,g$, and $q$. \begin{proposition}\label{prop:press-identification} Let $s\in(0,1)$. Suppose that $\nu$ satisfies \eqref{eq:press-hyp}, and that there exist $q\in(s,1)$ and $C_q<\infty$ such that \eqref{eq:press-fc} holds. Let $\nu^x$ be the law of $1/B$. \begin{enumerate}[label=\textup{(\roman*)}] \item There exists $C\ge1$, depending on $s,p,K,\nu,q$, and $C_q$, such that for every $L\ge1$, \[ C^{-1}\lambda(s)^L\le\int\mathcal A_s^Le_s\,\dd\nu^x\le C\lambda(s)^L . \] In particular, $\int\mathcal A_s^Le_s\,\dd\nu^x$ is finite for every $L$, and $\lim_{L\to\infty}L^{-1}\log\int\mathcal A_s^Le_s\,\dd\nu^x=\varphi(s)$. \item We have $\E|B|^{-s}<\infty$, and the norm moments satisfy \[ C_*^{-1}e^{(L+3)(\varphi(s)-\log p)}\le f_L(s)\le e^{L(\varphi(s)-\log p)},\qquad L\ge1, \] where $C_*$ is as in \Cref{cor:press-norm-moments}. In particular, $\lim_{L\to\infty}L^{-1}\log f_L(s)=\varphi(s)-\log p$. \item Let $\pi$ be a real solution at energy $E>0$. Then the hypotheses above hold for $\nu_g$ and $K_g=p/g^2$, for every $g>0$ and every $s\in(0,1)$. In particular, there exists $C\ge1$, depending on $s,p,E$, and $\pi$, such that \[ C^{-1}e^{L\varphi_\pi(s)}\le\Phi_L(s;E)\le Ce^{L\varphi_\pi(s)},\qquad L\ge1, \] and therefore \[ \lim_{L\to\infty}\frac1L\log\Phi_L(s;E)=\varphi_\pi(s). \] Further, $\varphi_\pi(1)=\lim_{s\uparrow1}\lim_{L\to\infty}L^{-1}\log\Phi_L(s;E)$, and the bounds in \textup{(ii)} hold for the norm moments of $\pi$ with every scaling. \end{enumerate} \end{proposition} \begin{proof} Since the weight function $h_s$ vanishes at $0$, the initial function $e_s=|x|^s$ of the path moment corresponds under \eqref{eq:press-conjugacy} to the function $u_s(y)=(1+y^2)^{s/2}$ (with $u_s(\infty)=\infty$), which is unbounded on $\RP$. We show that one application of $\mathcal Q_s$ turns $u_s$ into a bounded continuous positive function, using \eqref{eq:press-fc}; the uniform integrability needed for continuity comes from \eqref{eq:press-fc} at the larger exponent $q$. After this, the bounds follow from \Cref{thm:press-perron}. \proofstep{1}{A tilted moment bound.} For $r\in(0,1)$, we have $y^{r/2}e^{-y}\,\dd y=\Gamma(1+r/2)\,\P(T_r\in\dd y)$. Writing $\beta=Y/K$ with $Y\sim\Exp(1)$ independent of $B$, we therefore have, for $x\in\R$, \begin{equation}\label{eq:press-tilt} \E\big[\beta^{r/2}|B-\beta x|^{-r}\big]=K^{-r/2}\,\E\Big[Y^{r/2}\Big|B-\frac xKY\Big|^{-r}\Big]=K^{-r/2}\,\Gamma(1+r/2)\,\E\Big|B-\sigma\frac{|x|}KT_r\Big|^{-r}, \end{equation} where $\sigma=\sgn x$ (and $\sigma=1$ if $x=0$). Applying \eqref{eq:press-tilt} with $r=q$ and \eqref{eq:press-fc}, we obtain \[ (1+x^2)^{q/2}\,\E\big[\beta^{q/2}|B-\beta x|^{-q}\big]\le K^{-q/2}\,\Gamma(1+q/2)\,C_q\,(1+x^2)^{q/2}\min\{1,K^q|x|^{-q}\}. \] The right side is at most $(1+K^2)^{q/2}$ times the prefactor, uniformly in $x\in\R$: for $|x|\le K$ this is immediate, and for $|x|\ge K$ we have $(1+x^2)^{q/2}K^q|x|^{-q}=K^q(1+x^{-2})^{q/2}\le(K^2+1)^{q/2}$. Then there exists $C_1<\infty$, depending on $q,K$, and $C_q$, with \begin{equation}\label{eq:press-tilt-bound} \sup_{x\in\R}(1+x^2)^{q/2}\,\E\big[\beta^{q/2}|B-\beta x|^{-q}\big]\le C_1 . \end{equation} \proofstep{2}{The function $\mathcal Q_su_s$ is continuous and positive on $\RP$.} Let $x\in\R$. Almost surely $w=B-\beta x\neq0$, and then $f_{B,\beta}(x)=1/w$ and $u_s(1/w)=(1+w^{-2})^{s/2}=(1+w^2)^{s/2}|w|^{-s}$. By \eqref{eq:press-norm}, we have \[ \mathcal Q_su_s(x)=p\,\E Z_x,\qquad Z_x=\beta^{s/2}(1+x^2)^{s/2}|B-\beta x|^{-s}. \] At $\infty$, we have $f_{B,\beta}(\infty)=0$ and $u_s(0)=1$, which give $\mathcal Q_su_s(\infty)=p\,\E\beta^{-s/2}=pK^{s/2}\Gamma(1-s/2)$. Since $Z_x^{q/s}=\beta^{q/2}(1+x^2)^{q/2}|B-\beta x|^{-q}$ and $q/s>1$, \eqref{eq:press-tilt-bound} implies that the family $\{Z_x:x\in\R\}$ is bounded in $L^{q/s}$; in particular, it is uniformly integrable and bounded in $L^1$. In particular, $\sup_{x\in\R}\mathcal Q_su_s(x)<\infty$. Let $x_n\to x_0\in\R$. The event $\{B\neq\beta x_0\}$ has probability one: for $x_0=0$ because $\P(B=0)=0$, and for $x_0\neq0$ because $\beta$ has a density and is independent of $B$. On this event, $Z_{x_n}\to Z_{x_0}$, and by Vitali's convergence theorem, we have $\mathcal Q_su_s(x_n)\to\mathcal Q_su_s(x_0)$. Now let $x_n\in\R$ with $|x_n|\to\infty$. Since $|B-\beta x_n|(1+x_n^2)^{-1/2}\to\beta$ almost surely, we have $Z_{x_n}\to\beta^{s/2}\beta^{-s}=\beta^{-s/2}$ almost surely, and Vitali's theorem gives $\mathcal Q_su_s(x_n)\to p\,\E\beta^{-s/2}=\mathcal Q_su_s(\infty)$. This proves that $\mathcal Q_su_s\in C(\RP)$. Since $Z_x>0$ almost surely for every $x\in\R$ and $\E\beta^{-s/2}>0$, the function $\mathcal Q_su_s$ is strictly positive, and by compactness $c_1=\min\mathcal Q_su_s>0$. Let $c_2=\max\mathcal Q_su_s$. \proofstep{3}{Proof of (i).} Since $e_s=h_su_s$ on $\R$, \Cref{lem:press-conjugacy} with $G=u_s$ implies $\mathcal A_s^Le_s(x)=h_s(x)\,\mathcal Q_s^Lu_s(x)$ for $x\in\R\setminus\{0\}$. By the Chapman--Kolmogorov relations and Tonelli's theorem, $\mathcal Q_s^Lu_s=\mathcal Q_s^{L-1}(\mathcal Q_su_s)$, where on the right $\mathcal Q_s^{L-1}$ acts on the continuous function $\mathcal Q_su_s$. The law $\nu^x$ is concentrated on $\R\setminus\{0\}$, because $B$ is finite and $\P(B=0)=0$. We conclude that \begin{equation}\label{eq:press-endpoint-formula} \int\mathcal A_s^Le_s\,\dd\nu^x=\int h_s(x)\,\mathcal Q_s^{L-1}(\mathcal Q_su_s)(x)\,\nu^x(\dd x),\qquad L\ge1 . \end{equation} Let $\psi_s$ be the eigenfunction of \Cref{thm:press-perron}(ii), and let $a=\min\psi_s$ and $A=\max\psi_s$. As in Step 6 of the proof of \Cref{thm:press-perron}, positivity of $\mathcal Q_s$ and $c_1\le\mathcal Q_su_s\le c_2$ give \[ \frac{c_1a}{A}\,\lambda(s)^{L-1}\le\mathcal Q_s^{L-1}(\mathcal Q_su_s)\le\frac{c_2A}{a}\,\lambda(s)^{L-1} \] pointwise on $\RP$. Further, since $01$ is fixed. For a self-adjoint operator $K$ on a separable Hilbert space $\mathfrak h$ and a vector $\varphi\in\mathfrak h$, we write $\mu^K_\varphi$ for the spectral measure of $\varphi$, so that $\mu^K_\varphi(B)=\langle\varphi,\one_B(K)\varphi\rangle$ for Borel sets $B\subset\R$. We let $\mathfrak h_{\mathrm{pp}}(K)$ and $\mathfrak h_{\mathrm{ac}}(K)$ denote the closed subspaces of vectors whose spectral measures are pure point and absolutely continuous, respectively. For a Borel set $B\subset\R$, we say that $K$ has pure point spectrum in $B$ if $\Ran\one_B(K)\subset\mathfrak h_{\mathrm{pp}}(K)$, and that $K$ is purely absolutely continuous in $B$ if $\Ran\one_B(K)\subset\mathfrak h_{\mathrm{ac}}(K)$. Since $\mu^K_{\one_B(K)\varphi}$ is the restriction of $\mu^K_\varphi$ to $B$, the first property holds if and only if the restriction of $\mu^K_\varphi$ to $B$ is pure point for every $\varphi\in\mathfrak h$, and similarly for the second. We write $|B|$ for the Lebesgue measure of $B$, and $B^2=\{t^2:t\in B\}$. \section{Borel transforms and rank-one perturbations}\label{sec:rank-abstract} For a finite positive Borel measure $\mu$ on $\R$, its Borel transform is \[ F_\mu(z)=\int_\R\frac{\mu(\dd t)}{t-z},\qquad z\in\Hup . \] It is holomorphic on $\Hup$, and for $\lambda\in\R$ and $\eta>0$ we have \begin{equation}\label{eq:rank-poisson} \Im F_\mu(\lambda+\iu\eta)=\int_\R\frac{\eta\,\mu(\dd t)}{(t-\lambda)^2+\eta^2}. \end{equation} If $\mu$ is not the zero measure, then $\Im F_\mu>0$ on $\Hup$. We say that the vertical boundary value $F_\mu(\lambda+\iu0)$ exists if $\lim_{\eta\downarrow0}F_\mu(\lambda+\iu\eta)$ exists in $\C$; in that case $F_\mu(\lambda+\iu0)$ denotes the limit. We also define the reciprocal-square integral \begin{equation}\label{eq:rank-R} R_\mu(\lambda)=\int_\R\frac{\mu(\dd t)}{(t-\lambda)^2}\in[0,\infty], \end{equation} with the convention that the integrand equals $+\infty$ at $t=\lambda$, so that $R_\mu(\lambda)=\infty$ whenever $\mu(\{\lambda\})>0$. By \eqref{eq:rank-poisson}, the function $\eta\mapsto\eta^{-1}\Im F_\mu(\lambda+\iu\eta)$ is the integral of $((t-\lambda)^2+\eta^2)^{-1}$ against $\mu$, which increases as $\eta\downarrow0$. By the monotone convergence theorem, we have \begin{equation}\label{eq:rank-R-limit} R_\mu(\lambda)=\lim_{\eta\downarrow0}\frac{\Im F_\mu(\lambda+\iu\eta)}{\eta} =\sup_{n\in\N}\, n\Im F_\mu(\lambda+\iu/n). \end{equation} \begin{lemma}\label{lem:rank-borel} Let $\mu$ be a finite positive Borel measure on $\R$, and let $\mu=\mu_{\mathrm{ac}}+\mu_{\mathrm{sing}}$ be its decomposition into parts that are absolutely continuous and singular with respect to Lebesgue measure. \begin{enumerate}[label=\textup{(\roman*)}] \item For Lebesgue-almost every $\lambda\in\R$, the vertical boundary value $F_\mu(\lambda+\iu0)$ exists. \item The measure $\mu_{\mathrm{sing}}$ is carried by the set of $\lambda$ such that $\lim_{\eta\downarrow0}\Im F_\mu(\lambda+\iu\eta)=\infty$. In particular, $\mu_{\mathrm{sing}}(\R\setminus N_\mu)=0$, where $N_\mu$ is the set of $\lambda$ at which $F_\mu(\lambda+\iu0)$ does not exist. \item We have $\mu(\{\lambda\in\R:\lim_{\eta\downarrow0}\Im F_\mu(\lambda+\iu\eta)=0\})=0$. \item If $R_\mu(\lambda)<\infty$, then $\mu(\{\lambda\})=0$, the boundary value $F_\mu(\lambda+\iu0)$ exists, and it is real and equal to $\int_\R(t-\lambda)^{-1}\mu(\dd t)$. \end{enumerate} \end{lemma} \begin{proof} Part (i) is the classical theorem on the existence of finite boundary values of Borel transforms of finite measures; see \cite{Simon1995spectral} and \cite{AizenmanWarzel2015random}. (One proof applies Fatou's theorem on boundary values of bounded holomorphic functions to $(F_\mu-\iu)/(F_\mu+\iu)$; see \cite[Chapter~11]{Rudin1987real}.) For (ii), we first note that if $|t-\lambda|<\eta$, then $(t-\lambda)^2+\eta^2\le2\eta^2$. Restricting the integral in \eqref{eq:rank-poisson} to such $t$, we obtain the Poisson-kernel lower bound \begin{equation}\label{eq:rank-poisson-lower} \Im F_\mu(\lambda+\iu\eta)\ge\frac{\mu((\lambda-\eta,\lambda+\eta))}{2\eta}. \end{equation} By the differentiation theorem for positive measures that are singular with respect to Lebesgue measure \cite[Chapter~7]{Rudin1987real}, for $\mu_{\mathrm{sing}}$-almost every $\lambda$ we have \[ \lim_{\eta\downarrow0}\frac{\mu_{\mathrm{sing}}((\lambda-\eta,\lambda+\eta))}{2\eta}=\infty. \] Since $\mu\ge\mu_{\mathrm{sing}}$, \eqref{eq:rank-poisson-lower} implies that $\Im F_\mu(\lambda+\iu\eta)\to\infty$ for $\mu_{\mathrm{sing}}$-almost every $\lambda$. At such $\lambda$ the boundary value $F_\mu(\lambda+\iu0)$ does not exist, which proves (ii). For (iii), let $\mathcal A$ be the set in (iii); it is a Borel set, because the limit of the continuous function $\eta\mapsto\Im F_\mu(\lambda+\iu\eta)$ may be taken along rational $\eta$. By (ii), we have $\mu_{\mathrm{sing}}(\mathcal A)=0$. For $\lambda\in\mathcal A$, the bound \eqref{eq:rank-poisson-lower} implies that $\lim_{\eta\downarrow0}\mu((\lambda-\eta,\lambda+\eta))/(2\eta)=0$. By the differentiation theorem for measures \cite[Chapter~7]{Rudin1987real}, at Lebesgue-almost every $\lambda$ this limit exists and equals the density $f=\dd\mu_{\mathrm{ac}}/\dd\lambda$; the singular part contributes zero Lebesgue-almost everywhere. We conclude that $f=0$ almost everywhere on $\mathcal A$, and $\mu_{\mathrm{ac}}(\mathcal A)=\int_{\mathcal A}f(\lambda)\dd\lambda=0$. For (iv), the convention in \eqref{eq:rank-R} shows that $\mu(\{\lambda\})=0$. For $t\ne\lambda$ and $\eta>0$, we have $|t-\lambda-\iu\eta|^{-1}\le|t-\lambda|^{-1}$, and by the Cauchy--Schwarz inequality, \[ \int_\R\frac{\mu(\dd t)}{|t-\lambda|}\le\mu(\R)^{1/2}R_\mu(\lambda)^{1/2}<\infty . \] By the dominated convergence theorem, we have $F_\mu(\lambda+\iu\eta)\to\int_\R(t-\lambda)^{-1}\mu(\dd t)\in\R$ as $\eta\downarrow0$. \end{proof} The next lemma compares vertical boundary values at $E^2$ with limits along the curve $\eta\mapsto(E+\iu\eta)^2$, which is the curve that appears when the resolvent of $\mathcal H$ at $E+\iu\eta$ is expressed through resolvents of $\mathcal H^2$. \begin{lemma}\label{lem:rank-square} Let $\mu$ be a finite positive Borel measure on $\R$ and let $E>0$. Then $\lim_{\eta\downarrow0}F_\mu((E+\iu\eta)^2)$ exists in $\C$ if and only if $F_\mu(E^2+\iu0)$ exists, and in that case the two limits coincide. \end{lemma} \begin{proof} For $\eta>0$, set $w_\eta=(E+\iu\eta)^2=E^2-\eta^2+2\iu E\eta$ and $w'_\eta=E^2+2\iu E\eta$. Both lie in $\Hup$, they have the same imaginary part $2E\eta$, and $|w_\eta-w'_\eta|=\eta^2$. For $w\in\Hup$, we have $\int_\R|t-w|^{-2}\mu(\dd t)=\Im F_\mu(w)/\Im w$ by \eqref{eq:rank-poisson}. Using the Cauchy--Schwarz inequality, we find \begin{align} |F_\mu(w_\eta)-F_\mu(w'_\eta)| &=\left|\int_\R\frac{(w_\eta-w'_\eta)\,\mu(\dd t)}{(t-w_\eta)(t-w'_\eta)}\right| \le\eta^2\left(\int_\R\frac{\mu(\dd t)}{|t-w_\eta|^2}\right)^{1/2} \left(\int_\R\frac{\mu(\dd t)}{|t-w'_\eta|^2}\right)^{1/2}\notag\\ &=\frac{\eta}{2E}\bigl(\Im F_\mu(w_\eta)\,\Im F_\mu(w'_\eta)\bigr)^{1/2}. \label{eq:rank-curve} \end{align} Suppose that $F_\mu(E^2+\iu0)$ exists. Since $w'_\eta=E^2+\iu(2E\eta)$, we have $F_\mu(w'_\eta)\to F_\mu(E^2+\iu0)$, and there exist $M<\infty$ and $\eta_0\in(0,E]$ such that $\Im F_\mu(w'_\eta)\le M$ for $\eta\le\eta_0$. For such $\eta$, write $a=\Im F_\mu(w_\eta)$. By \eqref{eq:rank-curve} and the inequality $\sqrt{aM}\le(a+M)/2$, we have $a\le M+(a+M)/4$, which implies that $a\le2M$. Inserting this into \eqref{eq:rank-curve}, we obtain $|F_\mu(w_\eta)-F_\mu(w'_\eta)|\le\sqrt2M\eta/(2E)\to0$, and therefore $F_\mu(w_\eta)\to F_\mu(E^2+\iu0)$. The converse implication follows by exchanging the roles of $w_\eta$ and $w'_\eta$ in the same argument. \end{proof} We now turn to rank-one perturbations. Let $L$ be a self-adjoint operator on a separable Hilbert space $\mathfrak h$, and let $v\in\mathfrak h$ with $\|v\|=1$. For $u\in\R$, set \begin{equation}\label{eq:rank-family} L_u=L+u|v\rangle\langle v|,\qquad\Dom(L_u)=\Dom(L),\qquad \mu_u=\mu^{L_u}_v,\qquad F_u=F_{\mu_u}. \end{equation} Each $L_u$ is self-adjoint, since it is a bounded symmetric perturbation of $L$, and each $\mu_u$ is a probability measure. Let $N\subset\R$ be the set of $\lambda$ at which $F_0(\lambda+\iu0)$ does not exist, and for $u\in\R$ set \begin{equation}\label{eq:rank-pole-set} Z_u=\bigl\{\lambda\in\R\setminus N:\ F_0(\lambda+\iu0)\in\R,\ 1+uF_0(\lambda+\iu0)=0\bigr\}. \end{equation} Note that $Z_0=\emptyset$. \begin{lemma}\label{lem:rank-one} In the setting of \eqref{eq:rank-family} and \eqref{eq:rank-pole-set}, the following statements hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $u\in\R$ and $z\in\Hup$, we have $1+uF_0(z)\ne0$ and \begin{equation}\label{eq:rank-AK} F_u(z)=\frac{F_0(z)}{1+uF_0(z)}. \end{equation} \item The set $N$ is a Lebesgue-null Borel set, and for every $u\in\R$ the singular part of $\mu_u$ is carried by $N\cup Z_u$: \[ (\mu_u)_{\mathrm{sing}}\bigl(\R\setminus(N\cup Z_u)\bigr)=0. \] \item For every Borel set $B\subset\R$, the map $u\mapsto\mu_u(B)$ is Borel and \begin{equation}\label{eq:rank-averaging} \int_\R\mu_u(B)\dd u=|B|. \end{equation} \item Let $(\Omega',\mathscr A)$ be a measurable space, and suppose that for each $\omega\in\Omega'$ we are given $L=L(\omega)$ and $v=v(\omega)$ as above, possibly on a Hilbert space depending on $\omega$, such that $\omega\mapsto F_0(\omega,z)$ is $\mathscr A$-measurable for every $z\in\Hup$. Write $N(\omega)$, $Z_u(\omega)$, $\mu_u(\omega)$ for the corresponding objects. Then the sets \[ \{(\omega,\lambda):\lambda\in N(\omega)\}\in\mathscr A\otimes\mathcal B(\R), \qquad \{(\omega,u,\lambda):\lambda\in Z_u(\omega)\}\in\mathscr A\otimes\mathcal B(\R)\otimes\mathcal B(\R) \] are measurable, the function $(\omega,\lambda)\mapsto R_{\mu_0(\omega)}(\lambda)$ is $\mathscr A\otimes\mathcal B(\R)$-measurable, and for every $T\in\mathscr A\otimes\mathcal B(\R)$ the map $(\omega,u)\mapsto\mu_u(\omega)(T_\omega)$ is $\mathscr A\otimes\mathcal B(\R)$-measurable, where $T_\omega=\{\lambda:(\omega,\lambda)\in T\}$. \end{enumerate} \end{lemma} The identity \eqref{eq:rank-AK} is the Aronszajn--Krein formula (see \cite{Aronszajn1957problem}), and \eqref{eq:rank-averaging} is spectral averaging; see \cite{Simon1995spectral}. Part (iv) is what allows the null sets in (ii) and (iii) to be used inside conditional Fubini arguments in which $L$ and $v$ are random. \begin{proof} We prove the four parts in the order (i), (iv), (ii), (iii). \proofpart{Proof of (i).} Fix $z\in\Hup$. Since $\Ran(L-z)^{-1}=\Dom(L)=\Dom(L_u)$, the second resolvent identity yields \[ (L-z)^{-1}-(L_u-z)^{-1}=(L_u-z)^{-1}(L_u-L)(L-z)^{-1} =u\,(L_u-z)^{-1}|v\rangle\langle v|(L-z)^{-1}. \] Applying both sides to $v$ and taking the inner product with $v$, we obtain $F_0(z)-F_u(z)=uF_u(z)F_0(z)$, that is, $F_u(z)(1+uF_0(z))=F_0(z)$. If $u\ne0$ and $1+uF_0(z)=0$, then $F_0(z)=-1/u$ would be real, which is impossible because $\Im F_0(z)>0$. This proves (i). \proofpart{Proof of (iv).} We first show that $(\omega,z)\mapsto F_0(\omega,z)$ is $\mathscr A\otimes\mathcal B(\Hup)$-measurable. For $n\in\N$, partition $\Hup$ into countably many Borel sets $Q_{n,k}$ of diameter at most $1/n$, choose $z_{n,k}\in Q_{n,k}$, and set $F^{(n)}(\omega,z)=F_0(\omega,z_{n,k})$ for $z\in Q_{n,k}$. Each $F^{(n)}$ is jointly measurable, and $F^{(n)}\to F_0$ pointwise because $F_0(\omega,\cdot)$ is continuous. By \eqref{eq:rank-AK}, the function $(\omega,u,z)\mapsto F_u(\omega,z)$ is then jointly measurable as well. Let $\mathcal C$ be the set of $(\omega,\lambda)$ satisfying the rational Cauchy criterion \begin{equation}\label{eq:rank-rational-cauchy} \forall k\in\N\ \exists m\in\N\ \forall\eta,\eta'\in\mathbb Q\cap(0,1/m):\quad |F_0(\omega,\lambda+\iu\eta)-F_0(\omega,\lambda+\iu\eta')|<1/k . \end{equation} It is obtained from jointly measurable sets by countable unions and intersections, so $\mathcal C\in\mathscr A\otimes\mathcal B(\R)$. If $F_0(\omega,\lambda+\iu0)$ exists, then \eqref{eq:rank-rational-cauchy} holds. Conversely, if \eqref{eq:rank-rational-cauchy} holds, then by continuity of $\eta\mapsto F_0(\omega,\lambda+\iu\eta)$ the same inequality, with $\le$ in place of $<$, holds for all real $\eta,\eta'\in(0,1/m)$, and completeness of $\C$ shows that the limit exists. It follows that $\{(\omega,\lambda):\lambda\in N(\omega)\}$ is the complement of $\mathcal C$. On $\mathcal C$ we set $F_0^*(\omega,\lambda)=\lim_{n\to\infty}F_0(\omega,\lambda+\iu/n)$, and we set $F_0^*=0$ off $\mathcal C$; this function is jointly measurable, as a pointwise limit of measurable functions multiplied by $\one_{\mathcal C}$. Then \[ \{(\omega,u,\lambda):\lambda\in Z_u(\omega)\} =\bigl\{(\omega,u,\lambda):(\omega,\lambda)\in\mathcal C,\ \Im F_0^*(\omega,\lambda)=0,\ 1+uF_0^*(\omega,\lambda)=0\bigr\} \] is measurable. The measurability of $R_{\mu_0(\omega)}(\lambda)$ follows from \eqref{eq:rank-R-limit}. For the last assertion, let $P_\eta(x)=\pi^{-1}\eta/(x^2+\eta^2)$, so that $\pi^{-1}\Im F_u(\lambda+\iu\eta)=\int P_\eta(\lambda-t)\mu_u(\dd t)$ by \eqref{eq:rank-poisson}. For a continuous function $f$ with compact support, we have by Tonelli's theorem \[ \frac1\pi\int_\R f(\lambda)\Im F_u(\lambda+\iu\eta)\dd\lambda=\int_\R(P_\eta*f)(t)\,\mu_u(\dd t). \] Since $P_\eta*f\to f$ uniformly as $\eta\downarrow0$ and $\mu_u$ is a probability measure, the right side converges to $\int f\dd\mu_u$. The left side, with $\eta=1/n$, is a measurable function of $(\omega,u)$ by Fubini's theorem, because its integrand is jointly measurable and bounded by $n|f(\lambda)|$. This shows that $(\omega,u)\mapsto\int f\dd\mu_u(\omega)$ is measurable. Approximating the indicator of an open interval from below by such $f$ and applying the $\pi$--$\lambda$ theorem, we find that $(\omega,u)\mapsto\mu_u(\omega)(B)$ is measurable for every Borel set $B$. Finally, the class of $T\in\mathscr A\otimes\mathcal B(\R)$ for which $(\omega,u)\mapsto\mu_u(\omega)(T_\omega)$ is measurable contains the rectangles $A_0\times B$, for which the map is $\one_{A_0}(\omega)\mu_u(\omega)(B)$, and it is closed under complements and increasing unions because each $\mu_u(\omega)$ is a probability measure. By the $\pi$--$\lambda$ theorem it is all of $\mathscr A\otimes\mathcal B(\R)$; see also \cite[Chapter~1]{Kallenberg2002foundations}. \proofpart{Proof of (ii).} By (iv), applied with a one-point space $\Omega'$, the set $N$ is Borel, and it is Lebesgue-null by \Cref{lem:rank-borel}(i). We now fix $u\in\R$ and $\lambda\notin N\cup Z_u$, and write $a=F_0(\lambda+\iu0)$. If $1+ua\ne0$, then \eqref{eq:rank-AK} implies that $F_u(\lambda+\iu\eta)\to a/(1+ua)\in\C$; in particular, $\Im F_u(\lambda+\iu\eta)$ stays bounded as $\eta\downarrow0$. If $1+ua=0$, then $u\ne0$ and $a=-1/u$ is real, and $\lambda\in Z_u$, which was excluded. Hence $\R\setminus(N\cup Z_u)$ does not meet the set on which $\Im F_u(\lambda+\iu\eta)\to\infty$, and \Cref{lem:rank-borel}(ii), applied to $\mu_u$, gives $(\mu_u)_{\mathrm{sing}}(\R\setminus(N\cup Z_u))=0$. \proofpart{Proof of (iii).} The measurability of $u\mapsto\mu_u(B)$ is the case of (iv) with a one-point space $\Omega'$. Fix $z\in\Hup$ and write $F_0(z)=a+\iu b$ with $b>0$. By \eqref{eq:rank-AK}, \[ \Im F_u(z)=\frac{\Im\bigl(F_0(z)\,\overline{(1+uF_0(z))}\bigr)}{|1+uF_0(z)|^2} =\frac{b}{(1+ua)^2+u^2b^2}. \] With $c=a^2+b^2$, we have $(1+ua)^2+u^2b^2=c(u+a/c)^2+b^2/c$, and therefore \begin{equation}\label{eq:rank-pi-integral} \int_\R\Im F_u(z)\dd u=b\int_\R\frac{\dd u}{c(u+a/c)^2+b^2/c}=b\cdot\frac{\pi}{b}=\pi . \end{equation} Define the measure $\nu(B)=\int_\R\mu_u(B)\dd u$ on Borel sets $B\subset\R$. For $z=x+\iu y\in\Hup$, Tonelli's theorem and \eqref{eq:rank-pi-integral} give \begin{equation}\label{eq:rank-poisson-nu} \int_\R\frac{y\,\nu(\dd t)}{(t-x)^2+y^2}=\int_\R\Im F_u(x+\iu y)\dd u=\pi =\int_\R\frac{y\dd t}{(t-x)^2+y^2}. \end{equation} The measure $\nu$ is locally finite: for $t\in[x-y,x+y]$ we have $y/((t-x)^2+y^2)\ge1/(2y)$, and \eqref{eq:rank-poisson-nu} implies that $\nu([x-y,x+y])\le2\pi y$. Taking $z=\iu$ in \eqref{eq:rank-poisson-nu}, we also obtain $\int_\R(1+t^2)^{-1}\nu(\dd t)=\pi$. It remains to show that a locally finite measure $\nu$ with $\int_\R(1+t^2)^{-1}\nu(\dd t)<\infty$ is determined by its Poisson transform $(P_y*\nu)(x)=\pi^{-1}\int y((t-x)^2+y^2)^{-1}\nu(\dd t)$. Let $f$ be continuous with support in $[-r,r]$. For $00$. By the spectral theorem, \[ \|\psi_\eta-\psi_{\eta'}\|^2=\int_\R\left|\frac1{t-\lambda-\iu\eta}-\frac1{t-\lambda-\iu\eta'}\right|^2 \mu_0(\dd t). \] The integrand is bounded by $4(t-\lambda)^{-2}$, which is $\mu_0$-integrable because $R_{\mu_0}(\lambda)<\infty$, and it tends to zero for $t\ne\lambda$ as $\eta,\eta'\downarrow0$; since $\mu_0(\{\lambda\})=0$, the dominated convergence theorem shows that $(\psi_\eta)$ is Cauchy as $\eta\downarrow0$, with a limit $\psi$. We have $(L-\lambda)\psi_\eta=v+\iu\eta\psi_\eta\to v$. Since $L$ is closed, we obtain $\psi\in\Dom(L)$ and $(L-\lambda)\psi=v$. In particular $\psi\ne0$. Moreover, $\langle v,\psi\rangle=\lim_{\eta\downarrow0}F_0(\lambda+\iu\eta)=a$. Then \[ (L_u-\lambda)\psi=v+u\langle v,\psi\rangle v=(1+ua)v=0, \] which proves the claim. We conclude that $S\setminus T\subset\mathcal A_u\cup\sigma_{\mathrm p}(L_u)$, where $\sigma_{\mathrm p}(L_u)$ is the set of eigenvalues of $L_u$. By \Cref{lem:rank-borel}(iii), $\mu_u(\mathcal A_u)=0$. This implies that the restriction of $\mu_u$ to $S\setminus T$ is carried by $\sigma_{\mathrm p}(L_u)$. This set is countable, because eigenvectors of distinct eigenvalues are orthogonal and $\mathfrak h$ is separable. If moreover $\mu_u(T)=0$, the restriction of $\mu_u$ to $S$ is carried by a countable set, that is, it is pure point. Finally, if $|T|=0$, then \eqref{eq:rank-averaging} gives $\int_\R\mu_u(T)\dd u=|T|=0$, and $\mu_u(T)=0$ for Lebesgue-almost every $u$. For (ii), let $\lambda\in S\setminus T'$. Then $\lambda\notin N$ and $\Im F_0(\lambda+\iu0)>0$, and $\lambda\notin Z_u$ for every $u$. By \Cref{lem:rank-one}(ii), we have $(\mu_u)_{\mathrm{sing}}(S\setminus T')=0$. If $\mu_u(T')=0$, then $(\mu_u)_{\mathrm{sing}}(S)\le\mu_u(T')=0$. The last assertion follows from \eqref{eq:rank-averaging} as in (i). \end{proof} \section{Measurability of resolvent entries}\label{sec:rank-measurable} We use the realization of the marked tree in \Cref{def:tree-pgw}: vertices are words in the Ulam--Harris set $\mathbb U$, so that every vertex is a deterministic word, and the marked tree is built from the independent families $(N_u)_{u\in\mathbb U}$ and $(w_u)_{u\in\mathbb U_*}$. A quantity indexed by words that are not vertices of the relevant graph is set equal to zero; with this convention, events such as $\{x\in V\}$ and $\{x\sim y\}$ are measurable. For a vertex $y$, we write $\mathcal H^{(y)}$ for the operator with the vertex $y$ deleted and $\mu^{(y)}_x$ for the spectral measure of $\delta_x$ for $\mathcal H^{(y)}$, as in \Cref{thm:tree-esa} and \Cref{def:tree-resolvent}. By \Cref{lem:tree-branch}(ii), for a neighbor $x$ of $y$, the component of $x$ in $V\setminus\{y\}$ is the branch $V_{x\to y}$, so that \begin{equation}\label{eq:rank-deleted-diagonal} G^{(y)}_{xx}(z)=\Gamma_{x\to y}(z),\qquad z\in\C\setminus\R,\quad x\sim y . \end{equation} \begin{lemma}\label{lem:rank-measurable} Let $(F,D)$ be a finite removal given by a fixed finite set $F$ of words and a fixed finite set $D$ of edges, identified with words as in \Cref{def:tree-pgw} (on the event that these are vertices and edges of $\mathcal T$), and let $u,x$ be words. Then the map $(\omega,z)\mapsto G^{F,D}_{ux}(z)$ is measurable on $\Omega\times\Hup$. In particular, for all words $u,w$, the map $(\omega,z)\mapsto\Gamma_{u\to w}(z)$ is measurable. \end{lemma} \begin{proof} For fixed $z$, the map $\omega\mapsto G^{F,D}_{ux}(z)$ is measurable, as explained after \Cref{def:tree-resolvent}: by \Cref{thm:tree-esa}(iii), it is the limit of entries of inverses of finite matrices whose entries are measurable functions of finitely many of the variables $N_v$ and $w_v$. It is also continuous in $z\in\Hup$. For $n\in\N$, partition $\Hup$ into countably many Borel sets $Q_{n,k}$ of diameter at most $1/n$, choose $z_{n,k}\in Q_{n,k}$, and set $G^{(n)}(\omega,z)=G^{F,D}_{ux}(z_{n,k})$ for $z\in Q_{n,k}$. Each $G^{(n)}$ is jointly measurable, and $G^{(n)}\to G^{F,D}_{ux}$ pointwise by continuity. The last assertion holds because $\Gamma_{u\to w}=G^{\emptyset,\{e\}}_{uu}$, where $e$ is the edge between $u$ and $w$. \end{proof} \section{Row deletion and the square blocks}\label{sec:rank-rows} We now recall the row structure of \Cref{prop:tree-bipartite}, for $C=V$, in a form that treats both parities at once. Let $y$ be a vertex with $\deg(y)\ge1$, and let $\epsilon\in\{0,1\}$ be the parity of its neighbors, so that $y\in V_{1-\epsilon}$. Let $P_\epsilon$ be the orthogonal projection onto $\ell^2(V_\epsilon)$, and let $S_\epsilon$ denote the square block of $\mathcal H^2$ on $\ell^2(V_\epsilon)$, that is, $S_0=A^*A$ and $S_1=AA^*$. The row vector, squared row norm and row direction of $y$ are \begin{equation}\label{eq:rank-row} a_y=\sum_{x\sim y}h_{xy}\delta_x\in\ell^2(V_\epsilon),\qquad U_y=\|a_y\|^2=\sum_{x\sim y}|h_{xy}|^2, \qquad v_y=U_y^{-1/2}a_y . \end{equation} Almost surely every edge weight is nonzero, and then $U_y>0$. Set \begin{equation}\label{eq:rank-K} K^{(y)}=S_\epsilon-U_y|v_y\rangle\langle v_y| . \end{equation} By \Cref{prop:tree-bipartite}(iv) and (vii), $K^{(y)}$ is a nonnegative self-adjoint operator with $\Dom(K^{(y)})=\Dom(S_\epsilon)$; for $y\in V_1$ it is the operator $K_0^{(y)}$ of \Cref{prop:tree-bipartite}. Further, for $z\in\C\setminus\R$, \begin{equation}\label{eq:rank-square-resolvent} P_\epsilon(\mathcal H^{(y)}-z)^{-1}P_\epsilon=z\,(K^{(y)}-z^2)^{-1}, \end{equation} where both sides are regarded as operators on $\ell^2(V_\epsilon)$; this is meaningful because $y\notin V_\epsilon$. We set \begin{equation}\label{eq:rank-my} \nu_y=\mu^{K^{(y)}}_{v_y},\qquad m_y(w)=F_{\nu_y}(w)=\langle v_y,(K^{(y)}-w)^{-1}v_y\rangle, \quad w\in\Hup . \end{equation} \begin{lemma}\label{lem:rank-row} Almost surely, the following holds for every vertex $y$ with $\deg(y)\ge1$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $z\in\C$ with $\Re z>0$ and $\Im z>0$, \begin{equation}\label{eq:rank-my-formula} m_y(z^2)=\frac1{zU_y}\sum_{x\sim y}|h_{xy}|^2\,\Gamma_{x\to y}(z). \end{equation} \item For every neighbor $x$ of $y$, the measure $\mu^{(y)}_x$ is invariant under $\lambda\mapsto-\lambda$. Moreover, $\nu_y$ is the image of the measure $U_y^{-1}\sum_{x\sim y}|h_{xy}|^2\mu^{(y)}_x$ under the map $\lambda\mapsto\lambda^2$. \item For every $E>0$, \begin{equation}\label{eq:rank-R-bound} R_{\nu_y}(E^2)\le\frac{2}{E^2U_y}\sum_{x\sim y}|h_{xy}|^2R_{\mu^{(y)}_x}(E). \end{equation} \end{enumerate} \end{lemma} \begin{proof} We work on the probability-one event on which the conclusions of \Cref{thm:tree-esa} and \Cref{prop:tree-bipartite} hold and all edge weights are nonzero. Part (i) is the identity \eqref{eq:tree-square-resolvent-entry} (for $y\in V_1$; for $y\in V_0$ we use \Cref{prop:tree-bipartite}(vii)), divided by $z$, since $|v_y(x)|^2=|h_{xy}|^2/U_y$. For (ii), write $\mu^{(y)}_{v_y}$ for the spectral measure of $v_y$ for $\mathcal H^{(y)}$. By \Cref{lem:tree-branch}(ii), the neighbors of $y$ lie in distinct components of the graph $V\setminus\{y\}$, and $\mathcal H^{(y)}$ is the orthogonal sum of the operators of these components (\Cref{thm:tree-esa}(iv)). It follows that $\langle\delta_x,f(\mathcal H^{(y)})\delta_{x'}\rangle=0$ for distinct neighbors $x,x'$ of $y$ and every bounded Borel function $f$, and by \eqref{eq:rank-row}, \begin{equation}\label{eq:rank-mu-vy} \mu^{(y)}_{v_y}=\frac1{U_y}\sum_{x\sim y}|h_{xy}|^2\mu^{(y)}_x . \end{equation} By \Cref{prop:tree-symmetry}(i), applied to the components of $V\setminus\{y\}$, each $\mu^{(y)}_x$ is invariant under $\lambda\mapsto-\lambda$, and by \eqref{eq:rank-mu-vy} so is $\mu^{(y)}_{v_y}$. Let $\nu'$ be the image of $\mu^{(y)}_{v_y}$ under $\lambda\mapsto\lambda^2$. For $z$ with $\Re z>0$ and $\Im z>0$, the symmetry and the identity $(\lambda-z)^{-1}+(-\lambda-z)^{-1}=2z(\lambda^2-z^2)^{-1}$ give \[ \int_\R\frac{\mu^{(y)}_{v_y}(\dd\lambda)}{\lambda-z} =\int_\R\frac{z\,\mu^{(y)}_{v_y}(\dd\lambda)}{\lambda^2-z^2} =z\int_\R\frac{\nu'(\dd t)}{t-z^2}. \] By \eqref{eq:rank-square-resolvent}, the left side equals $z\,m_y(z^2)=z\int(t-z^2)^{-1}\nu_y(\dd t)$. Since $z\mapsto z^2$ maps the open first quadrant onto $\Hup$, the Borel transforms of $\nu'$ and $\nu_y$ agree on $\Hup$. By the Stieltjes inversion formula (or by the uniqueness argument in the proof of \Cref{lem:rank-one}(iii)), $\nu'=\nu_y$. This proves (ii). For (iii), we use (ii) and the factorization $\lambda^2-E^2=(\lambda-E)(\lambda+E)$: \[ R_{\nu_y}(E^2)=\int_\R\frac{\mu^{(y)}_{v_y}(\dd\lambda)}{(\lambda-E)^2(\lambda+E)^2} \le2\int_{[0,\infty)}\frac{\mu^{(y)}_{v_y}(\dd\lambda)}{(\lambda-E)^2(\lambda+E)^2} \le\frac2{E^2}R_{\mu^{(y)}_{v_y}}(E). \] In the first inequality, we used that the integrand is even in $\lambda$ and that $\mu^{(y)}_{v_y}$ is symmetric; in the second, we used that $(\lambda+E)^2\ge E^2$ for $\lambda\ge0$. Inserting \eqref{eq:rank-mu-vy}, we obtain \eqref{eq:rank-R-bound}. \end{proof} The next lemma passes from the spectral measures of the row directions to a square block. Together with \Cref{prop:tree-bipartite}(vi), which passes from the two square blocks to $\mathcal H$, it reduces the spectral type of $\mathcal H$ in $I\cup(-I)$ to that of the row measures in $I^2$. \begin{lemma}\label{lem:rank-span} Let $K$ be a self-adjoint operator on a separable Hilbert space $\mathfrak h$, let $B\subset\R$ be a Borel set with $0\notin B$, and let $(\varphi_j)_{j\in\N}$ be vectors whose closed linear span contains $(\ker K)^\perp$. If the restriction of $\mu^K_{\varphi_j}$ to $B$ is pure point (respectively absolutely continuous) for every $j$, then $K$ has pure point spectrum (respectively is purely absolutely continuous) in $B$. \end{lemma} \begin{proof} We treat the pure point case; the other is identical. Since $0\notin B$, we have $\one_B(K)\psi=\one_B(0)\psi=0$ for $\psi\in\ker K$. For each $j$, the spectral measure of $\one_B(K)\varphi_j$ is the restriction of $\mu^K_{\varphi_j}$ to $B$, so $\one_B(K)\varphi_j\in\mathfrak h_{\mathrm{pp}}(K)$. Since $\one_B(K)$ is bounded and $\mathfrak h_{\mathrm{pp}}(K)$ is closed, $\one_B(K)$ maps the closed span of the $\varphi_j$, and hence $(\ker K)^\perp$, into $\mathfrak h_{\mathrm{pp}}(K)$. Together with $\one_B(K)\ker K=\{0\}$, this gives $\Ran\one_B(K)\subset\mathfrak h_{\mathrm{pp}}(K)$. \end{proof} Finally, we record the conditional structure of a single row. For a word $y$, let $\mathcal G^y$ be the $\sigma$-algebra of \Cref{prop:tree-bipartite}(iv) (with $C=V$, and with the parities exchanged as in \Cref{prop:tree-bipartite}(vii) if $|y|$ is even). It is generated by the genealogy and auxiliary variables $\mathcal G_0$, by the row direction $v_y$, and by the marks of all edges not incident to $y$. We write $\Omega_y=\{y\in V,\ \deg(y)\ge1\}$, which belongs to $\mathcal G_0$. A \emph{probability kernel} from $(\Omega,\mathcal G^y)$ to $\R$ is a map $\omega\mapsto\kappa(\omega)$ into the Borel probability measures on $\R$ such that $\omega\mapsto\kappa(\omega)(B)$ is $\mathcal G^y$-measurable for every Borel set $B\subset\R$. For $d\ge1$, let \begin{equation}\label{eq:rank-gamma-density} g_d(u)=\frac{p^d}{\Gamma(d)}u^{d-1}e^{-pu}\one_{(0,\infty)}(u). \end{equation} \begin{lemma}\label{lem:rank-conditioning} Let $\mathsf Q\in\{\P,\P^\infty\}$ and let $y$ be a word. \begin{enumerate}[label=\textup{(\roman*)}] \item There exists a probability kernel $\nu'_y$ from $(\Omega,\mathcal G^y)$ to $\R$ such that, $\mathsf Q$-almost surely on $\Omega_y$, we have $\nu'_y=\nu_y$. \item For every $\mathcal G^y\otimes\mathcal B(\R)$-measurable function $\Phi\colon\Omega\times\R\to[0,\infty]$, \begin{equation}\label{eq:rank-disintegration} \E_{\mathsf Q}\bigl[\Phi(\cdot,U_y)\one_{\Omega_y}\bigr] =\E_{\mathsf Q}\left[\one_{\Omega_y}\int_0^\infty\Phi(\cdot,u)\,g_{\deg(y)}(u)\dd u\right]. \end{equation} \end{enumerate} \end{lemma} \begin{proof} We begin with (i). The operator $\mathcal H^{(y)}$ depends only on the genealogy and on the marks of edges not incident to $y$. For a neighbor $x$ of $y$, let $\widehat G_x(z)$ be the limit of the diagonal entries at $x$ of the resolvents at $z$ of the Dirichlet truncations of $\mathcal H^{(y)}$ to the balls of radius $n$ around $x$ in the component of $x$, when this limit exists, and zero otherwise. The argument of \Cref{lem:rank-measurable} shows that $\widehat G_x$ is $\mathcal G^y\otimes\mathcal B(\Hup)$-measurable, and $\widehat G_x=G^{(y)}_{xx}$ on $\Omega_{\mathrm{sa}}$ by \Cref{thm:tree-esa}(iii). For $w\in\Hup$, let $\sqrt w$ be the square root in the open first quadrant, and set \[ M_y(w)=\frac1{\sqrt w}\sum_{x\sim y}|v_y(x)|^2\,\widehat G_x(\sqrt w) \] on $\Omega_y$, and $M_y(w)=-1/w$ off $\Omega_y$. This function is $\mathcal G^y\otimes\mathcal B(\Hup)$-measurable, and by \eqref{eq:rank-my-formula} and \eqref{eq:rank-deleted-diagonal}, almost surely on $\Omega_y$ we have $M_y=m_y$ on $\Hup$. We recover $\nu_y$ from $M_y$ by the Stieltjes inversion formula: for a finite measure $\nu$ and $a\lambda\}$, and for $\omega\notin A$ let $\nu'_y(\omega)=\delta_0$. Then $\omega\mapsto\nu'_y(\omega)((-\infty,\lambda])$ is $\mathcal G^y$-measurable for every $\lambda$, and since the Borel sets $B$ for which $\omega\mapsto\nu'_y(\omega)(B)$ is measurable form a Dynkin system, $\nu'_y$ is a probability kernel. Almost surely on $\Omega_y$, the distribution functions of $\nu'_y$ and $\nu_y$ agree at every $\lambda$, by right continuity, and so $\nu'_y=\nu_y$. For (ii), \Cref{prop:tree-bipartite}(iv) (and (vii)) states that, under $\P$ and under $\P^\infty$, conditionally on $\mathcal G^y$, the squared row norm $U_y$ has the density $g_{\deg(y)}$ on $\Omega_y$; recall that the number of neighbors of $y$ in $V$ is $\deg(y)$, which is $\mathcal G^y$-measurable. The identity \eqref{eq:rank-disintegration} is the disintegration formula for this conditional law \cite[Theorem~6.4]{Kallenberg2002foundations}, applied with the $\mathcal G^y$-measurable random element $\omega\mapsto\omega$ of $(\Omega,\mathcal G^y)$. \end{proof} We now attach a rank-one family to the kernel $\nu'_y$. For $\omega\in\Omega$, let $L^y(\omega)$ be the operator of multiplication by the variable on $L^2(\nu'_y(\omega))$, and let $\mathbf 1$ be the constant function, whose spectral measure for $L^y$ is $\nu'_y$. Let $(\mu^y_u)_{u\in\R}$ be the family \eqref{eq:rank-family} with $L=L^y$ and $v=\mathbf 1$, so that $\mu^y_0=\nu'_y$. The Borel transform of $\nu'_y$ is $\mathcal G^y$-measurable, so \Cref{lem:rank-one}(iv) applies with $(\Omega',\mathscr A)=(\Omega,\mathcal G^y)$. By \eqref{eq:rank-AK}, the Borel transform of $\mu^y_u$ is determined by $u$ and the Borel transform of $\mu^y_0$, and the same holds for the spectral measures of $v_y$ for the operators $K^{(y)}+u|v_y\rangle\langle v_y|$, whose member with $u=0$ is $\nu_y$. Since a finite measure is determined by its Borel transform, we conclude from \Cref{lem:rank-conditioning}(i) and \eqref{eq:rank-K} that, $\mathsf Q$-almost surely on $\Omega_y$, \begin{equation}\label{eq:rank-family-y} \mu^y_u=\mu^{K^{(y)}+u|v_y\rangle\langle v_y|}_{v_y}\quad\text{for every }u\in\R, \qquad\text{in particular}\qquad\mu^y_{U_y}=\mu^{S_\epsilon}_{v_y}, \end{equation} where $\epsilon$ is the parity of the neighbors of $y$. \section{The pure point criterion}\label{sec:rank-pp-proof} \begin{proposition}\label{prop:rank-pp} Let $\mathsf Q\in\{\P,\P^\infty\}$ and let $I\subset(0,\infty)$ be a Borel set (for example, a compact interval). Suppose that $\mathsf Q$-almost surely, for Lebesgue-almost every $E\in I$, for every vertex $y$ and every neighbor $x$ of $y$, \begin{equation}\label{eq:rank-pp-hypothesis} \int_\R\frac{\mu^{(y)}_x(\dd\lambda)}{(\lambda-E)^2}<\infty . \end{equation} Then $\mathsf Q$-almost surely, $\mathcal H$ has pure point spectrum in $I\cup(-I)$. If the hypothesis holds $\P$-almost surely, then it holds $\P^\infty$-almost surely. \end{proposition} The order of quantifiers in the hypothesis is the following: there exists an event of $\mathsf Q$-probability one on which, outside a Lebesgue-null set of energies $E\in I$ that may depend on the realization, \eqref{eq:rank-pp-hypothesis} holds simultaneously for all pairs of adjacent vertices. The integral in \eqref{eq:rank-pp-hypothesis} is $R_{\mu^{(y)}_x}(E)$ in the notation of \eqref{eq:rank-R}; in particular, it is infinite if $E$ is an eigenvalue of the component of $x$ in $V\setminus\{y\}$ with $\mu^{(y)}_x(\{E\})>0$. \begin{proof} The last assertion holds because $\P^\infty=\P(\,\cdot\mid\mathcal S)$ is absolutely continuous with respect to $\P$. We now fix $\mathsf Q$ and assume the hypothesis under $\mathsf Q$. The proof isolates one row $y$ at a time. Conditional on all variables except the squared row norm $U_y$, the spectral measure of $v_y$ for the square block is the member at coupling $U_y$ of a rank-one family, and \Cref{lem:rank-row} converts the hypothesis into the finiteness of the reciprocal-square integral of the unperturbed member at almost every point of $I^2$. The Simon--Wolff criterion and spectral averaging then show that the perturbed measure is pure point on $I^2$ for almost every coupling, and the density of $U_y$ transfers this to the realized coupling. The row directions span the orthogonal complement of the kernel of each square block, which gives the conclusion for the square blocks and then for $\mathcal H$. \proofstep{1}{Reduction to a single row.} We claim that it suffices to prove the following statement for every word $y$: $\mathsf Q$-almost surely on $\{y\in V,\ \deg(y)\ge1\}$, the restriction of $\mu^{S_\epsilon}_{v_y}$ to $I^2$ is pure point, where $\epsilon$ is the parity of the neighbors of $y$. Indeed, the set of words is countable, and this then holds $\mathsf Q$-almost surely for all such $y$ simultaneously. Fix $\epsilon\in\{0,1\}$. The vertices whose neighbors lie in $V_\epsilon$ are those in $V_{1-\epsilon}$, and $A^*\delta_y=a_y$ for $y\in V_1$, while $A\delta_x=a_x$ for $x\in V_0$. By \Cref{prop:tree-bipartite}(ii) and (vii), the row directions $v_y$, $y\in V_{1-\epsilon}$ with $\deg(y)\ge1$, span $(\ker A)^\perp$ if $\epsilon=0$ and $(\ker A^*)^\perp$ if $\epsilon=1$. Since $\ker A^*A=\ker A$ and $\ker AA^*=\ker A^*$, they span $(\ker S_\epsilon)^\perp$. Since $0\notin I^2$, \Cref{lem:rank-span} shows that $S_\epsilon$ has pure point spectrum in $I^2$. By \Cref{prop:tree-bipartite}(vi), $\mathcal H$ has pure point spectrum in $I\cup(-I)$. \proofstep{2}{The exceptional set of the unperturbed row measure.} Fix a word $y$, let $\nu'_y$ be the kernel of \Cref{lem:rank-conditioning}(i), and set \[ T_y=\{\lambda\in I^2:R_{\nu'_y}(\lambda)=\infty\}. \] We claim that $\{(\omega,\lambda):\lambda\in T_y\}$ is $\mathcal G^y\otimes\mathcal B(\R)$-measurable and that $|T_y|=0$, $\mathsf Q$-almost surely on $\Omega_y$. The measurability follows from \Cref{lem:rank-one}(iv), applied with $(\Omega',\mathscr A)=(\Omega,\mathcal G^y)$, $L=L^y$ and $v=\mathbf 1$. For the second assertion, consider a realization in $\Omega_y$ and in the probability-one event of the hypothesis, of \Cref{lem:rank-row} and of \Cref{lem:rank-conditioning}(i), so that $\nu'_y=\nu_y$. Let $E\in I$ be an energy at which \eqref{eq:rank-pp-hypothesis} holds for all $x\sim y$. Then \eqref{eq:rank-R-bound} implies that $R_{\nu_y}(E^2)<\infty$. We conclude that $T_y\subset q(\mathcal N)$, where $q(E)=E^2$ and $\mathcal N\subset I$ is the Lebesgue-null set of energies at which the hypothesis fails. Since $q$ is locally Lipschitz, $|T_y|=0$. \proofstep{3}{Spectral averaging in the squared row norm.} Let $(\mu^y_u)_{u\in\R}$ be the rank-one family of \eqref{eq:rank-family-y}, so that $\mu^y_0=\nu'_y$ and, almost surely on $\Omega_y$, $\mu^y_{U_y}=\mu^{S_\epsilon}_{v_y}$. By \Cref{lem:rank-criteria}(i) with $L=L^y$ and $S=I^2$, the restriction of $\mu^y_u$ to $I^2$ is pure point whenever $\mu^y_u(T_y)=0$. Set \[ \Phi(\omega,u)=\one\{\mu^y_u(T_y)>0\}. \] By \Cref{lem:rank-one}(iv), with $T=\{(\omega,\lambda):\lambda\in T_y\}$, the function $\Phi$ is $\mathcal G^y\otimes\mathcal B(\R)$-measurable. For every $\omega$ with $|T_y|=0$, spectral averaging \eqref{eq:rank-averaging} gives $\int_\R\mu^y_u(T_y)\dd u=|T_y|=0$, and $\Phi(\omega,u)=0$ for Lebesgue-almost every $u$. Using \eqref{eq:rank-disintegration} and Step~2, we obtain \begin{align*} \mathsf Q\bigl(\{\mu^y_{U_y}(T_y)>0\}\cap\Omega_y\bigr) &\le\mathsf Q\bigl(\{|T_y|>0\}\cap\Omega_y\bigr)\\ &\quad+\E_{\mathsf Q}\left[\one_{\Omega_y}\one\{|T_y|=0\}\int_0^\infty\Phi(\cdot,u)\,g_{\deg(y)}(u)\dd u\right]=0 . \end{align*} Then, $\mathsf Q$-almost surely on $\Omega_y$, the restriction of $\mu^{S_\epsilon}_{v_y}=\mu^y_{U_y}$ to $I^2$ is pure point. By Step~1, this completes the proof. \end{proof} \begin{remark}\label{rem:rank-cyclic} The hypothesis of \Cref{prop:rank-pp} concerns spectral measures of the vertex-deleted operators at the real axis. In \Cref{sec:loc}, it is verified by constructing, for almost every energy $E$, a solution $f\in\ell^2$ of the equation $(\mathcal H^{(y)}-E)f=\delta_x$ (see \Cref{lem:loc-range}). \end{remark} \section{The absolutely continuous criterion}\label{sec:rank-ac-proof} \begin{proposition}\label{prop:rank-ac} Let $I\subset(0,\infty)$ be a Borel set (for example, an interval). Suppose that for Lebesgue-almost every $E\in I$, \begin{equation}\label{eq:rank-ac-hypothesis} \P\bigl(\Im\Gamma_o(E+\iu0)>0\bigr)>0, \end{equation} where the event is that the vertical boundary value $\Gamma_o(E+\iu0)$ exists and has positive imaginary part. Then $\P^\infty$-almost surely, $\mathcal H$ is purely absolutely continuous in $I\cup(-I)$. \end{proposition} By \Cref{prop:rde-boundary}(i), at every $E\in\mathcal E_p$ with $E>0$ the probability in \eqref{eq:rank-ac-hypothesis} is either $0$ or $\varrho_p$. Before the proof, we record two deterministic lemmas. The first propagates positivity of imaginary parts along paths, and the second describes the infinite components left after deleting a vertex. They are the pathwise form, at all energies outside a null set, of statements proved at a fixed regular energy in \Cref{sec:rde}. \begin{lemma}\label{lem:rank-propagate} Fix a realization in $\Omega_{\mathrm{sa}}$ with all edge weights nonzero, and fix $E\in\R$ such that $\Gamma_{u\to w}(E+\iu0)$ exists for every oriented edge $(u,w)$. If $(u,w)$ is an oriented edge and $\Im\Gamma_{x\to u}(E+\iu0)>0$ for some neighbor $x\ne w$ of $u$, then $\Im\Gamma_{u\to w}(E+\iu0)>0$. \end{lemma} \begin{proof} By \eqref{eq:tree-schur-cavity} with $C=V$, we have $\Gamma_{u\to w}(z)=-\Delta(z)^{-1}$ with \[ \Delta(z)=z+\sum_{x'\sim u,\,x'\ne w}|h_{ux'}|^2\Gamma_{x'\to u}(z). \] Since the sum is finite, the boundary value $\Delta(E+\iu0)$ exists. Every $\Gamma_{x'\to u}$ maps $\Hup$ to $\Hup$, and we have $\Im\Gamma_{x'\to u}(E+\iu0)\ge0$ and $\Im\Delta(E+\iu0)\ge|h_{ux}|^2\Im\Gamma_{x\to u}(E+\iu0)>0$. In particular $\Delta(E+\iu0)\ne0$, so $\Gamma_{u\to w}(E+\iu0)=-\Delta(E+\iu0)^{-1}$, and $\Im\Gamma_{u\to w}(E+\iu0)=\Im\Delta(E+\iu0)/|\Delta(E+\iu0)|^2>0$. \end{proof} \begin{lemma}\label{lem:rank-attach} Fix a realization as in \Cref{lem:rank-propagate}, and fix $E\in\R$ such that $\Gamma_{u\to w}(E+\iu0)$ exists for every oriented edge $(u,w)$ and $\Im\Gamma_v(E+\iu0)>0$ for every vertex $v$ whose forward subtree $\mathcal T_v$ is infinite. Let $y$ be a vertex and $x$ a neighbor of $y$ whose component in $V\setminus\{y\}$ is infinite. Then $\Im\Gamma_{x\to y}(E+\iu0)>0$. \end{lemma} \begin{proof} If $x$ is a child of $y$, then the component of $x$ is $V(\mathcal T_x)$ and $\Gamma_{x\to y}=\Gamma_x$; the claim is then part of the hypothesis. Otherwise $y\ne o$ and $x=y^-$. Let $o=a_0,a_1,\dots,a_k=y^-$ be the path from the root to $y^-$, and set $a_{k+1}=y$. The component of $y^-$ is $V\setminus V(\mathcal T_y)$, and it is the union of the path $\{a_0,\dots,a_k\}$ and the sets $V(\mathcal T_w)$, where $w$ ranges over the children of the $a_j$ ($0\le j\le k$) with $w\ne a_{j+1}$. There are finitely many such $w$, since the path is finite and degrees are finite. As the component is infinite, one of these forward subtrees, say $\mathcal T_w$ with $w$ a child of $a_j$ and $w\ne a_{j+1}$, is infinite, and $\Im\Gamma_w(E+\iu0)>0$ by hypothesis. We show by induction on $i=j,\dots,k$ that $\Im\Gamma_{a_i\to a_{i+1}}(E+\iu0)>0$. For $i=j$, the vertex $w$ is a neighbor of $a_j$ different from $a_{j+1}$, and $\Gamma_{w\to a_j}=\Gamma_w$, so \Cref{lem:rank-propagate} applies. For $j\le i0$. \end{proof} \begin{proof}[Proof of \Cref{prop:rank-ac}] The proof establishes, at almost every energy of $I$, that the row measures $\nu_y$ have Borel transforms with positive imaginary boundary values, and then applies the absolutely continuous criterion of \Cref{lem:rank-criteria}(ii) with spectral averaging in $U_y$. Positivity on every infinite forward subtree comes from \eqref{eq:rank-ac-hypothesis} and the zero--one law of \Cref{prop:rde-boundary}; \Cref{lem:rank-attach} transfers it to the attachment points of the infinite components of the vertex-deleted tree; and \Cref{lem:rank-row}(i) together with \Cref{lem:rank-square} transfers it to the Borel transform of $\nu_y$ at the point $E^2$. \proofstep{1}{Positivity on infinite forward subtrees.} Let $\mathcal B_1$ be the set of $(\omega,E)\in\Omega\times I$ such that, for every vertex $v$ with $\mathcal T_v$ infinite, the boundary value $\Gamma_v(E+\iu0)$ exists and has positive imaginary part. By \Cref{lem:rank-measurable} and the rational Cauchy criterion \eqref{eq:rank-rational-cauchy}, the set of $(\omega,E)$ at which $\Gamma_v(E+\iu0)$ exists is measurable and the boundary value is a measurable function on it. Since the event that $\mathcal T_v$ is infinite is measurable and there are countably many words $v$, the set $\mathcal B_1$ is jointly measurable. Let $E\in I\cap\mathcal E_p$ be an energy at which \eqref{eq:rank-ac-hypothesis} holds; this is the case for Lebesgue-almost every $E\in I$, since the complement of $\mathcal E_p$ is Lebesgue-null (\Cref{def:rde-regular}). By \Cref{prop:rde-boundary}(i), the probability in \eqref{eq:rank-ac-hypothesis} equals $\varrho_p$, and $\P^\infty((\omega,E)\in\mathcal B_1)=1$. By Tonelli's theorem, \[ \E^\infty\bigl|\{E\in I:(\omega,E)\notin\mathcal B_1\}\bigr| =\int_I\P^\infty\bigl((\omega,E)\notin\mathcal B_1\bigr)\dd E=0 . \] Further, each $\Gamma_{u\to w}$ is the Borel transform of a spectral measure. By \Cref{lem:rank-borel}(i), for every realization and every oriented edge, $\Gamma_{u\to w}(E+\iu0)$ exists for Lebesgue-almost every $E$; there are countably many oriented edges. We conclude that there exists an event $\Omega_1$ with $\P^\infty(\Omega_1)=1$, contained in $\Omega_{\mathrm{sa}}$, in the event of \Cref{lem:rank-row}, and in the event that all edge weights are nonzero and $\mathcal T$ is infinite, with the following property. For every $\omega\in\Omega_1$ there exists a Lebesgue-null set $\mathcal N(\omega)\subset I$ such that for every $E\in I\setminus\mathcal N(\omega)$, all boundary values $\Gamma_{u\to w}(E+\iu0)$ exist, and $\Im\Gamma_v(E+\iu0)>0$ for every $v$ with $\mathcal T_v$ infinite. \proofstep{2}{Positivity for the row measures.} Fix a word $y$, and let $\nu'_y$ be the kernel of \Cref{lem:rank-conditioning}(i). Let $T'_y$ be the set of $\lambda\in I^2$ such that the boundary value $F_{\nu'_y}(\lambda+\iu0)$ does not exist or has imaginary part zero. By \Cref{lem:rank-one}(iv), applied with $(\Omega',\mathscr A)=(\Omega,\mathcal G^y)$, $L=L^y$ and $v=\mathbf 1$, the set $\{(\omega,\lambda):\lambda\in T'_y\}$ is $\mathcal G^y\otimes\mathcal B(\R)$-measurable. We claim that $|T'_y|=0$, $\P^\infty$-almost surely on $\Omega_y$. To that end, fix $\omega\in\Omega_1\cap\Omega_y$ such that $\nu'_y=\nu_y$, so that $F_{\nu'_y}=m_y$, and fix $E\in I\setminus\mathcal N(\omega)$. The set $V\setminus\{y\}$ is the union of the components of the $\deg(y)<\infty$ neighbors of $y$, and $V$ is infinite; hence some neighbor $x_0$ of $y$ has an infinite component. By \Cref{lem:rank-attach}, $\Im\Gamma_{x_0\to y}(E+\iu0)>0$. By \eqref{eq:rank-my-formula} with $z=E+\iu\eta$, the limit \[ \ell=\lim_{\eta\downarrow0}m_y\bigl((E+\iu\eta)^2\bigr) =\frac1{EU_y}\sum_{x\sim y}|h_{xy}|^2\,\Gamma_{x\to y}(E+\iu0) \] exists, and $\Im\ell\ge(EU_y)^{-1}|h_{x_0y}|^2\Im\Gamma_{x_0\to y}(E+\iu0)>0$, because all terms have nonnegative imaginary part and $h_{x_0y}\ne0$. By \Cref{lem:rank-square}, applied to $\nu_y$, the boundary value $m_y(E^2+\iu0)$ exists and equals $\ell$, and $E^2\notin T'_y$. In particular, $T'_y$ is contained in $\{E^2:E\in\mathcal N(\omega)\}$, which is Lebesgue-null. Since $\P^\infty(\Omega_1)=1$ and $\nu'_y=\nu_y$ almost surely on $\Omega_y$, this proves the claim. \proofstep{3}{Spectral averaging.} Fix a word $y$, let $\epsilon$ be the parity of its neighbors, and let $(\mu^y_u)$ be the rank-one family of \eqref{eq:rank-family-y}, so that $\mu^y_0=\nu'_y$ and, almost surely on $\Omega_y$, $\mu^{S_\epsilon}_{v_y}=\mu^y_{U_y}$. By \Cref{lem:rank-criteria}(ii) with $L=L^y$ and $S=I^2$, the restriction of $\mu^y_u$ to $I^2$ is absolutely continuous whenever $\mu^y_u(T'_y)=0$. As in Step~3 of the proof of \Cref{prop:rank-pp}, the function $\Phi(\omega,u)=\one\{\mu^y_u(T'_y)>0\}$ is $\mathcal G^y\otimes\mathcal B(\R)$-measurable, it vanishes for Lebesgue-almost every $u$ when $|T'_y|=0$, and \eqref{eq:rank-disintegration} with $\mathsf Q=\P^\infty$ gives $\P^\infty(\{\mu^y_{U_y}(T'_y)>0\}\cap\Omega_y)=0$. It follows that, $\P^\infty$-almost surely, the restriction of $\mu^{S_\epsilon}_{v_y}$ to $I^2$ is absolutely continuous for every vertex $y$ with $\deg(y)\ge1$. \proofstep{4}{Conclusion.} As in Step~1 of the proof of \Cref{prop:rank-pp}, the row directions of the vertices in $V_{1-\epsilon}$ span $(\ker S_\epsilon)^\perp$, and $0\notin I^2$. By \Cref{lem:rank-span}, each square block $S_\epsilon$ is purely absolutely continuous in $I^2$, and by \Cref{prop:tree-bipartite}(vi), $\mathcal H$ is purely absolutely continuous in $I\cup(-I)$, $\P^\infty$-almost surely. \end{proof} \begin{remark}\label{rem:rank-ac-scope} The hypothesis of \Cref{prop:rank-ac} is a statement about the unconditioned law at almost every energy of $I$, and the conclusion is a statement about the survival-conditioned operator. On the extinction event the tree is finite and the spectrum is pure point, and the conclusion does not hold under $\P$. The energy $0$ is excluded: $\mathcal H$ has finitely supported eigenvectors with eigenvalue zero (\Cref{prop:tree-zero-modes}). \end{remark} \chapter{Negative free energy implies localization}\label{sec:loc} In this section, we prove that a real solution of the cavity equation with negative free energy at $s=1$ forces pure point spectrum. The precise statement is \Cref{thm:loc-criterion}: if for almost every energy $E$ in a set $U\subset(0,\infty)$ there exists a real solution $\pi$ at $E$ with $\varphi_\pi(1)<0$, then $\mathcal H$ has pure point spectrum in $U\cup(-U)$, almost surely under $\P$ and under $\P^\infty$. The real solution is not assumed to be related to the boundary values of the resolvent of $\mathcal H$, and the conclusion concerns only the operator. We now describe the argument. Fix $E>0$ and a real solution $\pi$ at $E$. By \Cref{lem:rde-rtp}, there exists a family of real denominators $(D_v)$ on the tree, with marginal law $\pi$, satisfying the cavity recursion $D_v=E-p^{-1}\sum_iY_{v,vi}/D_{vi}$ at every vertex. It is defined on an extension of the probability space of the marked tree. Setting $\gamma_v=-1/D_v$, the products \[ F(v)=\gamma_o\prod_{j=1}^L\bigl(-h_{v_jv_{j-1}}\gamma_{v_j}\bigr),\qquad o=v_0,v_1,\dots,v_L=v, \] define a function $F$ on the vertices that solves $(\mathcal H-E)F=\delta_o$ pointwise. We call it the formal inverse column. By the many-to-one formula, the expected generation-$L$ sum of $|F(v)|^s$ is the path moment $\Phi_L(s;E)$ of \Cref{def:press-path-moment}, whose exponential growth rate is the free energy $\varphi_\pi(s)$ by \Cref{prop:press-identification}. If $\varphi_\pi(1)<0$, we may choose $s<1$ with $\varphi_\pi(s)<0$, and then $F\in\ell^s\subset\ell^2$ almost surely. Using the maximal domain characterization of \Cref{thm:tree-esa}, we then obtain $\delta_o=(\mathcal H-E)F$, and this implies that $\int\mu_o(\dd\lambda)/(\lambda-E)^2<\infty$. The Simon--Wolff criterion of \Cref{prop:rank-pp} requires the same statement for every neighbor of every removed vertex. We obtain it by rerooting: on a component of the forest obtained by removing a vertex or an edge, rerooted at any of its vertices, the inverse column is computed from finitely many real cavity marks along an ancestral path together with finitely many forward columns hanging off that path. In \Cref{sec:loc-column}, we define the formal inverse columns and state the main results, \Cref{prop:loc-columns} and \Cref{thm:loc-criterion}. \Cref{sec:loc-palm} proves the many-to-one identity. \Cref{sec:loc-forward} proves summability of the forward columns, and \Cref{sec:loc-reroot} constructs the rerooted columns. \Cref{sec:loc-proofs} completes the proofs. Throughout this section, $p>1$ is fixed. \section{The formal inverse column}\label{sec:loc-column} We begin with a deterministic lemma, which is the real-energy analogue of the Schur complement formula of \Cref{prop:tree-schur} read as an equation for a column of the resolvent. \begin{lemma}\label{lem:loc-pointwise} Let $C$ be a countable, connected, locally finite tree with edge weights $h_{xy}=\overline{h_{yx}}$, let $r\in C$, and let $E\in\R$. For $x\in C\setminus\{r\}$, let $b(x)$ be the neighbor of $x$ on the path from $x$ to $r$, and let $\operatorname{ch}(x)$ be the set of neighbors $x'$ of $x$ with $x'\ne b(x)$; let $\operatorname{ch}(r)$ be the set of all neighbors of $r$. Suppose that real numbers $(\gamma_x)_{x\in C}$ satisfy, for every $x\in C$, \begin{equation}\label{eq:loc-cavity-relation} \Delta_x=E+\sum_{x'\in\operatorname{ch}(x)}|h_{xx'}|^2\gamma_{x'}\ne0,\qquad\gamma_x=-\frac1{\Delta_x}. \end{equation} Define $f(r)=\gamma_r$, and $f(x)=-\gamma_x\,h_{x\,b(x)}\,f(b(x))$ for $x\ne r$, recursively along the path from $r$ to $x$. Then for every $x\in C$, \begin{equation}\label{eq:loc-pointwise} \sum_{x'\sim x}h_{xx'}f(x')-Ef(x)=\delta_r(x). \end{equation} \end{lemma} \begin{proof} Fix $x\in C$. For $x'\in\operatorname{ch}(x)$ we have $b(x')=x$, hence $f(x')=-\gamma_{x'}h_{x'x}f(x)$ and $h_{xx'}f(x')=-|h_{xx'}|^2\gamma_{x'}f(x)$. The set $\operatorname{ch}(x)$ is finite, and by \eqref{eq:loc-cavity-relation}, \[ \sum_{x'\in\operatorname{ch}(x)}h_{xx'}f(x')-Ef(x)=-\Delta_xf(x)=\frac{f(x)}{\gamma_x}. \] If $x=r$, then $\operatorname{ch}(r)$ contains all neighbors of $r$, and the right side equals $\gamma_r/\gamma_r=1$. If $x\ne r$, the only neighbor of $x$ outside $\operatorname{ch}(x)$ is $b(x)$, and $f(x)/\gamma_x=-h_{x\,b(x)}f(b(x))$; the left side of \eqref{eq:loc-pointwise} then equals $-h_{x\,b(x)}f(b(x))+h_{x\,b(x)}f(b(x))=0$. \end{proof} We now fix $E>0$ and a real solution $\pi$ at energy $E$ (\Cref{def:rde-real}), and we let $(\omega,(D_v)_{v\in\mathbb U})$ be a recursive tree process with marginal $\pi$, as in \Cref{lem:rde-rtp}. We use the realization of the marked tree of \Cref{def:palm-config} and \Cref{sec:rde-ulam}: the edge joining $v$ and $vi$ carries the mark $w_{v,vi}=w_{vi}$, the weight $h_{v,vi}=w_{v,vi}/\sqrt p$ and the squared modulus $Y_{v,vi}=|w_{v,vi}|^2$. As in the proof of \Cref{lem:rde-rtp}, the innovation at a word $v$ is $\xi_v=(N_v,\zeta_v,(w_{v,vi})_{i\ge1})$, and for every word $w$ the marked tree $\mathcal T_w$ rooted at $w$ is built from the innovations at the words in $w\mathbb U=\{wu:u\in\mathbb U\}$. For $w\in V$, it is the forward subtree of $w$. By \Cref{lem:rde-rtp}, the configuration $\omega$ has the law $\P$, so the marked tree $\mathcal T$ has the law of \Cref{def:tree-pgw}. We keep writing $\P$ and $\E$ for the probability measure and the expectation on the space of \Cref{lem:rde-rtp}. We may do so because events concerning only the marked tree have the same probability there as under $\P$. For a word $w$, we write \[ \Theta_w=\bigl(\theta_w\omega,(D_{wu})_{u\in\mathbb U}\bigr) \] for the process seen from $w$, where $\theta_w$ is the shift of \Cref{def:palm-config}; it is in measurable bijection with $(\xi_{wu},D_{wu})_{u\in\mathbb U}$, and $\Theta_o=(\omega,(D_u)_{u\in\mathbb U})$ is the process itself. We use the following three properties of the process, which are the conclusions of \Cref{lem:rde-rtp}. \begin{enumerate}[label=\textup{(R\arabic*)}] \item Almost surely, every $D_v$ is a nonzero real number, and the recursion $D_v=E-p^{-1}\sum_{i\le N_v}Y_{v,vi}/D_{vi}$ holds at every word $v$. \item For every $n\ge0$, the variables $D_v$, $|v|=n$, are independent with law $\pi$, and they are independent of $\mathscr F_n=\sigma(\xi_u:|u|0$, \begin{equation}\label{eq:loc-Phi} \Phi_L(s;E)=g^s\int_\R(\mathcal A_s^Le_s)(x)\,\pi^x(\dd x), \end{equation} where $e_s(x)=|x|^s$, $\pi^x$ is the law of $X=1/(gD)$ with $D\sim\pi$, and $\mathcal A_s=\mathcal A_s^{(g)}$ is the operator of \Cref{def:press-operators} for the law of $B=gD$ and the rate $K=p/g^2$, so that $\beta=g^2Y/p$ with $Y\sim\Exp(1)$. By \Cref{lem:press-chain}(ii), $\Phi_L(s;E)$ does not depend on $g$. For a self-adjoint operator $T$ and a vector $\psi$, we use the reciprocal-square integral $R_{\mu^T_\psi}(E)=\int\mu^T_\psi(\dd\lambda)/(\lambda-E)^2$ of \eqref{eq:rank-R}, which is infinite if $\mu^T_\psi(\{E\})>0$. For a component $C$ of a forest obtained by a finite removal and $r\in C$, we write $\mu^C_r$ for the spectral measure of $\delta_r$ for $\mathcal H_C$ (\Cref{thm:tree-esa}). \begin{proposition}\label{prop:loc-columns} Let $E>0$, let $\pi$ be a real solution at energy $E$, let $(\xi_v,D_v)_{v\in\mathbb U}$ be a recursive tree process with marginal $\pi$ (\Cref{lem:rde-rtp}), and let $F$ be the formal inverse column of \Cref{def:loc-column}. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $s\in(0,1)$ and $L\ge0$, \begin{equation}\label{eq:loc-palm-identity} \E\sum_{v\in V:\,|v|=L}|F(v)|^s=\Phi_L(s;E). \end{equation} \end{enumerate} Suppose in addition that $\varphi_\pi(1)<0$. Then the following hold. \begin{enumerate}[label=\textup{(\roman*)},resume] \item There exists $s_0\in(0,1)$ such that $\varphi_\pi(s)<0$ for $s\in[s_0,1)$. For every such $s$, almost surely $F\in\ell^s(V)$; moreover $F\in\Dom(\mathcal H)$ and $(\mathcal H-E)F=\delta_o$. \item Almost surely under $\P$ and under $\P^\infty$, the following holds. For every vertex $v$, we have $\delta_v\in\Ran(\mathcal H-E)$ and $\int\mu_v(\dd\lambda)/(\lambda-E)^2<\infty$. For every vertex $y$ and every edge $e$ of $\mathcal T$, every component $C$ of the reduced forest obtained by removing $y$ (or $e$), and every $r\in C$, we have $\delta_r\in\Ran(\mathcal H_C-E)$ and $\int\mu^C_r(\dd\lambda)/(\lambda-E)^2<\infty$. In particular, for every vertex $y$ and every neighbor $x$ of $y$, \begin{equation}\label{eq:loc-neighbor} \int_\R\frac{\mu^{(y)}_x(\dd\lambda)}{(\lambda-E)^2}<\infty . \end{equation} \end{enumerate} \end{proposition} We note that the assertions in (iii) concern only the marked tree and do not refer to $\pi$. Only their proof uses the recursive tree process. Combining \Cref{prop:loc-columns} with the Simon--Wolff criterion of \Cref{prop:rank-pp}, we obtain the main result of this section. \begin{theorem}\label{thm:loc-criterion} Let $U\subset(0,\infty)$ be a Borel set (for example, an open set). Suppose that for Lebesgue-almost every $E\in U$ there exists a real solution $\pi_E$ at energy $E$ with $\varphi_{\pi_E}(1)<0$. Then, almost surely under $\P$ and under $\P^\infty$, the operator $\mathcal H$ has pure point spectrum in $U\cup(-U)$. \end{theorem} \section{The many-to-one identity}\label{sec:loc-palm} We prove \Cref{prop:loc-columns}(i). For a vertex $v$ with $|v|=L$ and path $o=v_0,\dots,v_L=v$, and for $s\in(0,1)$, set \begin{equation}\label{eq:loc-weight} W_L(v)=\prod_{j=1}^L\Bigl(\frac{Y_{v_{j-1},v_j}}p\Bigr)^{s/2}|D_{v_j}|^{-s},\qquad W_0(o)=1 . \end{equation} Since $|h_{v_jv_{j-1}}|^s=(Y_{v_{j-1},v_j}/p)^{s/2}$ and $|\gamma_v|=|D_v|^{-1}$, \eqref{eq:loc-column-product} gives \begin{equation}\label{eq:loc-F-weight} |F(v)|^s=|D_o|^{-s}\,W_L(v). \end{equation} \begin{lemma}\label{lem:loc-palm} Let $s\in(0,1)$ and $g>0$, and set $X_o=1/(gD_o)$. For every $L\ge0$ and every Borel function $k\colon\R\to[0,\infty]$, \begin{equation}\label{eq:loc-palm-general} \E\sum_{v\in V:\,|v|=L}k(X_o)\,W_L(v)=\int_\R(\mathcal A_s^Lk)(x)\,\pi^x(\dd x). \end{equation} \end{lemma} The identity \eqref{eq:loc-palm-general} is the many-to-one formula of \Cref{prop:palm-spine} for the recursive tree process; compare \Cref{cor:palm-chain} and \Cref{rem:palm-rtp}. We give a direct proof by induction on $L$, using the Mecke formula at the root, which works with the consistent process of \Cref{lem:rde-rtp} on the whole tree and so requires no truncation. (Alternatively, the left side of \eqref{eq:loc-palm-general} is a function of $(\xi_u)_{|u|0$ and let $\pi$ be a real solution at energy $E$ with $\varphi_\pi(1)<0$. There exists $s_0\in(0,1)$ such that $\varphi_\pi(s)<0$ for every $s\in[s_0,1)$. For every such $s$, almost surely, for every word $w$, \[ \sum_{u\in V(\mathcal T_w)}|F_w(u)|^s<\infty . \] \end{lemma} \begin{proof} By \Cref{prop:press-properties}(i), the free energy $\varphi_\pi$ is real-analytic, and in particular continuous, on $(0,2)$. Since $\varphi_\pi(1)<0$, there exists $s_0\in(0,1)$ with $\varphi_\pi<0$ on $[s_0,1)$. (By \Cref{prop:press-properties}(v), $\varphi_\pi$ is strictly decreasing on $(0,1]$, so the set of $s\in(0,1)$ with $\varphi_\pi(s)<0$ is an interval with right endpoint $1$.) Fix $s\in[s_0,1)$. By \Cref{prop:press-identification}(iii), there exists $C<\infty$ such that $\Phi_L(s;E)\le Ce^{L\varphi_\pi(s)}$ for every $L\ge1$, and $\Phi_0(s;E)=\E|D|^{-s}<\infty$ by \Cref{prop:rde-properties}. By \Cref{prop:loc-columns}(i) and Tonelli's theorem, \[ \E\sum_{v\in V}|F(v)|^s=\sum_{L\ge0}\Phi_L(s;E)\le\E|D|^{-s}+C\sum_{L\ge1}e^{L\varphi_\pi(s)}<\infty , \] since $\varphi_\pi(s)<0$. We conclude that $\sum_v|F(v)|^s<\infty$ almost surely. For a word $w$, the column $F_w$ is the same function of $\Theta_w$ as $F$ is of $\Theta_o$, and $\Theta_w$ has the law of $\Theta_o$ by (R3). Then $\sum_u|F_w(u)|^s<\infty$ almost surely, and a countable intersection over words $w$ completes the proof. \end{proof} \section{Rerooted columns}\label{sec:loc-reroot} We now construct inverse columns at arbitrary vertices of the full tree and of the components left after removing one vertex or one edge. A \emph{removed set} is one of the following: the empty set; a word $y$, standing for the vertex $y$; or a word $y\ne o$, standing for the edge $\{y^-,y\}$. Fix a removed set and a word $r$. On the event that $r\in V$, that $y\in V$ (if the removed set is not empty), and that $r\ne y$ (for a removed vertex), let $C$ be the component containing $r$ of the reduced forest obtained from $\mathcal T$ by the removal (with $C=V$ for the empty removed set). For a word $x$, let $\operatorname{anc}(x)$ be the set of its prefixes, including $o$ and $x$. We define \[ S=\operatorname{anc}(r)\cup\operatorname{anc}(y) \] (with $S=\operatorname{anc}(r)$ for the empty removed set), $S_C=S\cap C$, and \[ W_C=\{w\in C\setminus S:\ w^-\in S\}. \] The set $S$ is a finite set of words which contains $o$ and all prefixes of its elements, and it is determined by the removed set and $r$. For $x\in C\setminus\{r\}$, let $b(x)$ be the neighbor of $x$ on the path in $C$ from $x$ to $r$, and let $\operatorname{ch}_C(x)$ be the set of neighbors $x'\ne b(x)$ of $x$ in $C$; let $\operatorname{ch}_C(r)$ be the set of all neighbors of $r$ in $C$. \begin{lemma}\label{lem:loc-skeleton} On the event above, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item The set $S_C$ contains $r$ and induces a finite connected subtree of $C$. \item The set $W_C$ is finite. For every $w\in W_C$, the set $V(\mathcal T_w)$ is contained in $C\setminus S$, and $C\setminus S_C$ is the disjoint union of the sets $V(\mathcal T_w)$, $w\in W_C$. \item Let $w\in W_C$ and $x\in V(\mathcal T_w)$. Then $b(x)=x^-$, and $\operatorname{ch}_C(x)$ is the set of children of $x$ in $\mathcal T$. \item If $x\in S_C$ and $x'\in\operatorname{ch}_C(x)\setminus S_C$, then $x'$ is a child of $x$ and $x'\in W_C$. \end{enumerate} \end{lemma} \begin{proof} For (i), we have $r\in S\cap C$. The set $S$ is a union of ancestral paths starting at $o$, and it induces a subtree of $\mathcal T$. If $x,x'\in S_C$, then the path in $\mathcal T$ between them lies in $S$, and it lies in $C$ because $C$ induces a subtree of $\mathcal T$ (\Cref{lem:tree-branch}(i)). This shows that $S_C$ is connected. It is finite because $S$ is. We next prove (ii). Let $w\in W_C$. If some $u\in V(\mathcal T_w)$ belonged to $S$, then its prefix $w$ would belong to $S$, which is false. We find that $V(\mathcal T_w)\cap S=\emptyset$. In particular, $V(\mathcal T_w)$ contains neither the removed vertex $y$ nor an endpoint of the removed edge, since both $y$ and $y^-$ lie in $S$. Every edge of $\mathcal T_w$ is therefore retained, and $V(\mathcal T_w)$ is connected in the reduced forest. Since it contains $w\in C$, we obtain $V(\mathcal T_w)\subset C\setminus S$. Conversely, let $u\in C\setminus S_C$, so that $u\notin S$. Since $o\in S$, there exists a shortest prefix $w$ of $u$ with $w\notin S$. Then $w\ne o$, $w^-\in S$, and $u\in V(\mathcal T_w)$. By the previous argument, $V(\mathcal T_w)$ lies in one component, which is $C$ because it contains $u$. In particular $w\in C$. It remains to check that $w^-\in C$, that is, that the edge $\{w^-,w\}$ is retained and $w^-\ne y$. If the edge $\{w^-,w\}$ were the removed edge, then $w=y\in S$. If $w^-=y$ were the removed vertex, then the component of $w$ would be $V(\mathcal T_w)$, which contains $r$; then $w$ would be a prefix of $r$ and would lie in $S$. Both are impossible, and we conclude that $w^-\in S_C$ and $w\in W_C$. If $w,w'\in W_C$ are distinct and $w$ is a prefix of $w'$, then $w'^-\in V(\mathcal T_w)\cap S$, which is impossible. We conclude that the sets $V(\mathcal T_w)$, $w\in W_C$, are disjoint. Finally, $W_C$ is finite, since its elements are children of the finitely many vertices in $S$. For (iii), we have $r\in S$ and $r\notin V(\mathcal T_w)$. The path from $x$ to $r$ therefore leaves $V(\mathcal T_w)$, and since $V(\mathcal T_w)$ is joined to the rest of $\mathcal T$ only by the edge $\{w^-,w\}$, the path first moves from $x$ to $x^-$. All children of $x$ belong to $V(\mathcal T_w)\subset C$, and the edges to them are retained. The neighbors of $x$ in $C$ other than $x^-$ are its children. For (iv), the parent of $x$ belongs to $S$ (if $x\ne o$), since $S$ is closed under prefixes. It follows that $x'$ is a child of $x$, and $x'\in C\setminus S$ with $x'^-=x\in S$, that is, $x'\in W_C$. \end{proof} We now define real cavity marks on $C$, adapted to the root $r$. For $x\in C\setminus S_C$, we set $\gamma^C_x=\gamma_x$. For $x\in S_C$, we define, recursively from the leaves of the finite tree $S_C$ rooted at $r$ toward $r$, \begin{equation}\label{eq:loc-reroot-marks} \Delta^C_x=E+\sum_{x'\in\operatorname{ch}_C(x)}|h_{xx'}|^2\gamma^C_{x'},\qquad \gamma^C_x=-\frac1{\Delta^C_x}. \end{equation} This is well defined as long as the denominators $\Delta^C_x$ do not vanish: the elements of $\operatorname{ch}_C(x)$ are either in $S_C$, where they are closer to the leaves, or in $W_C$ by \Cref{lem:loc-skeleton}(iv). For the empty removed set and $r=o$, we have $S_C=\{o\}$, $W_C$ is the set of children of $o$, and $\gamma^C_o=\gamma_o$. In general, the numbers $\gamma^C_x$ with $x\in S_C$ are real counterparts of the cavity resolvents $\Gamma^C_{x\to b(x)}$ of the rerooted component (compare \Cref{rem:tree-reroot}). \begin{lemma}\label{lem:loc-denominators} Fix a removed set and a word $r$. Almost surely, on the event above, every denominator $\Delta^C_x$, $x\in S_C$, is finite and nonzero; hence every $\gamma^C_x$, $x\in C$, is finite and nonzero. \end{lemma} \begin{proof} The proof uses that the squared modulus of the mark of each edge of the skeleton $S_C$ is a fresh exponential variable for the denominator at its endpoint farther from the leaves. Let $W(S)=\{si:\ s\in S,\ i\in\N\}\setminus S$. Its elements are not prefixes of each other: if $w\in W(S)$ were a proper prefix of $w'=si\in W(S)$, then $w$ would be a prefix of $s\in S$ and would belong to $S$. By (R3), applied to finite subsets of $W(S)$, and since the coordinates of $\omega$ are independent under $\P$, we see that the following random elements are independent: the variables $N_s$ and $w_{s,si}$ for $s\in S$ and $i\in\N$, and the processes $\Theta_w$ for $w\in W(S)$. The event in question, the sets $S_C$ and $W_C\subset W(S)$, and the maps $b$ and $\operatorname{ch}_C$ are functions of $(N_s)_{s\in S}$. The weights of the edges with an endpoint in $S_C$ are functions of the variables $w_{s,si}$, and $\gamma_w$ is a function of $\Theta_w$. For $x\in S_C$ and $x'\in\operatorname{ch}_C(x)$, let $C_{x'\to x}$ be the branch of $C$ at $x'$ away from $x$. By induction from the leaves, $\gamma^C_{x'}$ is a function of $(N_s)_{s\in S}$, of the marks of the edges with both endpoints in $C_{x'\to x}$, and of the processes $\Theta_w$ with $w\in W_C\cap C_{x'\to x}$. We now prove by induction from the leaves of $S_C$ toward $r$ that, almost surely, $\Delta^C_x$ is finite and nonzero for every $x\in S_C$. Fix $x\in S_C$ and suppose the claim holds for the elements of $\operatorname{ch}_C(x)\cap S_C$. Then $\gamma^C_{x'}$ is finite and nonzero for every $x'\in\operatorname{ch}_C(x)$: for $x'\in S_C$ by the induction hypothesis, and for $x'\in W_C$ because $\gamma^C_{x'}=-1/D_{x'}$ with $D_{x'}\in\R\setminus\{0\}$. In particular $\Delta^C_x$ is finite. If $\operatorname{ch}_C(x)$ is empty, then $\Delta^C_x=E\ne0$. Otherwise, let $x_*$ be the element of $\operatorname{ch}_C(x)$ that comes first in a fixed enumeration of $\mathbb U$, which is a function of $(N_s)_{s\in S}$. Since there are countably many words, it suffices to show that $\Delta^C_x\ne0$ almost surely on the event $\{x_*=x'\}$, for a fixed word $x'$. On this event, the edge $e=\{x,x'\}$ joins two vertices of $S\cup W(S)$, one of which is the parent of the other and lies in $S$. Its mark is therefore one of the variables $w_{s,si}$, with $s$ and $i$ determined by $x$ and $x'$; we denote it by $\omega_e$. We write \[ \Delta^C_x=A+\frac{|\omega_e|^2}p\,\gamma^C_{x'},\qquad A=E+\sum_{x''\in\operatorname{ch}_C(x)\setminus\{x'\}}|h_{xx''}|^2\gamma^C_{x''}. \] Let $\mathcal R_e$ be the $\sigma$-algebra generated by all the random elements listed in the first paragraph except $\omega_e$. Since the edge $e$ has an endpoint at $x$, it is not an edge of any branch $C_{x''\to x}$. By the previous paragraph, the event $\{x_*=x'\}$ and the variables $A$ and $\gamma^C_{x'}$ are $\mathcal R_e$-measurable. The variable $\omega_e$ is independent of $\mathcal R_e$, and $|\omega_e|^2$ has, conditionally on $\mathcal R_e$, the $\Exp(1)$ law, which has no atoms. Since $\gamma^C_{x'}\ne0$ almost surely on $\{x_*=x'\}$, we have \[ \P\bigl(\{\Delta^C_x=0\}\cap\{x_*=x'\}\mid\mathcal R_e\bigr) =\one_{\{x_*=x'\}}\,\P\bigl(|\omega_e|^2=-pA/\gamma^C_{x'}\mid\mathcal R_e\bigr)=0 \] almost surely. This completes the induction, and the last assertion follows from \eqref{eq:loc-reroot-marks} and $\gamma^C_x=\gamma_x$ for $x\notin S_C$. \end{proof} \begin{lemma}\label{lem:loc-reroot} Let $s\in(0,1)$ be such that the conclusion of \Cref{lem:loc-forward} holds. Fix a removed set and a word $r$, and work on the intersection of the almost sure events of \Cref{lem:loc-forward}, \Cref{lem:loc-denominators} and (R1) with the event above. Define $F^{C,r}$ on $C$ by $F^{C,r}(r)=\gamma^C_r$ and $F^{C,r}(x)=-\gamma^C_x\,h_{x\,b(x)}\,F^{C,r}(b(x))$ for $x\ne r$. Then \begin{equation}\label{eq:loc-reroot-equation} (\mathcal H_{C,\mathrm{form}}F^{C,r})(x)-EF^{C,r}(x)=\delta_r(x),\qquad x\in C, \end{equation} and $\sum_{x\in C}|F^{C,r}(x)|^s<\infty$. \end{lemma} \begin{proof} We verify the hypotheses of \Cref{lem:loc-pointwise} on the tree $C$ with root $r$ and the numbers $\gamma^C_x$. The neighbors of $x$ in $C$ are $b(x)$ (if $x\ne r$) and the elements of $\operatorname{ch}_C(x)$. For $x\in S_C$, \eqref{eq:loc-cavity-relation} is the definition \eqref{eq:loc-reroot-marks}, with nonzero denominators by \Cref{lem:loc-denominators}. For $x\in V(\mathcal T_w)$ with $w\in W_C$, \Cref{lem:loc-skeleton}(iii) shows that $\operatorname{ch}_C(x)$ is the set of children of $x$, all of which lie in $V(\mathcal T_w)$, so that $\gamma^C_{x'}=\gamma_{x'}$ for $x'\in\operatorname{ch}_C(x)$. The relation \eqref{eq:loc-cavity-relation} then holds by the computation after \Cref{def:loc-column}. By \Cref{lem:loc-skeleton}(ii), these two cases cover $C$. Then \Cref{lem:loc-pointwise} implies \eqref{eq:loc-reroot-equation}. For summability, let $w\in W_C$. For $x\in V(\mathcal T_w)$ we have $b(x)=x^-$ and $\gamma^C_x=\gamma_x$, so $F^{C,r}$ and $F_w$ satisfy the same recursion $f(x)=-\gamma_xh_{xx^-}f(x^-)$ on $V(\mathcal T_w)\setminus\{w\}$. Comparing their values at $w$, we find \[ F^{C,r}(x)=\frac{F^{C,r}(w)}{\gamma_w}\,F_w(x),\qquad x\in V(\mathcal T_w). \] By \Cref{lem:loc-skeleton}(ii), \[ \sum_{x\in C}|F^{C,r}(x)|^s=\sum_{x\in S_C}|F^{C,r}(x)|^s +\sum_{w\in W_C}\Bigl|\frac{F^{C,r}(w)}{\gamma_w}\Bigr|^s\sum_{x\in V(\mathcal T_w)}|F_w(x)|^s . \] The sets $S_C$ and $W_C$ are finite, all values of $F^{C,r}$ are finite, $\gamma_w\ne0$, and each inner sum is finite by \Cref{lem:loc-forward}. This proves that the left side is finite. \end{proof} \section{Proofs of the main results}\label{sec:loc-proofs} We use the following characterization of the range of $T-E$ for a self-adjoint operator $T$. \begin{lemma}\label{lem:loc-range} Let $T$ be a self-adjoint operator on a Hilbert space, let $\psi$ be a vector, and let $E\in\R$. If $f\in\Dom(T)$ and $(T-E)f=\psi$, then $\mu^T_\psi(\{E\})=0$ and $R_{\mu^T_\psi}(E)\le\|f\|^2$. Conversely, if $R_{\mu^T_\psi}(E)<\infty$, then $\psi\in\Ran(T-E)$. \end{lemma} \begin{proof} Suppose $(T-E)f=\psi$. By the spectral theorem, for every bounded Borel function $k$, \[ \langle\psi,k(T)\psi\rangle=\int_\R k(\lambda)(\lambda-E)^2\,\mu^T_f(\dd\lambda), \] that is, $\mu^T_\psi(\dd\lambda)=(\lambda-E)^2\mu^T_f(\dd\lambda)$. In particular, $\mu^T_\psi(\{E\})=0$ and $R_{\mu^T_\psi}(E)=\mu^T_f(\R\setminus\{E\})\le\|f\|^2$. Conversely, suppose $R_{\mu^T_\psi}(E)<\infty$. Then $\mu^T_\psi(\{E\})=0$. Let $k(\lambda)=(\lambda-E)^{-1}$ for $\lambda\ne E$ and $k(E)=0$. Then $\int|k|^2\dd\mu^T_\psi=R_{\mu^T_\psi}(E)<\infty$, and since $|\lambda k(\lambda)|\le1+|E||k(\lambda)|$, also $\int|\lambda k(\lambda)|^2\dd\mu^T_\psi<\infty$. By the functional calculus, $f=k(T)\psi$ belongs to $\Dom(T)$, and $(T-E)f=\one_{\R\setminus\{E\}}(T)\psi=\psi$, since $\|\one_{\{E\}}(T)\psi\|^2= \mu^T_\psi(\{E\})=0$. \end{proof} \begin{proof}[Proof of \Cref{prop:loc-columns}(ii) and (iii)] Part (i) was proved in \Cref{sec:loc-palm}. We assume that $\varphi_\pi(1)<0$, let $s_0$ be as in \Cref{lem:loc-forward}, and fix $s\in[s_0,1)$. \proofstep{1}{The root column.} For the empty removed set and $r=o$, we have $S_C=\{o\}$, $\gamma^C_o=\gamma_o$ and $\gamma^C_x=\gamma_x$ for $x\ne o$, and $F^{V,o}=F$. By \Cref{lem:loc-reroot}, almost surely $F\in\ell^s(V)$, and $F\in\ell^2(V)$ because $\|f\|_2\le\|f\|_s$ for $s\le2$. By \eqref{eq:loc-reroot-equation}, $\mathcal H_{\mathrm{form}}F=EF+\delta_o\in\ell^2(V)$. By the maximal domain characterization \Cref{thm:tree-esa}(ii), in the form of \Cref{rem:tree-maximal}, $F\in\Dom(\mathcal H)$ and $(\mathcal H-E)F=\delta_o$. This proves (ii). \proofstep{2}{All columns.} Fix a removed set and a word $r$. On the intersection of the almost sure events of \Cref{lem:loc-reroot} and \Cref{thm:tree-esa} with the event on which $C$ is defined (see \Cref{sec:loc-reroot}), \Cref{lem:loc-reroot} yields $F^{C,r}\in\ell^s(C)\subset\ell^2(C)$ and $\mathcal H_{C,\mathrm{form}}F^{C,r}=EF^{C,r}+\delta_r\in \ell^2(C)$. By \Cref{rem:tree-maximal}, $F^{C,r}\in\Dom(\mathcal H_C)$ and $(\mathcal H_C-E)F^{C,r}=\delta_r$. In particular $\delta_r\in\Ran(\mathcal H_C-E)$, and \Cref{lem:loc-range} gives $R_{\mu^C_r}(E)\le\|F^{C,r}\|^2<\infty$. There are countably many removed sets and words $r$, so these conclusions hold almost surely for all of them simultaneously. For the empty removed set, $C=V$ and $\mu^C_r=\mu_r$. For the removal of a vertex $y$, the operator $\mathcal H^{(y)}$ is the orthogonal sum of the operators of the components (\Cref{thm:tree-esa}(iv)), and $\mu^{(y)}_x=\mu^C_x$ for $x\in C$; this gives \eqref{eq:loc-neighbor}. \proofstep{3}{Transfer to $\P$ and $\P^\infty$.} Let $\mathcal A_E$ be the event that $R_{\mu^C_r}(E)<\infty$ for every removed set, every word $r$ and the corresponding component $C$ (whenever these are defined). By \eqref{eq:rank-R-limit}, $R_{\mu^C_r}(E)=\sup_nn\Im G^C_{rr}(E+\iu/n)$, and by \Cref{lem:rank-measurable} this is a measurable function of the marked tree. It follows that $\mathcal A_E$ is an event of the marked tree. By Step~2, it has probability one on the space of \Cref{lem:rde-rtp}, where the marked tree has the law $\P$, and we have $\P(\mathcal A_E)=1$. Since $\P^\infty\le\varrho_p^{-1}\P$, also $\P^\infty(\mathcal A_E)=1$. By \Cref{lem:loc-range}, on $\mathcal A_E$ every $\delta_r$ belongs to $\Ran(\mathcal H_C-E)$. This proves (iii). \end{proof} \begin{proof}[Proof of \Cref{thm:loc-criterion}] Let $\mathcal N_0\subset U$ be a Lebesgue-null set such that a real solution $\pi$ with $\varphi_\pi(1)<0$ exists at every $E\in U\setminus\mathcal N_0$. Let $\mathcal B$ be the set of pairs $(\omega,E)\in\Omega\times U$ such that, for every vertex $y$ and every neighbor $x$ of $y$, $R_{\mu^{(y)}_x}(E)<\infty$, where $\Omega$ is the probability space of the marked tree. For fixed words $y,x$, we have $R_{\mu^{(y)}_x}(E)=\sup_nn\Im G^{(y)}_{xx}(E+\iu/n)$ by \eqref{eq:rank-R-limit}, which is jointly measurable in $(\omega,E)$ by \Cref{lem:rank-measurable}. Since there are countably many pairs of words, $\mathcal B$ is jointly measurable. For $E\in U\setminus\mathcal N_0$, \Cref{prop:loc-columns}(iii), applied with a real solution $\pi$ at $E$ with $\varphi_\pi(1)<0$, gives $\P((\omega,E)\in\mathcal B)=1$. The event $\mathcal B$ does not involve the real solution, and no measurable dependence of the solution on $E$ is needed. By Tonelli's theorem, \[ \E\bigl|\{E\in U:(\omega,E)\notin\mathcal B\}\bigr|=\int_U\P\bigl((\omega,E)\notin\mathcal B\bigr)\dd E=0, \] since the integrand vanishes outside the null set $\mathcal N_0$. This implies that, $\P$-almost surely, for Lebesgue-almost every $E\in U$, we have $R_{\mu^{(y)}_x}(E)<\infty$ for every vertex $y$ and every neighbor $x$ of $y$. This is the hypothesis of \Cref{prop:rank-pp} with $I=U$ and $\mathsf Q=\P$, and it also holds $\P^\infty$-almost surely. By \Cref{prop:rank-pp}, $\mathcal H$ has pure point spectrum in $U\cup(-U)$, almost surely under $\P$ and under $\P^\infty$. \end{proof} \begin{remark}\label{rem:loc-scope} The energy $0$ is excluded in \Cref{thm:loc-criterion}, and indeed $\mathcal H$ has finitely supported eigenvectors with eigenvalue zero (\Cref{prop:tree-zero-modes}). In \Cref{prop:loc-columns}, the real solution $\pi$ is arbitrary, and the recursive tree process need not be endogenous, that is, the denominators $D_v$ need not be functions of the marked tree. This causes no difficulty, because the assertions of \Cref{prop:loc-columns}(iii) are events of the marked tree; see Step~3 of the proof. If $E\in\mathcal E_p$, the no-ac hypothesis holds at $E$, and $\pi$ is the real boundary law, then the boundary denominators of \Cref{prop:rde-boundary}(ii) form an endogenous recursive tree process with marginal $\pi$ (\Cref{rem:rde-endogeny}). For this process, \eqref{eq:tree-path-column} shows that $F(v)$ is the limit of $G_{vo}(E+\iu\eta)$ as $\eta\downarrow0$. \end{remark} \chapter{Resonances along paths}\label{sec:res} This section and the next prove that the boundary values of the cavity variables cannot all be real at an energy where the real boundary law has positive free energy at $s=1$. The argument follows the resonance strategy of Aizenman and Warzel \cite{AizenmanWarzel2013resonant}; variants of it appear in \cite{AggarwalBordenaveLopatto2022mobility,AggarwalLopatto2025mobility}. We look for vertices $x$ at a large depth whose Green function entry $G_{ox}$ is exponentially large, and we use the Ward identity to convert such a vertex into a large imaginary part at the root. In this section, we collect the deterministic identities and the estimates along one or two paths that this argument uses. The counting argument itself is given in \Cref{sec:deloc}. In \Cref{sec:res-paths}, we express the Green function between the endpoints of a path through a product of transfer matrices, and we record the Schur complement formula for two vertices. In \Cref{sec:res-terminal}, we study the end of a path: we show that two vertices with a single child each bring the rootward cavity variable into a fixed disc, uniformly in its incoming value, and we estimate the probability that a pendant leaf creates a large diagonal entry of the Green function, at one vertex and at two vertices simultaneously. \Cref{sec:res-blocks} selects blocks of transfer matrices whose truncated free energy approximates the free energy and estimates the probability of a regular growth event, on which the growth of the transfer products along a path stays close to its typical value. \Cref{sec:res-buffers} averages bilinear forms of products of transfer matrices over four groups of three independent factors. Finally, \Cref{sec:res-limits} records a semicontinuity property of probabilities used to pass from the real axis to a small positive imaginary part. The results of \Cref{sec:res-paths,sec:res-terminal} are deterministic, or concern one or two squared moduli with an explicit density, and they hold under the hypotheses on $p$ and $z$ stated in each lemma; in particular, they do not use a real solution. From \Cref{sec:res-blocks} on, we fix $p>1$, an energy $E>0$, and a real solution $\pi$ at energy $E$; in \Cref{sec:deloc}, $\pi$ is the real boundary law. Whenever an energy $E$ is fixed, we write $z=E+\iu\eta$ with $\eta\ge0$. Statements about the tree hold on the almost sure event $\Omega_{\mathrm{sa}}$ of \Cref{thm:tree-esa}, intersected with the almost sure event on which all edge weights are nonzero. \section{Transfer matrices along a path}\label{sec:res-paths} In this section, we prove two deterministic identities for the resolvent of a component of a forest obtained by a finite removal. The first, \Cref{lem:res-path}, expresses the Green function between the endpoints of a path, and its ratio to the diagonal entry at the far endpoint, through the product of the transfer matrices along the path. In \Cref{sec:deloc}, it is applied in \Cref{lem:deloc-gain} to the path from the root to a resonant vertex, where the ratio converts a large diagonal entry into a large entry $G_{ox}$, and in \Cref{lem:deloc-pair-real} to the path between the parents of two resonant vertices, where the Green function measures the interaction between them. The second, \Cref{lem:res-schur-two}, is the Schur complement formula for the $2\times2$ block of the resolvent at two vertices; it is used in \Cref{lem:deloc-second} to write the two diagonal entries at a pair of resonant vertices as functions of the weights of their pendant leaves. We use the transfer matrices of \Cref{def:press-operators} with the scaling $g=1$, in which the denominator is $B=D$ and $\beta=Y/p$ is the squared weight, and we allow complex denominators, as in \Cref{prop:press-three-rows-complex}. \begin{definition}\label{def:res-rows} \begin{enumerate}[label=\textup{(\roman*)}] \item For $d\in\C$ and $t>0$, the \emph{factor} with denominator $d$ and squared weight $t$ is the matrix \[ M(d,t)=t^{-1/2}\begin{pmatrix}0&1\\-t&d\end{pmatrix}. \] For real $d$, this is the matrix $M(b,\beta)$ of \Cref{def:press-operators} with $b=d$ and $\beta=t$. We write $e_2=(0,1)^{\mathsf T}$. \item Let $C$ be a component of a forest obtained by a finite removal, let $w_0,w_1,\dots,w_r$ be a path in $C$ with $r\ge0$, and let $z\in\C\setminus\R$. For $0\le k\le r$, the \emph{side denominator} at $w_k$ is \[ d_k=z+\sum_{a}|h_{w_ka}|^2\,\Gamma^C_{a\to w_k}(z), \] where the sum runs over the neighbors $a$ of $w_k$ in $C$ other than $w_{k-1}$ and $w_{k+1}$ (only those of them that exist). For $1\le k\le r$, we set $t_k=|h_{w_{k-1}w_k}|^2$. The \emph{rootward denominators} are defined by $R_0=d_0$ and $R_k=d_k-t_k/R_{k-1}$ for $1\le k\le r$, and the \emph{transfer product} of the path is \[ A=M(d_0,t_1)M(d_1,t_2)\cdots M(d_{r-1},t_r), \] with $A=I$ if $r=0$. \end{enumerate} \end{definition} In words, the $k$-th factor of a path pairs the side denominator at $w_k$ with the squared weight of the edge leaving $w_k$ towards $w_{k+1}$. Every factor has determinant one, since $\det M(d,t)=t^{-1}(0\cdot d+t)=1$, and so does every product of factors. Let $A$ be a complex $2\times2$ matrix with $\det A=1$, and let $\sigma_1\ge\sigma_2\ge0$ be its singular values. Then $\sigma_1\sigma_2=|\det A|=1$, so that $\sigma_2>0$, $\sigma_1=\sigma_2^{-1}$, and $\sigma_1\ge1$. Since $\|A\|=\sigma_1$ and $\|A^{-1}\|=\sigma_2^{-1}$, we have \begin{equation}\label{eq:res-inverse-norm} \|A^{-1}\|=\|A\|\ge1. \end{equation} Bounding the operator norm by the Hilbert--Schmidt norm, we have \begin{equation}\label{eq:res-row-norm} \|M(d,t)\|\le t^{-1/2}\big(1+t^2+|d|^2\big)^{1/2}. \end{equation} The following algebraic identity relates products of factors to continuants. \begin{lemma}\label{lem:res-continuant} Let $r\ge1$, let $d_0,\dots,d_r\in\C$, and let $t_1,\dots,t_r>0$. Define $K_{-2}=0$, $K_{-1}=1$, $K_0=d_0$, and $K_k=d_kK_{k-1}-t_kK_{k-2}$ for $1\le k\le r$. Let $A=M(d_0,t_1)\cdots M(d_{r-1},t_r)$ and $\tau=(t_1\cdots t_r)^{1/2}$. Then \[ \tau\,e_2^{\mathsf T}A=\big(-t_rK_{r-2},\,K_{r-1}\big). \] In particular, $\tau\,e_2^{\mathsf T}Ae_2=K_{r-1}$ and $\tau\,e_2^{\mathsf T}A(1,d_r)^{\mathsf T}=K_r$. If the numbers $R_0=d_0$ and $R_k=d_k-t_k/R_{k-1}$, $1\le k\le r$, are well defined and nonzero, then $K_k=R_0R_1\cdots R_k$ for $0\le k\le r$. \end{lemma} \begin{proof} Let $N_k=t_{k+1}^{1/2}M(d_k,t_{k+1})$, which has the rows $(0,1)$ and $(-t_{k+1},d_k)$. We show by induction on $l\in\{1,\dots,r\}$ that $e_2^{\mathsf T}N_0\cdots N_{l-1}=(-t_lK_{l-2},K_{l-1})$. For $l=1$, we have $e_2^{\mathsf T}N_0=(-t_1,d_0)=(-t_1K_{-1},K_0)$. If the claim holds for $l0$. By \eqref{eq:tree-path-column}, applied to the path from $u=w_r$ to $v=w_0$, that is, to the vertices $v_l=w_{r-l}$, we have \[ G^C_{w_0w_r}(z)=(-1)^rG^C_{w_rw_r}(z)\prod_{k=0}^{r-1}h_{w_kw_{k+1}}\,\Gamma^C_{w_k\to w_{k+1}}(z). \] Taking absolute values and using \eqref{eq:res-rootward}, we find \[ \big|G^C_{w_0w_r}(z)\big|=\frac{(t_1\cdots t_r)^{1/2}}{|R_0R_1\cdots R_r|},\qquad \bigg|\frac{G^C_{w_0w_r}(z)}{G^C_{w_rw_r}(z)}\bigg|=\frac{(t_1\cdots t_r)^{1/2}}{|R_0\cdots R_{r-1}|}. \] By \Cref{lem:res-continuant}, $R_0\cdots R_r=K_r=\tau e_2^{\mathsf T}A(1,d_r)^{\mathsf T}$ and $R_0\cdots R_{r-1}=K_{r-1}=\tau e_2^{\mathsf T}Ae_2$, where $\tau=(t_1\cdots t_r)^{1/2}$; here the weights $t_k$ are positive, since the edge weights are nonzero. This proves \eqref{eq:res-cross-formula} and the identity in \eqref{eq:res-ratio-formula}; in particular, $e_2^{\mathsf T}Ae_2=\tau^{-1}R_0\cdots R_{r-1}\neq0$. The inequality in \eqref{eq:res-ratio-formula} follows from the Cauchy--Schwarz inequality $|e_2^{\mathsf T}Ae_2|=|\langle e_2,Ae_2\rangle|\le\|Ae_2\|\le\|A\|$, which holds for every complex $2\times2$ matrix (the condition $\det A=1$ is not needed here). \end{proof} In \eqref{eq:res-ratio-formula}, the side denominator $d_r$ at the endpoint does not enter; the ratio depends only on the side denominators $d_0,\dots,d_{r-1}$ and on the weights of the path. In \Cref{sec:deloc}, we apply \eqref{eq:res-ratio-formula} to a path from the root to a vertex $x$ of the full tree, and \eqref{eq:res-cross-formula} to the path between the parents of two vertices $x$ and $y$, in the tree from which $x$ and $y$ have been deleted. The next lemma is the Schur complement formula for the $2\times2$ block of the resolvent at two vertices. For a component $C$ of the forest obtained by a finite removal $(F,D)$ and $x,y\in C$, we write $G^{C\setminus\{x,y\}}_{ab}(z)$ for the entries of the resolvent of the reduced operator of the finite removal $(F\cup\{x,y\},D)$ at vertices $a,b\in C\setminus\{x,y\}$. These vanish if $a$ and $b$ lie in different components of the corresponding reduced forest. \begin{lemma}\label{lem:res-schur-two} Let $C$ be a component of a forest obtained by a finite removal, let $x,y\in C$ be distinct and not adjacent, and let $z\in\C\setminus\R$. For $u,v\in\{x,y\}$, set \[ S_{uv}(z)=-z\one_{\{u=v\}}-\sum_{a\sim_Cu}\ \sum_{b\sim_Cv}h_{ua}\,G^{C\setminus\{x,y\}}_{ab}(z)\,h_{bv}, \] where $a\sim_Cu$ means that $a$ is a neighbor of $u$ in $C$. Then the matrix $S(z)=(S_{uv}(z))_{u,v\in\{x,y\}}$ is invertible, and \begin{equation}\label{eq:res-schur-two} \begin{pmatrix}G^C_{xx}(z)&G^C_{xy}(z)\\G^C_{yx}(z)&G^C_{yy}(z)\end{pmatrix}=S(z)^{-1}. \end{equation} Let $x'$ and $y'$ be the neighbors of $x$ and $y$ on the path from $x$ to $y$ in $C$. Then \begin{equation}\label{eq:res-schur-offdiag} S_{xy}(z)=-h_{xx'}\,G^{C\setminus\{x,y\}}_{x'y'}(z)\,h_{y'y},\qquad |S_{xy}(z)S_{yx}(z)|=|h_{xx'}|^2\,|h_{yy'}|^2\,\big|G^{C\setminus\{x,y\}}_{x'y'}(z)\big|^2, \end{equation} and $S_{xx}(z)=-z-\sum_{a\sim_Cx}|h_{xa}|^2\,G^{C\setminus\{x,y\}}_{aa}(z)$; the analogous formula holds for $S_{yy}(z)$. Finally, if $\Im z>0$, then the Hermitian matrix $\Im S(z)=(S(z)-S(z)^*)/(2\iu)$ satisfies $\Im S(z)\le-(\Im z)I$. \end{lemma} \begin{proof} Let $D'$ be the set of edges of $C$ with an endpoint in $\{x,y\}$. By \Cref{lem:tree-decouple}, the components contained in $C$ of the reduced forest obtained by adding $D'$ to the removed edges are $\{x\}$, $\{y\}$, and the components of $C\setminus\{x,y\}$. With respect to $\ell^2(C)=\C^{\{x,y\}}\oplus\mathfrak h$, where $\mathfrak h=\ell^2(C\setminus\{x,y\})$, the same lemma gives $\mathcal H_C=(0\oplus K)+W_{D'}$ and $\Dom(\mathcal H_C)=\C^{\{x,y\}}\oplus\Dom(K)$, where $K$ is the direct sum of the operators of the components of $C\setminus\{x,y\}$. These components are also the components contained in $C$ of the reduced forest of the finite removal $(F\cup\{x,y\},D)$, and in both reduced forests the operator of such a component is the closure of the weighted adjacency operator of the same induced subtree (\Cref{lem:tree-branch}(i)). Then $\langle\delta_a,(K-z)^{-1}\delta_b\rangle=G^{C\setminus\{x,y\}}_{ab}(z)$ for $a,b\in C\setminus\{x,y\}$. Define the linear map $J\colon\C^{\{x,y\}}\to\mathfrak h$ by $J\delta_u=\sum_{a\sim_Cu}h_{au}\delta_a$; it is bounded, since $x$ and $y$ have finitely many neighbors. Since $x$ and $y$ are not adjacent, we have, for $f\in\C^{\{x,y\}}$ and $g\in\Dom(K)$, \[ \mathcal H_C(f\oplus g)=J^*g\oplus(Jf+Kg). \] Indeed, the coordinate of $W_{D'}g$ at $u\in\{x,y\}$ is $\sum_{a\sim_Cu}h_{ua}g(a)=\langle J\delta_u,g\rangle$, because $h_{ua}=\overline{h_{au}}$. Fix $f\in\C^{\{x,y\}}$ and set $g=-(K-z)^{-1}Jf\in\Dom(K)$. Then $f\oplus g\in\Dom(\mathcal H_C)$ and \[ (\mathcal H_C-z)(f\oplus g)=\big(-zf-J^*(K-z)^{-1}Jf\big)\oplus0=(S(z)f)\oplus0, \] where we used that the matrix of $J^*(K-z)^{-1}J$ has the entries $\langle J\delta_u,(K-z)^{-1}J\delta_v\rangle=\sum_{a,b}h_{ua}G^{C\setminus\{x,y\}}_{ab}(z)h_{bv}$. Since $\mathcal H_C$ is self-adjoint, $\mathcal H_C-z$ is a bijection from $\Dom(\mathcal H_C)$ onto $\ell^2(C)$ with inverse $(\mathcal H_C-z)^{-1}$. Then $(\mathcal H_C-z)^{-1}\big((S(z)f)\oplus0\big)=f\oplus g$. Projecting onto $\C^{\{x,y\}}$, we find $\Pi(\mathcal H_C-z)^{-1}\Pi\,S(z)f=f$, where $\Pi$ is the orthogonal projection onto $\C^{\{x,y\}}$. In particular, $S(z)$ is injective, hence invertible, and its inverse is the matrix on the left side of \eqref{eq:res-schur-two}. For \eqref{eq:res-schur-offdiag}, let $a\sim_Cx$ and $b\sim_Cy$ lie in the same component of $C\setminus\{x,y\}$. A path from $a$ to $b$ in this component, preceded by $x$ and followed by $y$, is a path from $x$ to $y$ in the tree $C$. Since paths in a tree are unique, we have $a=x'$ and $b=y'$. All other terms in the sum defining $S_{xy}(z)$ vanish, which gives the first identity. The same argument shows that $S_{yx}(z)=-h_{yy'}G^{C\setminus\{x,y\}}_{y'x'}(z)h_{x'x}$, and \Cref{lem:tree-gauge}, applied to the component of $C\setminus\{x,y\}$ containing $x'$ and $y'$, gives $|G^{C\setminus\{x,y\}}_{y'x'}(z)|=|G^{C\setminus\{x,y\}}_{x'y'}(z)|$. This proves the second identity. Finally, two distinct neighbors of $x$ lie in different components of $(F\cup\{x\},D)$, by \Cref{lem:tree-branch}(ii), and a fortiori in different components of $C\setminus\{x,y\}$, since deleting $y$ as well only splits components further; the cross terms in the sum defining $S_{xx}(z)$ vanish, and we obtain the formula for $S_{xx}(z)$. For the last claim, let $\Im z>0$ and $f\in\C^{\{x,y\}}$, and set $g=Jf$. Since $K$ is self-adjoint, we have $\Im\langle g,(K-z)^{-1}g\rangle=\Im z\,\|(K-z)^{-1}g\|^2\ge0$. We have $\langle f,\Im S(z)f\rangle=\Im\langle f,S(z)f\rangle=-\Im z\,\|f\|^2-\Im\langle g,(K-z)^{-1}g\rangle\le-\Im z\,\|f\|^2$. \end{proof} \section{Endpoint estimates}\label{sec:res-terminal} In this section, we study the last few vertices of a path. The path to a resonant vertex of \Cref{sec:deloc} ends with a single-child segment, that is, a sequence of vertices with exactly one child each, and its endpoint has a pendant leaf and a surviving child. \Cref{lem:res-reset,lem:res-terminal} show that, with a probability bounded below uniformly in the rest of the tree, the single-child segment places the denominator with the leaf removed at the endpoint in a fixed region; they are used in \Cref{lem:deloc-gain,lem:deloc-first}. \Cref{lem:res-tuning} bounds the probability that the weight of the pendant leaf makes the diagonal resolvent entry at the endpoint large, from above and from below; the lower bound is used in the first moment (\Cref{lem:deloc-first}). \Cref{lem:res-two-terminal} bounds the probability that this happens at two vertices simultaneously; it is used in the second moment (\Cref{lem:deloc-second}). Finally, \Cref{lem:res-quantile} records the properties of the quantile $\zeta_\alpha(\eta)$ of $\Im\Gamma_o$ on which the contradiction argument of \Cref{sec:deloc-proof} is based. We fix $E>0$ and write $z=E+\iu\eta$. We first show that two consecutive vertices without side children bring the rootward cavity variable into a fixed disc, uniformly in its incoming value. Consider a vertex $w$ with a single neighbor $w'$ other than the next vertex of a path, and let $X=-\Gamma_{w'\to w}(z)$ be the incoming rootward variable. If the squared weight of the edge $ww'$ is $t$, then by \eqref{eq:tree-schur-cavity} the outgoing variable is $(z-tX)^{-1}$. We set \begin{equation}\label{eq:res-reset-constants} \rho_1=e^{-1}-e^{-2},\qquad\eps_E=\frac{\rho_1E}{16},\qquad b_2=\frac{\eps_EE}8,\qquad b_3=\frac{3E^2}{32}, \end{equation} and we let $\mathfrak D_E=\{X\in\C:|X-E^{-1}|\le(3E)^{-1}\}$. Every $X\in\mathfrak D_E$ satisfies $|X|\le4/(3E)$. \begin{lemma}\label{lem:res-reset} Let $E>0$ and $0\le\eta\le E/8$. \begin{enumerate}[label=\textup{(\roman*)}] \item Let $X\in\C$, and let $t$ be a random variable with density $pe^{-pt}$ on $(0,\infty)$. Then \[ \P\Big(\frac1p\eps_E\Big)\ge\frac{\rho_1}2. \] \item If $|X|<\eps_E^{-1}$ and $0E-E/2=E/2>\eps_E$, and the event is empty. If $|X|>pE/4$, then $|z-tX|=|X|\,|t-z/X|$, and the set of real $t$ with $|z-tX|\le\eps_E$ is contained in an interval of length $2\eps_E/|X|<8\eps_E/(pE)$. Since the density of $t$ is at most $p$, the event has probability at most $8\eps_E/E=\rho_1/2$. \end{proof} In applications, $X$ is random and $t=Y/p$, where $Y$ is the squared modulus of an edge mark. Part (i) then holds conditionally, in the following sense. Let $\mathscr G$ be a $\sigma$-field, let $X$ be a $\mathscr G$-measurable complex random variable, and suppose that $Y$ is fresh given $\mathscr G$ in the sense of \Cref{def:palm-fresh}, with $M=1$; that is, $Y$ has the law $\Exp(1)$ conditionally on $\mathscr G$. We apply the substitution identity \eqref{eq:palm-freeze} with $m=1$, $c=X$, $a_1=0$, and \[ \Phi(c,a_1,y)=\one\Big\{1\eps_E\Big\}. \] For each fixed value of $X$, the integral of $\Phi(X,0,\cdot)$ against $\Exp(1)$ is the probability in (i) with $t=Y/p$, which has density $pe^{-pt}$. We conclude that \[ \P\Big(\frac1p\eps_E\Bigm|\mathscr G\Big)\ge\frac{\rho_1}2\qquad\text{almost surely}. \] In the same way, \Cref{lem:res-tuning,lem:res-two-terminal} below hold conditionally on a $\sigma$-field $\mathscr G$ given which the squared moduli of the marks of the leaf edges are fresh and the parameters are revealed: the parameter $\mathfrak p$ in \Cref{lem:res-tuning}, and in \Cref{lem:res-two-terminal} the three numbers $\mathsf B_{xx}$, $\mathsf B_{yy}$, and $\mathsf B_{xy}\mathsf B_{yx}$, through which alone the event and the bound depend. In each case, \eqref{eq:palm-freeze} is applied with $c$ and $a_1,\dots,a_m$ equal to these revealed numbers. The next lemma describes the end of a path. In \Cref{sec:deloc}, the endpoint $x$ of a path has three neighbors: its parent, whose rootward variable $X$ has been brought into $\mathfrak D_E$ by the single-child segment, as in \Cref{lem:res-reset}; a child $y$ with a bounded cavity variable; and a pendant leaf. The \emph{denominator with the leaf removed} at $x$ is the quantity $\Delta$ below. \begin{lemma}\label{lem:res-terminal} Let $E>0$, $0\le\eta\le E/8$, and $R'>0$. Let $X\in\mathfrak D_E$, $00$, then $\Im\Delta\ge\eta>0$ and $\Im\mathfrak p=E\Im\Delta+\eta\Re\Delta$. \begin{lemma}\label{lem:res-tuning} Let $p>0$, $E>0$, $0<\eta\le1$, and $T\ge4(E+1)/E^2$, and let $t$ be a random variable with density $pe^{-pt}$ on $(0,\infty)$. Let $\mathfrak p\in\C\setminus\R$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\P(|z|\ge T|t-\mathfrak p|)\le2p|z|/T$. \item If $E^2/2\le\Re\mathfrak p\le3E^2/2$ and $|\Im\mathfrak p|\le|z|/(2T)$, then \[ \P\big(|z|\ge T|t-\mathfrak p|\big)\ge\frac{c_{\mathrm{tune}}}T,\qquad c_{\mathrm{tune}}=pEe^{-2pE^2}. \] \end{enumerate} \end{lemma} \begin{proof} The set of real $t$ with $|t-\mathfrak p|\le|z|/T$ is empty if $|\Im\mathfrak p|>|z|/T$, and otherwise it is the interval of center $\Re\mathfrak p$ and half-length $\varrho=((|z|/T)^2-(\Im\mathfrak p)^2)^{1/2}\le|z|/T$. Since the density of $t$ is at most $p$, this proves (i). Under the assumptions of (ii), we have $\varrho\ge(\sqrt3/2)|z|/T\ge(\sqrt3/2)E/T$. Further, $\varrho\le|z|/T\le(E+1)/T\le E^2/4$, so that the interval is contained in $[E^2/4,7E^2/4]$, where the density of $t$ is at least $pe^{-7pE^2/4}$. The probability in (ii) is therefore at least $2\varrho\,pe^{-7pE^2/4}\ge\sqrt3\,pEe^{-7pE^2/4}/T\ge c_{\mathrm{tune}}/T$. \end{proof} By \eqref{eq:res-leaf}, the event in \Cref{lem:res-tuning} with $\mathfrak p=z\Delta$ is the event $|G_{xx}(z)|\ge T$. For two vertices $x$ and $y$ with pendant leaves, \Cref{lem:res-schur-two} expresses the $2\times2$ block of the resolvent at $\{x,y\}$ as a function of the squared weights of the two leaf edges. The following lemma bounds the probability that both diagonal entries are large. \begin{lemma}\label{lem:res-two-terminal} Let $p\ge1$, $z\in\C\setminus\R$, and $T>0$, and let $\mathsf B=(\mathsf B_{uv})_{u,v\in\{x,y\}}$ be a complex $2\times2$ matrix such that $\operatorname{diag}(t_x,t_y)-\mathsf B$ is invertible for all $t_x,t_y>0$. Set \[ \mathsf R(t_x,t_y)=z\big(\operatorname{diag}(t_x,t_y)-\mathsf B\big)^{-1}, \] and let $t_x,t_y$ be independent random variables with density $pe^{-pt}$ on $(0,\infty)$. Then \[ \P\big(|\mathsf R_{xx}|\ge T,\ |\mathsf R_{yy}|\ge T\big)\le\frac{8p^2|z|^2}{T^2}+\frac{8p^2|z|}T\min\big\{1,|\mathsf B_{xy}\mathsf B_{yx}|^{1/2}\big\}. \] \end{lemma} \begin{proof} Set $\theta=|z|/T$, $X=t_x-\mathsf B_{xx}$, $Y=t_y-\mathsf B_{yy}$, and $c=\mathsf B_{xy}\mathsf B_{yx}$. The determinant $XY-c$ is nonzero, and \[ \mathsf R_{xx}=\frac{zY}{XY-c},\qquad\mathsf R_{yy}=\frac{zX}{XY-c}. \] Let $\mathcal E$ be the event in the statement. We first bound the probability of $\{|\mathsf R_{xx}|\ge T\}$, which contains $\mathcal E$. Fix $t_y$. If $Y=0$, then $\mathsf R_{xx}=0$. If $Y\neq0$, then $\mathsf R_{xx}=z/(X-c/Y)$, and $|\mathsf R_{xx}|\ge T$ holds if and only if $|t_x-\mathsf B_{xx}-c/Y|\le\theta$, which confines $t_x$ to an interval of length at most $2\theta$. By Tonelli's theorem and the bound $p$ on the density of $t_x$, we obtain $\P(\mathcal E)\le2p\theta$. Next, $|\mathsf R_{xx}|\ge T$ and $|\mathsf R_{yy}|\ge T$ mean that $|XY-c|\le\theta|Y|$ and $|XY-c|\le\theta|X|$, so on $\mathcal E$ we have $|XY-c|\le\theta\min\{|X|,|Y|\}$. Consider the part of $\mathcal E$ on which $|X|\le|Y|$. There $X\neq0$: otherwise the last inequality would give $|XY-c|\le0$, contradicting $XY-c\neq0$. Dividing by $|X|$, we find $|Y-c/X|\le\theta$. Then $|X|\le|Y|\le|c|/|X|+\theta$, that is, $|X|^2-\theta|X|\le|c|$, which implies $|X|\le\theta+|c|^{1/2}$; indeed, if $|X|>\theta+|c|^{1/2}$, then $|X|(|X|-\theta)>(\theta+|c|^{1/2})|c|^{1/2}\ge|c|$. Thus $t_x$ lies in an interval of length at most $2(\theta+|c|^{1/2})$, and for each such $t_x$, the condition $|t_y-\mathsf B_{yy}-c/X|\le\theta$ confines $t_y$ to an interval of length at most $2\theta$. The part of $\mathcal E$ on which $|X|\le|Y|$ is therefore contained in a set of area at most $4\theta(\theta+|c|^{1/2})$, and by symmetry the same holds for the part on which $|Y|\le|X|$. Since the joint density of $(t_x,t_y)$ is at most $p^2$, we find $\P(\mathcal E)\le8p^2\theta(\theta+|c|^{1/2})$. Combining the two bounds, and using $2p\theta\le8p^2\theta$ for $p\ge1$ when $|c|\ge1$, we find \[ \P(\mathcal E)\le\min\big\{2p\theta,\ 8p^2\theta^2+8p^2\theta|c|^{1/2}\big\}\le8p^2\theta^2+8p^2\theta\min\{1,|c|^{1/2}\}, \] which is the claim. \end{proof} Finally, we record the properties of the quantiles of the imaginary part of the root cavity variable that we use. For $p>1$, $E>0$, $\eta>0$, and $\alpha\in(0,1)$, we define \begin{equation}\label{eq:res-quantile} \zeta_\alpha(\eta)=\sup\big\{t\ge0:\P^\infty\big(\Im\Gamma_o(E+\iu\eta)\ge t\big)\ge\alpha\big\}. \end{equation} \begin{lemma}\label{lem:res-quantile} Let $p>1$, $E>0$, and $\alpha\in(0,1)$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $\eta>0$, we have $0<\zeta_\alpha(\eta)<\infty$, $\P^\infty(\Im\Gamma_o(E+\iu\eta)\ge\zeta_\alpha(\eta))\ge\alpha$, and $\P^\infty(\Im\Gamma_o(E+\iu\eta)\ge t)<\alpha$ for every $t>\zeta_\alpha(\eta)$. \item Let $r\in(0,1)$ and $C_r=\sup_{0<\eta\le1}\E^\infty|\Gamma_o(E+\iu\eta)|^r$, which is finite by \Cref{lem:rde-moments}. Set $R_{\mathrm{aux}}=(4C_r/\alpha)^{1/r}$. Then for every $\eta\in(0,1]$, \[ \P^\infty\big(|\Gamma_o(E+\iu\eta)|0$, write $I=\Im\Gamma_o(E+\iu\eta)$, and let $\psi(t)=\P^\infty(I\ge t)$. By \Cref{lem:tree-ward}(i), $I>0$ almost surely, so $\psi(t)\to1>\alpha$ as $t\downarrow0$, and the set in \eqref{eq:res-quantile} contains some $t>0$. Since $\psi(t)\le\P^\infty(|\Gamma_o(E+\iu\eta)|\ge t)\to0$ as $t\to\infty$, the set is bounded. We conclude that $0<\zeta_\alpha(\eta)<\infty$. The function $\psi$ is nonincreasing, and it is left-continuous because $\{I\ge t\}=\bigcap_{t'1$, $E>0$, and a real solution $\pi$ at energy $E$, and we write $\varphi=\varphi_\pi$ for its free energy. The \emph{factor law} $\nu_{\mathrm{fac}}$ is the law of $(d,t)$, where $d$ has law $\pi$, $t=Y/p$ with $Y\sim\Exp(1)$, and $d$ and $Y$ are independent. Factors $M(d_i,t_i)$ with independent $(d_i,t_i)\sim\nu_{\mathrm{fac}}$ are independent copies of the matrix $M(B,\beta)$ of \Cref{sec:press} with the scaling $g=1$, in which $B=D$ has law $\pi$ and $\beta$ has the exponential law of rate $K=p$. For $u\in(0,1)$ and $r\ge0$, we let \[ f_r(u)=\E\big\|M(d_1,t_1)\cdots M(d_r,t_r)\big\|^{-u} \] be the norm moments of \Cref{def:press-pressure}, computed with independent factors of law $\nu_{\mathrm{fac}}$; the order of the factors does not affect the law of the product. By \Cref{prop:rde-properties}(iii), $\E|d|^{-u}<\infty$, and \Cref{cor:press-norm-moments} gives \begin{equation}\label{eq:res-norm-moments} f_r(u)\le e^{r(\varphi(u)-\log p)},\qquad r\ge0,\ u\in(0,1). \end{equation} \begin{definition}\label{def:res-block} Let $m\ge1$ and $R>1$. \begin{enumerate}[label=\textup{(\roman*)}] \item A \emph{block} of length $m$ is a family $(d_i,t_i)_{i=1}^m$ with $d_i\in\C$ and $t_i>0$. Its \emph{growth} is \[ Z=\log\big\|M(d_1,t_1)M(d_2,t_2)\cdots M(d_m,t_m)\big\|\ge0. \] The block is \emph{retained} if $R^{-1}\le|d_i|\le R$ and $R^{-1}\le t_i\le R$ for $1\le i\le m$, and \emph{strictly retained} if these inequalities are strict. We write $H_R$ and $H_R^\circ$ for the corresponding events. \item For independent factors $(d_i,t_i)\sim\nu_{\mathrm{fac}}$, the \emph{block free energy} is \[ \varphi_{m,R}(u)=\log p+\frac1m\log\E\big[\one_{H_R}e^{-uZ}\big],\qquad u\in\R. \] \end{enumerate} \end{definition} The growth is nonnegative by \eqref{eq:res-inverse-norm}. By \eqref{eq:res-row-norm}, a factor with $R^{-1}\le|d|\le R$ and $R^{-1}\le t\le R$ has norm at most $C_R=R^{1/2}(1+2R^2)^{1/2}$, and a retained block satisfies \begin{equation}\label{eq:res-block-bound} 0\le Z\le m\log C_R. \end{equation} The probability $\P(H_R)$ is positive. Indeed, since $\pi$ has full support (\Cref{prop:rde-properties}(iv)), the set $\{R^{-1}\le|d|\le R\}$, which contains the open interval $(R^{-1},R)$, has positive $\pi$-probability; the event $\{R^{-1}\le t\le R\}$ has positive probability since $R>1$; and $H_R$ is an intersection of $2m$ independent such events. Let $\Lambda_{m,R}(u)=\log\E[\one_{H_R}e^{-uZ}]$, so that $\varphi_{m,R}(u)=\log p+m^{-1}\Lambda_{m,R}(u)$. Since $0\le Z\le m\log C_R$ on $H_R$ and $\P(H_R)>0$, the function $u\mapsto\E[\one_{H_R}e^{-uZ}]$ is finite and positive, and it is smooth on $\R$, with derivatives $\E[\one_{H_R}(-Z)^ke^{-uZ}]$ obtained by differentiating under the expectation (the integrands are bounded uniformly for $u$ in compact sets). Then $\varphi_{m,R}$ is finite and smooth on $\R$. It is convex by H\"older's inequality: for $u_0,u_1\in\R$ and $\tau\in(0,1)$, applying H\"older's inequality with the exponents $1/(1-\tau)$ and $1/\tau$ to $\one_{H_R}e^{-(1-\tau)u_0Z}\cdot\one_{H_R}e^{-\tau u_1Z}$, we find \[ \Lambda_{m,R}\big((1-\tau)u_0+\tau u_1\big)\le(1-\tau)\Lambda_{m,R}(u_0)+\tau\Lambda_{m,R}(u_1). \] The next lemma replaces the free energy by a block free energy, with a small loss in the quantities of \Cref{lem:press-gap}. \begin{lemma}\label{lem:res-block-selection} Suppose that $\varphi(1)>0$, and let $s\in(1/2,1)$, $\gamma=-\varphi'(s)>0$, and $F=\varphi(s)-(1-s)\gamma>0$ be as in \Cref{lem:press-gap}. There exist $m\ge1$ and $R>\max\{E,E^{-1}\}$ with the following properties. Write $\widehat\varphi=\varphi_{m,R}$, and set \[ \widehat\gamma=-\widehat\varphi'(s),\qquad\widehat F=\widehat\varphi(s)-(1-s)\widehat\gamma,\qquad\iota=\log p-\widehat\varphi(s)-s\widehat\gamma. \] \begin{enumerate}[label=\textup{(\roman*)}] \item $\widehat\varphi(s)\le \varphi(s)$. \item $\widehat\gamma\ge\gamma/2$, $\widehat F\ge F/2$, and $(2s-1)\widehat\gamma-2(\varphi(s)-\widehat\varphi(s))\ge(2s-1)\gamma/2$. \item $\iota\ge0$ and $\log p-\iota=\widehat\varphi(s)+s\widehat\gamma>0$. \item $\P(H_R\setminus H_R^\circ)=0$. \end{enumerate} \end{lemma} \begin{proof} The proof has three steps. We first let $m\to\infty$ without truncation, then let $R\to\infty$ at fixed $m$, and finally verify the inequalities. \proofstep{1}{Long blocks.} Let $Z$ be the growth of a block of $m$ independent factors, and set $\varphi^{(m)}(u)=\log p+m^{-1}\log\E e^{-uZ}=\log p+m^{-1}\log f_m(u)$ for $u\in(0,1)$. By \Cref{cor:press-norm-moments}, $m^{-1}\log f_m(u)\to\varphi(u)-\log p$ as $m\to\infty$, that is, $\varphi^{(m)}(u)\to\varphi(u)$ for every $u\in(0,1)$; by \eqref{eq:res-norm-moments}, $\varphi^{(m)}(u)\le \varphi(u)$. By H\"older's inequality, as in the discussion after \eqref{eq:res-block-bound}, $\varphi^{(m)}$ is convex on $(0,1)$. It is differentiable, and the derivative of $u\mapsto\E e^{-uZ}$ is $-\E[Ze^{-uZ}]$. Indeed, for $u,u+h\in(0,1)$, the mean value theorem gives $|e^{-(u+h)Z}-e^{-uZ}|\le|h|Z$, since $Z\ge0$, and we may differentiate under the expectation by dominated convergence with the dominating function $Z$. This function is integrable, since $Z\le\sum_{i=1}^m\log\|M(d_i,t_i)\|$ by submultiplicativity, and $\E\log\|M(d_1,t_1)\|<\infty$ by \Cref{prop:press-properties}(iii). For $0\max\{E,E^{-1}\}$ such that $|\widehat\varphi(s)-\varphi(s)|\le\eps$ and $|\widehat\varphi'(s)-\varphi'(s)|\le\eps$. Then (i) holds by Step 2, and \[ \widehat\gamma\ge\gamma-\eps\ge\frac\gamma2,\qquad \widehat F\ge \varphi(s)-\eps-(1-s)(\gamma+\eps)\ge F-2\eps\ge\frac F2, \] \[ (2s-1)\widehat\gamma-2\big(\varphi(s)-\widehat\varphi(s)\big)\ge(2s-1)(\gamma-\eps)-2\eps\ge(2s-1)\gamma-3\eps\ge\frac{(2s-1)\gamma}2, \] where we used $(2s-1)\eps\le\eps$. This proves (ii). For (iii), let $\Lambda=\Lambda_{m,R}$ for the chosen $m$ and $R$, so that $\Lambda(u)=m(\widehat\varphi(u)-\log p)$. It is convex and differentiable on $\R$, with $\Lambda'(s)=-m\widehat\gamma$, and \[ m\iota=m\log p-m\widehat\varphi(s)-sm\widehat\gamma=-\Lambda(s)+s\Lambda'(s). \] A convex differentiable function lies above its tangent lines; at the point $0$, the tangent line at $s$ gives $\Lambda(0)\ge\Lambda(s)+\Lambda'(s)(0-s)$. Then $m\iota\ge-\Lambda(0)=-\log\P(H_R)\ge0$. (Equivalently, $m\iota$ is the relative entropy of the tilted block law $\widehat\mu$ of the proof of \Cref{lem:res-corridor} with respect to the law of a block, and $-\log\P(H_R)$ is a lower bound for it.) Further, $\log p-\iota=\widehat\varphi(s)+s\widehat\gamma=\widehat F+\widehat\gamma>0$. For (iv), the only atom of $\pi$ is at $E$ (\Cref{prop:rde-properties}(i)), and $R,R^{-1}\neq E$; thus $\P(|d_i|\in\{R,R^{-1}\})=0$, and $\P(t_i\in\{R,R^{-1}\})=0$ since $t_i$ has a density. \end{proof} From now on in this section, $m$, $R$, $\widehat\varphi$, $\widehat\gamma$, $\widehat F$, and $\iota$ are as in \Cref{lem:res-block-selection}. The number $\iota$ is the exponential cost per factor of the regular growth events of the next lemma, on which the partial sums of the block growths stay close to $m\widehat\gamma$ per block, their typical value under the exponentially tilted block law. \begin{lemma}\label{lem:res-corridor} Let $(d_i,t_i)_{i\ge1}$ be independent factors with law $\nu_{\mathrm{fac}}$, and for $l\ge1$ let $Z_l$, $H_l$, and $H_l^\circ$ be the growth and the retention events of the block $(d_i,t_i)_{(l-1)m0$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have \[ \P\big(H_1\cap\dots\cap H_k\cap\{S_k\le km\widehat\gamma+b\}\big)\le e^{-\iota mk+sb}. \] \item We have \[ \P\big(H_1^\circ\cap\dots\cap H_k^\circ\cap\{|S_i-im\widehat\gamma|0$, and $a_+,a_-\ge0$, and let $C_*$ be the constant of \Cref{prop:press-three-rows}. Let $\mathcal G$ be a $\sigma$-algebra, and let $G_1,G_2,G_3,G_4$ be random matrices such that $\mathcal G,G_1,G_2,G_3,G_4$ are independent, and each $G_i$ is a product of three factors as in \Cref{prop:press-three-rows-complex}, with denominators satisfying the moment bounds with $a_+$ and $a_-$, and with parameters $\beta$ of the exponential law of rate $K$. Let $A_1,A_2,A_3$ be $\mathcal G$-measurable random complex $2\times2$ matrices with determinant one, and let $w$ be a $\mathcal G$-measurable random vector in $\C^2\setminus\{0\}$. Then, almost surely, \[ \E\Big[\big|e_2^{\mathsf T}G_1A_1G_2A_2G_3A_3G_4w\big|^{-s}\,\Big|\,\mathcal G\Big]\le C_*^4\,\|A_1\|^{-s}\|A_3\|^{-s}\|w\|^{-s}. \] \end{lemma} \begin{proof} The proof averages first over the averaging triples $G_1$ and $G_4$ next to the endpoints, with everything else fixed, and then over the averaging triples $G_2$ and $G_3$ next to the fork. Let $\mathcal G'$ be the $\sigma$-algebra generated by $\mathcal G$, $G_2$, and $G_3$. Since $\mathcal G,G_1,\dots,G_4$ are independent, the $\sigma$-algebra $\mathcal G'$ and the matrices $G_1$ and $G_4$ are independent (by grouping independent families); in particular, $\mathcal G'$ is independent of $(G_1,G_4)$, and $G_1$ and $G_4$ are independent. The matrix $A=A_1G_2A_2G_3A_3$ is $\mathcal G'$-measurable and has determinant one, so it is nonzero, and the vector $w$ is $\mathcal G'$-measurable and nonzero. By \Cref{cor:press-buffers}(i) and (iii), applied with the $\sigma$-algebra $\mathcal G'$, with $u=e_2$, $v=w$, and with the averaging triples $G=G_1$ and $G'=G_4$, we have \[ \E\Big[\big|e_2^{\mathsf T}G_1AG_4w\big|^{-s}\,\Big|\,\mathcal G'\Big]\le C_*^2\|A\|^{-s}\|w\|^{-s}. \] Here we used $\|e_2\|=1$. Next, $\mathcal G$ is independent of $(G_2,G_3)$, and $G_2$ and $G_3$ are independent. By \Cref{cor:press-buffers}(ii) and (iii), applied with the $\sigma$-algebra $\mathcal G$, with $C=A_1$, $G=G_2$, $F=A_2$, $G'=G_3$, and $A=A_3$, and since $\|A_2\|\ge1$ by \eqref{eq:res-inverse-norm}, we have \[ \E\big[\|A_1G_2A_2G_3A_3\|^{-s}\,\big|\,\mathcal G\big]\le C_*^2\|A_1\|^{-s}\|A_3\|^{-s}. \] Since $\mathcal G\subset\mathcal G'$, the tower property gives \[ \E\Big[\big|e_2^{\mathsf T}G_1AG_4w\big|^{-s}\,\Big|\,\mathcal G\Big]=\E\Big[\E\Big[\big|e_2^{\mathsf T}G_1AG_4w\big|^{-s}\,\Big|\,\mathcal G'\Big]\,\Big|\,\mathcal G\Big]\le C_*^2\|w\|^{-s}\,\E\big[\|A\|^{-s}\,\big|\,\mathcal G\big], \] where we used that $w$ is $\mathcal G$-measurable. Inserting the second bound, we obtain the claim. \end{proof} The factors $A_1$ and $A_3$ will be products of factors along the two arms of a pair of paths. Some of these factors have a deterministic denominator, and the next lemma removes them at a fixed cost. \begin{lemma}\label{lem:res-arm} Let $u\in(0,1)$, $r,k\ge0$, and let $c_1,\dots,c_k\in\C$ with $0<|c_i|\le E+1$. Let $V$ be a product of $r$ independent factors of law $\nu_{\mathrm{fac}}$ (with $V=I$ if $r=0$), and let $H=M(c_1,t'_1)\cdots M(c_k,t'_k)$, where $t'_1,\dots,t'_k$ are independent with density $pe^{-pt}$ and independent of $V$. Set $C_{\mathrm{arm}}=\E\big[(t_1')^{-u/2}(1+(t_1')^2+(E+1)^2)^{u/2}\big]$, which is finite and at least one. Then \[ \E\|HV\|^{-u}\le C_{\mathrm{arm}}^k\,e^{r(\varphi(u)-\log p)},\qquad\E\|VH\|^{-u}\le C_{\mathrm{arm}}^k\,e^{r(\varphi(u)-\log p)}. \] \end{lemma} \begin{proof} By \eqref{eq:res-inverse-norm}, $\|V\|=\|H^{-1}HV\|\le\|H\|\,\|HV\|$, which implies that $\|HV\|^{-u}\le\|H\|^u\|V\|^{-u}$. By submultiplicativity and \eqref{eq:res-row-norm}, $\|H\|^u\le\prod_{i=1}^k(t_i')^{-u/2}(1+(t_i')^2+(E+1)^2)^{u/2}$. Taking expectations, and using that $t_1',\dots,t_k'$ and $V$ are independent together with \eqref{eq:res-norm-moments}, we obtain the first bound. The constant $C_{\mathrm{arm}}$ is finite, since $(1+(t_1')^2+(E+1)^2)^{u/2}\le1+(t_1')^u+(E+1)^u$ for $u/2\le1$ and since $\E(t_1')^{-u/2}$ and $\E(t_1')^{u/2}$ are finite; it is at least one, since $(t_1')^{-1}(1+(t_1')^2)\ge2$. The second bound follows in the same way from $\|V\|\le\|VH\|\,\|H^{-1}\|$. \end{proof} \section{Passage to the real axis}\label{sec:res-limits} In \Cref{sec:deloc}, the events whose probabilities we bound are defined at $E+\iu\eta$ with a small $\eta>0$, while the probabilities are computed for the real recursive tree process at $\eta=0$. The passage between the two uses only almost sure convergence, through the following form of the portmanteau theorem. Part (i) gives lower bounds for events defined by strict inequalities, and it is used in \Cref{lem:deloc-channel,lem:deloc-regularize} and in the proof of \Cref{lem:deloc-first}. Part (ii) gives upper bounds for events defined by non-strict inequalities, weighted by a bounded factor, and it is used in \Cref{lem:deloc-channel} and in the proof of \Cref{lem:deloc-second}. Neither part requires the limiting law to give zero mass to the boundary of the event, and neither requires uniform integrability. \begin{lemma}\label{lem:res-semicontinuity} Let $(\Omega',\mathscr A,\mathbb Q)$ be a probability space, let $k\ge1$, and for $0\le\eta\le1$ let $V_\eta$ be random vectors in $\C^k$ such that $V_\eta\to V_0$ almost surely as $\eta\downarrow0$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every open set $O\subset\C^k$, we have $\liminf_{\eta\downarrow0}\mathbb Q(V_\eta\in O)\ge\mathbb Q(V_0\in O)$. \item Let $\mathsf F\subset\C^k$ be closed, and let $h_\eta$, $0\le\eta\le1$, be random variables with $0\le h_\eta\le C$ for a constant $C$ and $h_\eta\to h_0$ almost surely. Then \[ \limsup_{\eta\downarrow0}\E_{\mathbb Q}\big[\one_{\mathsf F}(V_\eta)h_\eta\big]\le\E_{\mathbb Q}\big[\one_{\mathsf F}(V_0)h_0\big]. \] \end{enumerate} \end{lemma} \begin{proof} It suffices to consider a sequence $\eta_j\downarrow0$. For (i), if $V_0\in O$ and $V_{\eta_j}\to V_0$, then $V_{\eta_j}\in O$ for all large $j$, since $O$ is open. This implies that $\liminf_j\one_O(V_{\eta_j})\ge\one_O(V_0)$ almost surely, and (i) follows from Fatou's lemma. For (ii), if $V_0\notin\mathsf F$, then $V_{\eta_j}\notin\mathsf F$ for all large $j$, since the complement of $\mathsf F$ is open. Hence $\limsup_j\one_{\mathsf F}(V_{\eta_j})h_{\eta_j}\le\one_{\mathsf F}(V_0)h_0$ almost surely, and Fatou's lemma applied to $C-\one_{\mathsf F}(V_{\eta_j})h_{\eta_j}\ge0$ gives (ii). \end{proof} We apply \Cref{lem:res-semicontinuity} under the spine laws of \Cref{def:palm-spine-laws}, with $V_\eta$ a finite collection of coordinates of the depth-$m$ data $\mathcal L_m(E+\iu\eta)$ of \Cref{def:rde-local}, or of continuous functions of them. Under the no-ac hypothesis, \Cref{prop:rde-fixed-depth}(i) provides the required almost sure convergence. Events defined by strict inequalities between continuous functions of the coordinates are of the form $\{V_\eta\in O\}$ with $O$ open, and events defined by non-strict inequalities are of the form $\{V_\eta\in\mathsf F\}$ with $\mathsf F$ closed. The growth of a block is a continuous function of the factors only where the weights $t_i$ are positive. In \Cref{sec:deloc}, it appears only together with the retention conditions of \Cref{def:res-block}: a strict inequality for partial sums of growths together with strict retention defines an open set, since the set of strictly retained blocks is open and the growths are continuous on it; and a non-strict inequality together with retention defines a closed set, since the set of retained blocks is closed and contained in the open set $\{t_i>0\text{ for all }i\}$, on which the growths are continuous. Further, if $A_\eta$, $0\le\eta\le1$, are events with $\one_{A_\eta}\to\one_{A_0}$ almost surely as $\eta\downarrow0$ (for instance, events that do not depend on $\eta$), we may add $\one_{A_\eta}$ as a coordinate of $V_\eta$. Since $\one_{A_\eta}\in\{0,1\}$, the event $A_\eta\cap\{V_\eta\in O\}$ is then of the form $\{(V_\eta,\one_{A_\eta})\in O\times\{w\in\C:\Re w>1/2\}\}$, with an open set, and $A_\eta\cap\{V_\eta\in\mathsf F\}$ is of the form $\{(V_\eta,\one_{A_\eta})\in\mathsf F\times\{w\in\C:\Re w\ge1/2\}\}$, with a closed set. In \Cref{sec:deloc}, lower bounds are proved for strict inequalities and upper bounds for non-strict ones, and no condition on the boundaries of the events is needed. \chapter{Positive free energy implies delocalization}\label{sec:deloc} In this section, we prove that the boundary values of the cavity variables cannot all be real at a regular energy where the real boundary law has positive free energy at $s=1$. Together with the spectral averaging criterion of \Cref{prop:rank-ac}, this gives purely absolutely continuous spectrum on intervals where, for almost every regular energy, all possible real boundary laws have positive free energy at $s=1$. Recall from \Cref{def:rde-boundary-law} that, for $E\in\mathcal E_p$ with $E>0$, the no-ac hypothesis holds at $E$ if $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$; in this case, the law $\pi_E$ of $D_o=-1/\Gamma_o(E+\iu0)$ under $\P$ is a real solution at energy $E$, by \Cref{prop:rde-boundary}(ii), and we call it the real boundary law. \begin{theorem}\label{thm:deloc-criterion} There exists a numerical constant $p_*>1$ such that the following holds for every $p\ge p_*$. Let $E\in\mathcal E_p$ with $E>0$, and suppose that $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Then the real boundary law $\pi_E$ satisfies $\varphi_{\pi_E}(1)\le0$. \end{theorem} \begin{corollary}\label{cor:deloc-ac} Let $p\ge p_*$, where $p_*$ is as in \Cref{thm:deloc-criterion}, and let $I\subset(0,\infty)$ be an interval. Suppose that for Lebesgue-almost every $E\in I\cap\mathcal E_p$, either $\P^\infty(\Im\Gamma_o(E+\iu0)=0)<1$, or $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$ and the real boundary law $\pi_E$ satisfies $\varphi_{\pi_E}(1)>0$. Then $\P^\infty$-almost surely, $\mathcal H$ is purely absolutely continuous in $I\cup(-I)$. \end{corollary} In particular, if $\P^\infty(\Im\Gamma_o(E+\iu0)=0)<1$ for every $E\in I\cap\mathcal E_p$, then the conclusion of \Cref{cor:deloc-ac} holds. \begin{proof}[Proof of \Cref{cor:deloc-ac}, assuming \Cref{thm:deloc-criterion}] By \Cref{lem:rde-regular}, the set $\mathcal E_p$ has full Lebesgue measure. Let $E\in I\cap\mathcal E_p$ be an energy at which the hypothesis holds; this is the case for almost every $E\in I$. If $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$, then \Cref{thm:deloc-criterion} implies that $\varphi_{\pi_E}(1)\le0$, contradicting the hypothesis. Thus $\P^\infty(\Im\Gamma_o(E+\iu0)=0)<1$. By \Cref{prop:rde-boundary}(i), this probability equals $1-u/\varrho_p\in\{0,1\}$, where $u=\P(\Im\Gamma_o(E+\iu0)>0)\in\{0,\varrho_p\}$. Hence it equals zero, and $\P(\Im\Gamma_o(E+\iu0)>0)=u=\varrho_p>0$. This verifies the hypothesis of \Cref{prop:rank-ac} at almost every $E\in I$, and \Cref{prop:rank-ac} gives the conclusion. \end{proof} \section{Outline of the proof}\label{sec:deloc-outline} The proof of \Cref{thm:deloc-criterion} follows the resonance strategy of Aizenman and Warzel, who used it in \cite{AizenmanWarzel2013resonant} to prove absolutely continuous spectrum for the Anderson model on regular trees beyond the band of the hopping operator. Versions of this strategy for tree operators were developed by Aggarwal, Bordenave, and Lopatto for the local limits of L\'evy matrices \cite{AggarwalBordenaveLopatto2022mobility}, and by Aggarwal and Lopatto for the Anderson model on the Bethe lattice \cite{AggarwalLopatto2025mobility}. We adapt it to the Poisson tree with complex Gaussian weights. The argument uses only the properties of real solutions established in \Cref{sec:rde,sec:press}, so it applies to an arbitrary real boundary law. We now describe its steps. \paragraph{The quantile.} Fix $p\ge p_*$ and $E\in\mathcal E_p$ with $E>0$, suppose that the no-ac hypothesis holds at $E$, and suppose, for a contradiction, that $\varphi(1)>0$, where $\varphi=\varphi_{\pi_E}$. Let $\alpha=q_*/2$ be the numerical level fixed in \eqref{eq:deloc-qstar} below. For $\eta>0$, let $\zeta_\alpha(\eta)$ be the quantile of $\Im\Gamma_o(E+\iu\eta)$ under $\P^\infty$ defined in \eqref{eq:res-quantile}. By \Cref{lem:res-quantile}(i), we have $0<\zeta_\alpha(\eta)<\infty$, and $\P^\infty(\Im\Gamma_o(E+\iu\eta)\ge t)<\alpha$ for every $t>\zeta_\alpha(\eta)$. Under the no-ac hypothesis, $\Im\Gamma_o(E+\iu\eta)\to0$ almost surely, so that $\zeta_\alpha(\eta)\to0$ as $\eta\downarrow0$, at an unknown rate. For this reason, we always compare $\Im\Gamma_o(E+\iu\eta)$ with the quantile at the same imaginary part $\eta$. We show that there exist $\delta>0$ and $C<\infty$ such that, for every large $n$, there exists $\eta_n>0$ with \begin{equation}\label{eq:deloc-outline-goal} \P^\infty\big(\Im\Gamma_o(E+\iu\eta_n)\ge e^{2\delta n-Cn^{3/4}}\zeta_\alpha(\eta_n)\big)\ge q_*>\alpha. \end{equation} For large $n$, the threshold in \eqref{eq:deloc-outline-goal} is larger than $\zeta_\alpha(\eta_n)$, which contradicts \Cref{lem:res-quantile}(i). \paragraph{Exponents.} By \Cref{lem:press-gap}, there exists $s\in(1/2,1)$ such that $\gamma=-\varphi'(s)>0$ and $F=\varphi(s)-(1-s)\gamma>0$. The number $F$ is the value at $u=1$ of the tangent line to $\varphi$ at $s$. Since $\varphi'(1)=0$, it tends to $\varphi(1)>0$ as $s\uparrow1$, and this is how $s$ is chosen. In \Cref{sec:res-blocks}, the free energy is replaced by the free energy $\widehat\varphi$ of blocks of $m$ factors with entries in a compact set, with the corresponding numbers $\widehat\gamma\approx\gamma$ and $\widehat F\approx F$ (\Cref{lem:res-block-selection}). We fix $\delta>0$ small in terms of $F$ and $(2s-1)\gamma$, and we set $N=n+10$, $b_n=n^{3/4}$, and $T_n=e^{(\widehat\gamma+\delta)n}$. \paragraph{The gain from a resonance.} Let $x$ be a vertex of generation $N$, and let $A$ be the product of the transfer matrices (factors, in the sense of \Cref{def:res-rows}) along the path from $o$ to $x$, evaluated at $z=E+\iu\eta$. By \Cref{lem:res-path}, we have $|G_{ox}(z)|\ge|G_{xx}(z)|/\|A\|$. If $y$ is a child of $x$, the Ward identity of \Cref{lem:tree-ward}(iii) gives $\Im\Gamma_o(z)\ge|G_{ox}(z)|^2|h_{xy}|^2\Im\Gamma_y(z)$. Suppose that $\|A\|\le e^{\widehat\gamma n+O(b_n)}$, that $|G_{xx}(z)|\ge T_n$, that $|h_{xy}|^2$ is bounded below, and that $\Im\Gamma_y(z)\ge\zeta_\alpha(\eta)/2$. Then $\Im\Gamma_o(z)\ge e^{2\delta n-O(b_n)}\zeta_\alpha(\eta)$. We call the factor $e^{2\delta n-O(b_n)}$ by which $\Im\Gamma_o(z)$ exceeds the quantile the \emph{gain}; the precise statement is \Cref{lem:deloc-gain}. \paragraph{Resonant vertices.} We count the vertices $x$ of generation $N$ for which the conditions of the previous paragraph hold; they are called resonant (\Cref{def:deloc-candidate}). The conditions are the following. The factors of the first $n$ generations satisfy a regular growth condition: the logarithms of the norms of the products of consecutive blocks of $m$ factors have partial sums within $b_n$ of their typical values (R3). The last ten vertices of the path have no side children (R1), and the squared weights of their edges lie in fixed intervals (R4). By \Cref{lem:res-reset}, this forces the rootward cavity variable at the parent of $x$ into a fixed disc, whatever its value at generation $n$. The vertex $x$ has exactly one leaf child and one surviving child $y$ (R2), and the cavity variable of $y$ is bounded and has imaginary part at least $\zeta_\alpha(\eta)/2$ (R5); we call $y$ the \emph{seed}, since its imaginary part is the one that the resonance amplifies. By \eqref{eq:res-leaf}, the diagonal entry $G_{xx}(z)$ is then a M\"obius function of the squared weight $t$ of the leaf edge, with a pole whose real part lies in $[E^2/2,3E^2/2]$ (\Cref{lem:res-terminal}) and whose imaginary part, which we call the \emph{pole width}, is small (R6). Since $t$ has the density $pe^{-pt}$, the event $|G_{xx}(z)|\ge T_n$ (R7) has probability of order $T_n^{-1}$ (\Cref{lem:res-tuning}). \paragraph{The first moment.} By the many-to-one formula for the backbone, the expected number of resonant vertices is $p^N$ times the probability that the endpoint of a spine of length $N$ is resonant. The structural conditions (R1), (R2) and the conditions (R4), (R5) cost a factor independent of $n$, the tuning (R7) costs a factor of order $T_n^{-1}$, and the regular growth condition (R3) costs a factor $e^{-\iota n+O(b_n)}$, where $\iota=\log p-\widehat\varphi(s)-s\widehat\gamma$ (\Cref{lem:res-corridor}, which computes this probability after an exponential tilt of the law of a block at the exponent $s$). Since $\log p-\iota-\widehat\gamma-\delta=\widehat F-\delta$, we obtain in \Cref{lem:deloc-first} that \[ \E^\infty N_n(\eta)\ge e^{(\widehat F-\delta)n-O(b_n)}. \] The probability of the regular growth event is computed for the one-spine real recursive tree process, at $\eta=0$, where the factors along a spine are independent with the law $\nu_{\mathrm{fac}}$ determined by $\pi_E$ (\Cref{lem:rde-comparison-law}). The growth of their products is then governed by $\varphi$. The bound is transferred to a small imaginary part $\eta>0$, with $n$ fixed, by \Cref{lem:res-semicontinuity}: lower bounds are proved for events defined by strict inequalities, which are open, and upper bounds for events defined by non-strict inequalities, which are closed. \paragraph{The second moment.} By the many-to-two formula, the expected number of ordered pairs $(u,v)$ of resonant vertices whose paths split at generation $j$ is $\varrho_pp^{2N-j}$ times a probability under a two-spine law. The two leaf weights are fresh, and by \Cref{lem:res-schur-two} the $2\times2$ block of the resolvent at $\{u,v\}$ is a function of them. By \Cref{lem:res-two-terminal}, the probability that both diagonal entries exceed $T_n$ is at most of order $T_n^{-2}+T_n^{-1}\min\{1,\mathfrak D\}$, where $\mathfrak D$ measures the interaction of $u$ and $v$ through the path that joins their parents. The first term is the one for independent vertices. With the upper bound of \Cref{lem:res-corridor}(i) for regular growth events on the common part and on the two arms of the paths, it contributes $e^{O(b_n)}(\E^\infty N_n)^2$ after summation over $j$. For the second term, we bound $\E\mathfrak D^s$ by averaging over four groups of three fresh factors, placed next to the two ends and next to the fork of the joining path (\Cref{lem:res-cross}). What remains are the negative moments of the norms of the products along the two arms, which decay like $e^{(n-j)(\varphi(s)-\log p)}$, where $\varphi(s)<\log p$. The resulting exponent is affine in $j$, and its values at $j=0$ and $j=n$ are negative by the choice of $\delta$. We obtain $\E^\infty[N_n(N_n-1)]\le e^{O(b_n)}(\E^\infty N_n)^2$ (\Cref{lem:deloc-second}), and the Paley--Zygmund inequality gives $\P^\infty(N_n\ge1)\ge e^{-O(b_n)}$. On this event, $\Im\Gamma_o(E+\iu\eta)\ge e^{2\delta n-O(b_n)}\zeta_\alpha(\eta)$. \paragraph{Amplification.} It remains to raise the probability $e^{-O(b_n)}$ to the fixed level $q_*>\alpha$. In \Cref{sec:deloc-amplification}, we show that if $\Im\Gamma_o\ge y$ with probability at least $e^{-a}$, then $\Im\Gamma_o\ge c_3e^{-C_3(1+a)}y$ with probability at least $q_*$ (\Cref{cor:deloc-amplify}). The main step uses the Poisson structure of the children of the root. A random set of children, whose number has a Poisson law with mean about $64$, carries independent copies of an event of probability $q$, while the other children keep $|D_o|$ in a fixed annulus. If one of the copies occurs, then so does the event at the root, with a fixed loss in the imaginary part, and this happens with probability at least $4q$. After $O(1+a)$ iterations, the probability reaches $q_*$. This is the only step that requires $p\ge p_*$, and $p_*$ does not depend on $E$, since it uses only \Cref{lem:rde-guard}, which holds uniformly over all real solutions. With $a=O(b_n)$, we obtain \eqref{eq:deloc-outline-goal}. \paragraph{Order of choices.} The constants are fixed in the order (a)--(f) listed in \Cref{sec:deloc-candidates}. The numerical constants come first; then the constants that depend on $p$ and $E$; then the depth $n$; and finally the imaginary part $\eta_n$, which depends on $n$. Each estimate at positive $\eta$ is obtained from an estimate at $\eta=0$ with $n$ fixed, and the two limits $n\to\infty$ and $\eta\downarrow0$ are never taken in the same estimate. The quantile $\zeta_\alpha(\eta)$ enters only through the seed condition (R5), whose probability is bounded at each fixed $\eta$ by \Cref{lem:res-quantile}(ii), and through the gain. \paragraph{Organization.} \Cref{sec:deloc-amplification} contains the amplification. \Cref{sec:deloc-candidates} fixes the constants, defines the resonant vertices, and proves the gain. \Cref{sec:deloc-first,sec:deloc-second} contain the first and second moments, and \Cref{sec:deloc-proof} completes the proof. \section{Amplification at the root}\label{sec:deloc-amplification} In this section, we show that an event of small probability on which $\Im\Gamma_o$ is large can be converted into an event of a fixed probability, at the cost of a moderate factor in the imaginary part. We begin with the numerical constants. Let \begin{equation}\label{eq:deloc-a0} a_0=\frac1{32\sqrt2}, \end{equation} so that $4a_0=a(1/32)$ in the notation of \Cref{lem:rde-guard}. Let $N_0\in\N$ be such that a $\Pois(128)$ variable exceeds $N_0-1$ with probability at most $1/32$. Let $\lambda\in[32,128]$ and $N\sim\Pois(\lambda)$. Since a $\Pois(128)$ variable is the sum of $N$ and an independent $\Pois(128-\lambda)$ variable, we have $\P(N\ge N_0)\le1/32$, and the identity $\E[N\one\{N>N_0\}]=\lambda\P(N\ge N_0)$ gives \begin{equation}\label{eq:deloc-N0} \P(N>N_0)\le\frac1{32},\qquad\E\big[N\one\{N\le N_0\}\big]\ge\frac\lambda2\ge16. \end{equation} Set $y_+=a_0^2/(4N_0)$, $y_-=y_+/2$, $J=[y_-,y_+]$, and $\rho_J=\P(Y\in J)$ for $Y\sim\Exp(1)$. We let $p_*\ge32\log64+1$ be the smallest number such that \begin{equation}\label{eq:deloc-pstar} \frac7{16}\,p_*\rho_J\ge64,\qquad e^{-p_*/2+128}\le\frac1{16}, \end{equation} and we set \begin{equation}\label{eq:deloc-qstar} q_0=\frac1{2N_0},\qquad q_*=\frac1{4N_0},\qquad\alpha=\frac{q_*}2. \end{equation} All these constants are numerical. Let $p\ge p_*$. Since $1-e^{-p/2}>1/2$, the concave function $\varrho\mapsto1-e^{-p\varrho}-\varrho$ is positive at $\varrho=1/2$, and therefore $\varrho_p>1/2$. By \Cref{lem:rde-guard} with $\eps=1/32$, every real solution at every energy $E>0$ satisfies $\P(|D|\le4a_0/\sqrt p)<1/32$. By \eqref{eq:deloc-pstar}, we have $p(\varrho_p-1/16)\rho_J\ge64$, $e^{-(p\varrho_p-128)}\le1/16$, and $e^{-(p-1)\varrho_p}\le1/16$. \paragraph{Standing assumptions.} From now on until the end of this section, we fix $p\ge p_*$ and $E\in\mathcal E_p$ with $E>0$, and we assume that the no-ac hypothesis holds at $E$. We write $\pi=\pi_E$ for the real boundary law, $D_o(z)=-1/\Gamma_o(z)$ for $z\in\C\setminus\R$, and $D_o(E)=-1/\Gamma_o(E+\iu0)$. By \Cref{prop:rde-boundary}(ii), $\P$-almost surely $\Gamma_v(E+\iu\eta)\to\Gamma_v(E+\iu0)\in\R\setminus\{0\}$ as $\eta\downarrow0$ for every vertex $v$. The law of $D_o(E)$ is $\pi$ under $\P$ and $\pi^\infty_E$ under $\P^\infty$, and $\pi^\infty_E$ has no atoms (see the discussion after \Cref{def:rde-boundary-law}). We set \begin{equation}\label{eq:deloc-chart-constants} d_-=\frac{a_0}{\sqrt p},\qquad h_0=\frac{d_-^2}4, \end{equation} we let $d_+\ge16d_-$ be such that $\P(|D_o(E)|>d_+/2)\le1/32$, and we set $c_0=\min\{1,y_-/(pd_+^2)\}$. Then $\P(|D_o(E)|\le4d_-)<1/32$ by the choice of $a_0$, and $N_0y_+/(pd_-)=d_-/4$. \begin{definition}\label{def:deloc-charts} For $\eta\in(0,1]$ and $y>0$, let $\mathcal A_y(\eta)$ be the event that \[ d_-\le|D_o(E+\iu\eta)|\le d_+\qquad\text{and}\qquad y\le\Im\Gamma_o(E+\iu\eta)\le1. \] \end{definition} Recall from \Cref{lem:palm-branching} that the children of the root form the Poisson process $\Pi_o=\sum_k\delta_{(w_k,\theta_k\omega)}$ on $\mathsf M=\C\times\Omega$ with intensity measure $p\,\mathfrak m$, where $\mathfrak m=\gamma\otimes\P$. The auxiliary mark $\zeta_k$ of the child $k$ is the root coordinate of $\theta_k\omega$; its coordinates other than the rank $\zeta_k(1)$ are independent uniform variables, independent of the marked tree (\Cref{rem:palm-auxiliary}). We use them to color children. \begin{lemma}\label{lem:deloc-channel} There exists $\eta_{\mathrm{ch}}\in(0,1]$, depending on $p$ and $E$, such that the following holds for every $\eta\in(0,\eta_{\mathrm{ch}}]$, every $y>0$, and every $q\in(0,q_0]$. If $\P^\infty(\mathcal A_y(\eta))\ge q$, then $\P^\infty(\mathcal A_{c_0y}(\eta))\ge4q$. \end{lemma} \begin{proof} We write $z=E+\iu\eta$. The proof splits the children of the root into a bounded number of \emph{amplifying children}, which carry copies of the event $\mathcal A_y(\eta)$, and the remaining \emph{reserve children}, which keep $|D_o|$ in the interval $[d_-,d_+]$ of \Cref{def:deloc-charts}. \proofstep{1}{Amplifying children.} Let \[ \tau=\frac{64}{p\rho_J\,\P\big(\mathcal S\cap\{d_-\le|D_o(E)|\le d_+\}\big)}. \] Since $\P(|D_o(E)|\notin[d_-,d_+])\le\P(|D_o(E)|\le4d_-)+\P(|D_o(E)|>d_+/2)\le1/16$, the probability in the denominator is at least $\varrho_p-1/16$, and $\tau\le1$ by the choice of $p_*$. For $0\le\eta\le1$, let $\mathsf C_\eta$ be the set of $(w,\omega')\in\mathsf M$ such that $\omega'\in\mathcal S$, $|w|^2\in J$, $\zeta_o(\omega')(2)\le\tau$, and \[ d_-\le|D_o(E+\iu\eta)(\omega')|\le d_+,\qquad\Im\Gamma_o(E+\iu\eta)(\omega')\le1, \] where for $\eta=0$ the second condition is void. A child $k$ is an amplifying child at imaginary part $\eta$ if $(w_k,\theta_k\omega)\in\mathsf C_\eta$, and a reserve child otherwise. As in the proof of \Cref{lem:palm-survival}(i), the restrictions $\Pi^{\mathrm{ch}}_\eta$ and $\Pi^{\mathrm{res}}_\eta$ of $\Pi_o$ to $\mathsf C_\eta$ and to its complement are independent Poisson processes under $\P$, with intensity measures $p\,\mathfrak m(\cdot\cap\mathsf C_\eta)$ and $p\,\mathfrak m(\cdot\setminus\mathsf C_\eta)$ \cite[Theorem~5.2]{LastPenrose2018lectures}. Since $|w|^2$, the color $\zeta_o(\omega')(2)$, and the remaining coordinates of $\omega'$ are independent under $\mathfrak m$, the number $N^{\mathrm{ch}}_\eta$ of amplifying children has the law $\Pois(\lambda(\eta))$, where \[ \lambda(\eta)=p\tau\rho_J\,\P\big(\mathcal S\cap\{d_-\le|D_o(E+\iu\eta)|\le d_+,\ \Im\Gamma_o(E+\iu\eta)\le1\}\big), \] and $\lambda(0)=64$ by the choice of $\tau$. Under $\P^\infty$, we have $D_o(E+\iu\eta)\to D_o(E)$ and $\Im\Gamma_o(E+\iu\eta)\to0$ almost surely. We apply \Cref{lem:res-semicontinuity} under $\P^\infty$ to the vectors $V_\eta=(D_o(E+\iu\eta),\Im\Gamma_o(E+\iu\eta))$ and $V_0=(D_o(E),0)$, with the closed set $\{(x,y):d_-\le|x|\le d_+,\ \Re y\le1\}$ in part (ii) (with $h_\eta=1$) and the open set $\{(x,y):d_-<|x|1/2\}$, with $V_0=(D^{\mathrm{res}}_0,0,\varsigma^{\mathrm{res}}_0)$, so that $V_\eta\to V_0$ almost surely. Since $00$, and every $a\ge0$. If $\P^\infty(\Im\Gamma_o(E+\iu\eta)\ge y)\ge e^{-a}$, then $\P^\infty(\mathcal A_{c_2e^{-2a}y}(\eta))\ge c_1e^{-3a}$. \end{lemma} \begin{proof} We write $z=E+\iu\eta$. Let $C_\natural=\sup_{0<\eta'\le1}\E^\infty|\Gamma_o(E+\iu\eta')|^{1/2}$, which is finite by \Cref{lem:rde-moments}, and set \[ \lambda_{\mathrm r}=\frac1{100},\qquad\kappa=\min\Big\{1,\frac{pd_-}8,\frac{pd_-^2}8\Big\},\qquad R_a=\max\{1,4C_\natural^2e^{2a}\}. \] By Markov's inequality, $\P^\infty(|\Gamma_o(z)|>R_a)\le C_\natural R_a^{-1/2}\le e^{-a}/2$. \proofstep{1}{Candidate and reserve.} A child $k$ of the root is a \emph{candidate} if $\zeta_k(3)\le\lambda_{\mathrm r}/p$, and a reserve child otherwise. As in the proof of \Cref{lem:deloc-channel}, the candidates and the reserve children form independent Poisson processes $\Pi^{\mathrm c}$ and $\Pi'$ with intensity measures $\lambda_{\mathrm r}\mathfrak m$ and $(p-\lambda_{\mathrm r})\mathfrak m$. This coloring does not depend on $\eta$. Let $D'_\eta=z+p^{-1}\sum_kY_k\Gamma_k(z)$ and $h'_\eta=\eta+p^{-1}\sum_kY_k\Im\Gamma_k(z)$, where the sums run over the reserve children, and let $\mathcal R'_\eta$ be the event that $2d_-<|D'_\eta|<3d_+/4$, that $h'_\eta0$, and every $a\ge0$. If $\P^\infty(\Im\Gamma_o(E+\iu\eta)\ge y)\ge e^{-a}$, then \[ \P^\infty\big(\Im\Gamma_o(E+\iu\eta)\ge c_3e^{-C_3(1+a)}y\big)\ge q_*. \] \end{corollary} \begin{proof} By \Cref{lem:deloc-regularize}, the event $\mathcal A_{y_1}(\eta)$ with $y_1=c_2e^{-2a}y$ has $\P^\infty$-probability at least $q_1=c_1e^{-3a}$. We claim that $\P^\infty(\mathcal A_{c_0^iy_1}(\eta))\ge\min\{4^iq_1,q_*\}$ for every $i\ge0$. For $i=0$ this holds. If it holds for $i$, we apply \Cref{lem:deloc-channel} with $q=\min\{4^iq_1,q_*\}$, which is at most $q_0$, and obtain $\P^\infty(\mathcal A_{c_0^{i+1}y_1}(\eta))\ge4\min\{4^iq_1,q_*\}\ge\min\{4^{i+1}q_1,q_*\}$. Let $k$ be the smallest integer $i\ge0$ with $4^iq_1\ge q_*$. Since $q_*\le1$, we have $k\le1+\max\{0,\log_4(q_*/q_1)\}\le1+(3a+\log(1/c_1))/\log4$, and since $c_0\le1$ and $\mathcal A_{c_0^ky_1}(\eta)\subset\{\Im\Gamma_o(E+\iu\eta)\ge c_0^ky_1\}$, we obtain the claim with $c_3=c_2c_0^{1+\log(1/c_1)/\log4}$ and $C_3=2+3\log(1/c_0)/\log4$. \end{proof} \section{Resonant vertices}\label{sec:deloc-candidates} We keep the standing assumptions of \Cref{sec:deloc-amplification}, and we assume in addition that $\varphi(1)>0$, where $\varphi=\varphi_\pi$ is the free energy of the real boundary law. We use the notation of \Cref{sec:res-blocks} for this real solution: the factor law $\nu_{\mathrm{fac}}$ and the norm moments $f_r(u)$. We fix the remaining constants in the following order. \begin{enumerate}[label=\textup{(\alph*)}] \item The numerical constants $a_0$, $N_0$, $J$, $p_*$, $q_0$, $q_*$, and $\alpha=q_*/2$ of \Cref{sec:deloc-amplification}, and the constants $d_\pm$, $h_0$, $c_0$, $\eta_{\mathrm{ch}}$, $\eta_{\mathrm{reg}}$, $c_1$, $c_2$, $c_3$, and $C_3$ of \Cref{lem:deloc-channel,lem:deloc-regularize,cor:deloc-amplify}, which depend on $p$ and $E$. \item The cutoff $R_{\mathrm{aux}}$ of \Cref{lem:res-quantile}(ii) for the level $\alpha$ and $r=1/2$, and $b_{\mathrm{aux}}=E/(8R_{\mathrm{aux}})$. \item The exponent $s\in(1/2,1)$ and the numbers $\gamma=-\varphi'(s)>0$ and $F=\varphi(s)-(1-s)\gamma>0$ of \Cref{lem:press-gap}; then $m$, $R$, $\widehat\varphi$, $\widehat\gamma$, $\widehat F$, and $\iota$ of \Cref{lem:res-block-selection}; and \begin{equation}\label{eq:deloc-delta} \delta=\min\Big\{\frac F4,\frac{(2s-1)\gamma}4\Big\}. \end{equation} \item The length $c=10$ of the single-child segment and the constants $\rho_1$, $\eps_E$, $b_2$, and $b_3$ of \eqref{eq:res-reset-constants}. \item A depth $n\in m\N$, large depending on all previous constants, together with \begin{equation}\label{eq:deloc-n-constants} N=n+c,\qquad L=\frac nm,\qquad b_n=n^{3/4},\qquad T_n=e^{(\widehat\gamma+\delta)n}. \end{equation} \item An imaginary part $\eta_n>0$, small depending on $n$ and all previous constants. \end{enumerate} The constants in (a)--(d) depend only on $p$ and $E$, through the real boundary law $\pi$; they do not depend on $n$ or on $\eta$, and those listed as numerical in (a) do not depend on $E$ either. The quantile $\zeta_\alpha(\eta)$ is a function of $\eta$ and is not one of the constants. By \Cref{lem:res-block-selection}(ii) and \eqref{eq:deloc-delta}, the numbers \begin{equation}\label{eq:deloc-kappa} \kappa_0=(2s-1)\widehat\gamma-2\big(\varphi(s)-\widehat\varphi(s)\big)-\delta,\qquad\widehat F-\delta \end{equation} satisfy $\kappa_0\ge(2s-1)\gamma/4>0$ and $\widehat F-\delta\ge F/4>0$. We now define the vertices that we count. For a vertex $x$ of generation $N$, we write $o=x_0,x_1,\dots,x_N=x$ for its ancestors, and for $z=E+\iu\eta$ with $\eta>0$, we set \begin{equation}\label{eq:deloc-path-rows} t_k=\frac{Y_{x_k}}p,\qquad d_{k-1}(z)=z+\frac1p\sum_{u}Y_u\Gamma_u(z),\qquad X_{k-1}(z)=-\Gamma_{x_{k-1}\to x_k}(z),\qquad1\le k\le N, \end{equation} where the sum runs over the children $u$ of $x_{k-1}$ other than $x_k$. Thus $t_k$ is the squared weight of the edge from $x_{k-1}$ to $x_k$, $d_k(z)$ is the side denominator at $x_k$ in the sense of \Cref{def:res-rows}(ii) for the path from $o$ to $x$ in the whole tree, and $X_k(z)$ is the rootward variable at $x_k$. The \emph{factors} of $x$ are $(d_k(z),t_{k+1})$, $0\le k\eps_E$; $b_2/20$, depending on $p$ and $E$, such that the following holds for every $n\in m\N$ and $\eta\in(0,\min\{1,E/8\}]$. If $x\in\mathcal V_N$ is resonant at $E+\iu\eta$, then \begin{equation}\label{eq:deloc-gain} \Im\Gamma_o(E+\iu\eta)\ge c_{\mathrm{gain}}e^{2\delta n-2b_n}\zeta_\alpha(\eta). \end{equation} Further, on the event that (R1)--(R5) hold, the pole parameter $z\Delta_x(z)$ of \eqref{eq:res-leaf} satisfies $E^2/2\le\Re(z\Delta_x(z))\le3E^2/2$. \end{lemma} \begin{proof} Let $z=E+\iu\eta$. By \eqref{eq:res-ratio-formula} for the path $o=x_0,\dots,x_N=x$ in the whole tree, whose side denominators are the numbers $d_k(z)$ of \eqref{eq:deloc-path-rows}, we have \[ \bigg|\frac{G_{ox}(z)}{G_{xx}(z)}\bigg|\ge\|A\|^{-1},\qquad A=\prod_{k=0}^{N-1}M(d_k(z),t_{k+1}). \] By (R1), the last $c$ factors are $M(z,t_{k+1})$, $n\le k\eps_E$, so that $|X_{n+1}(z)|<\eps_E^{-1}$. Since $t_{n+2}0$ and $n_1\in\N$, depending on $p$, $E$, and the constants (a)--(d), such that for every $n\in m\N$ with $n\ge n_1$ there exists $\eta_1(n)\in(0,\min\{1,E/8\}]$ with \[ \E^\infty N_n(\eta)\ge c_4\,e^{(\widehat F-\delta)n-sb_n}\qquad\text{for every }\eta\in(0,\eta_1(n)]. \] \end{lemma} \begin{proof} Let $n_1$ be such that $T_n\ge4(E+1)/E^2$ and $L(m\log C_R)^2/b_n^2=m(\log C_R)^2n^{-1/2}\le1/2$ for $n\ge n_1$, where $C_R$ is the constant of \eqref{eq:res-block-bound}. Fix $n\in m\N$ with $n\ge n_1$, let $\eta\in(0,\min\{1,E/8\}]$, and write $z=E+\iu\eta$ and $\mathbb Q=\mathbb Q_N^\infty$. The proof first integrates the leaf weight, then the squared moduli $Y_{\mathsf v_{n+1}},\dots,Y_{\mathsf v_N}$, and finally passes to the real axis in the regular growth condition. \proofstep{1}{Many-to-one formula and structure.} Every resonant vertex belongs to $\mathcal V_N^\infty$. By \eqref{eq:palm-backbone}, we have \[ \E^\infty N_n(\eta)=p^N\,\mathbb Q(\mathsf v\text{ is resonant at }E+\iu\eta). \] By \Cref{lem:palm-decorations}(iii) and (v), we have $\mathbb Q(\mathcal T)=Q_*$. Here $S_0,\dots,S_{N-1}$ denote the side sequences of \Cref{lem:palm-spine-structure}; the sequences $S_n,\dots,S_{N-1}$ are empty on $\mathcal T$. \proofstep{2}{The leaf weight.} Let $\mathcal A=\{\ell(\mathsf v)\}$ if the endpoint $\mathsf v$ has a leaf child, and $\mathcal A=\emptyset$ otherwise, where $\ell(\mathsf v)$ is the selected leaf child of \Cref{sec:palm-decorations}; on $\mathcal T$, we have $\mathcal A=\{\ell_{\mathsf v}\}$. By \Cref{lem:palm-decorations}(iv) with $X=\{\mathsf v\}$, applied under $\mathbb Q$, the squared modulus $Y_{\ell(\mathsf v)}$ is fresh given the $\sigma$-field $\mathscr H_{\mathcal A}$ of \Cref{lem:palm-fresh-edges}(iii), which is generated by $\mathcal A$, the endpoint $\mathsf v$, the configuration $\omega^{\mathcal A\leftarrow\mathbf 1}$, and the phase of $w_{\ell(\mathsf v)}$. We claim that the indicator $F$ of the event that (R1)--(R6) hold for $\mathsf v$ is $\mathscr H_{\mathcal A}$-measurable, and that so is $\Delta_{\mathsf v}(z)$ on this event. Indeed, by the discussion after \Cref{def:deloc-candidate}, $F$ is a function of the configuration which does not change when the mark of the edge from $\mathsf v$ to $\ell_{\mathsf v}$ is replaced by another value. The same holds for $\Delta_{\mathsf v}(z)$, since it is computed in the tree from which $\ell_{\mathsf v}$ is deleted. Hence $F(\omega)=F(\omega^{\mathcal A\leftarrow\mathbf 1})$, and similarly for $\Delta_{\mathsf v}(z)$, which proves the claim. Let $t=Y_{\ell(\mathsf v)}/p$ and $\mathfrak p=z\Delta_{\mathsf v}(z)$. On $\mathcal T$, \eqref{eq:res-leaf} gives $G_{\mathsf v\mathsf v}(z)=z/(t-\mathfrak p)$, so that (R7) is the event $|z|\ge T_n|t-\mathfrak p|$. On the event that (R1)--(R6) hold, we have $E^2/2\le\Re\mathfrak p\le3E^2/2$ by \Cref{lem:deloc-gain}, and $|\Im\mathfrak p|\le|z|/(2T_n)$ by (R6). Further, $\mathfrak p$ is not real. Indeed, $\Im\Delta_{\mathsf v}(z)>0$, since $\Delta_{\mathsf v}(z)=-1/G^{\{\ell_{\mathsf v}\}}_{\mathsf v\mathsf v}(z)$ and $\Im G^{\{\ell_{\mathsf v}\}}_{\mathsf v\mathsf v}(z)>0$, and $\Re\Delta_{\mathsf v}(z)\ge5E/8$ by \Cref{lem:res-terminal}; thus $\Im\mathfrak p=E\Im\Delta_{\mathsf v}(z)+\eta\Re\Delta_{\mathsf v}(z)>0$. Let $\mathsf K$ be the set of $\mathfrak p\in\C\setminus\R$ with $E^2/2\le\Re\mathfrak p\le3E^2/2$ and $|\Im\mathfrak p|\le|z|/(2T_n)$. We apply the substitution identity \eqref{eq:palm-freeze} (consequence (c) after \Cref{def:palm-fresh}) with $m=1$, the revealed variable $X=\mathfrak p$, and $\Phi(\mathfrak p,y)=\one_{\mathsf K}(\mathfrak p)\one\{|z|\ge T_n|y/p-\mathfrak p|\}$. Since $y/p$ has the density $pe^{-pt}$ when $y$ has law $\Exp(1)$, \Cref{lem:res-tuning}(ii) gives, almost surely on the event that (R1)--(R6) hold, \[ \mathbb Q\big(\text{(R7)}\bigm|\mathscr H_{\mathcal A}\big)\ge\int\Phi(\mathfrak p,y)\,e^{-y}\,\dd y\ge\frac{c_{\mathrm{tune}}}{T_n}; \] here we used $T_n\ge4(E+1)/E^2$, which holds since $n\ge n_1$. Multiplying by $F$ and taking expectations, we conclude that \begin{equation}\label{eq:deloc-first-1} \mathbb Q(\mathsf v\text{ is resonant})\ge\frac{c_{\mathrm{tune}}}{T_n}\,\mathbb Q\big(\text{(R1)--(R6)}\big)\ge\frac{c_{\mathrm{tune}}}{T_n}\Big(\mathbb Q\big(\text{(R1)--(R5)}\big)-\mathbb Q\big(\mathcal T\setminus\text{(R6)}\big)\Big). \end{equation} \proofstep{3}{The single-child segment and the seed.} We first integrate over the squared moduli $Y_{\mathsf v_{n+1}},\dots,Y_{\mathsf v_N}$, so that $t_{n+i}=Y_{\mathsf v_{n+i}}/p$ for $1\le i\le c$. Let $\mathscr G_4$ be the $\sigma$-field of \Cref{lem:rde-spine}(ii) for the set $\{\mathsf v_{n+1},\dots,\mathsf v_N\}$, which is a function of the endpoint. It is generated by the offspring numbers, the auxiliary marks, the endpoint $\mathsf v$, the phases of all marks, and the marks $w_u$ with $u\notin\{\mathsf v_{n+1},\dots,\mathsf v_N\}$, and the $c$ squared moduli $Y_{\mathsf v_{n+1}},\dots,Y_{\mathsf v_N}$ are fresh given $\mathscr G_4$. By the discussion after \Cref{def:deloc-candidate}, the event $\mathcal T\cap\text{(R3)}\cap\text{(R5)}$ and the variable $X_n(z)$ do not depend on these squared moduli, and they are therefore $\mathscr G_4$-measurable. Condition (R4) is the event $\Phi(X_n(z),Y_{\mathsf v_{n+1}},\dots,Y_{\mathsf v_N})=1$, where \[ \begin{aligned} \Phi(X,y_1,\dots,y_c)={}&\one\Big\{\frac1p<\frac{y_1}p<\frac2p,\ \Big|z-\frac{y_1}pX\Big|>\eps_E\Big\}\\ &\times\one\Big\{\frac{b_2}2<\frac{y_2}p0$. By dominated convergence, $\mathbb Q(\mathcal T\setminus\text{(R6) at }E+\iu\eta')\to0$ as $\eta'\downarrow0$. \proofstep{6}{Conclusion.} By Steps 4 and 5, there exists $\eta_1(n)\in(0,\min\{1,E/8\}]$ such that for every $\eta\in(0,\eta_1(n)]$, we have $\mathbb Q(\text{(R3)})\ge e^{-\iota n-sb_n}/4$ and $\mathbb Q(\mathcal T\setminus\text{(R6)})\le c_{\mathrm{seg}}Q_*\alpha\rho_{\mathrm{aux}}e^{-\iota n-sb_n}/16$. Inserting these bounds into \eqref{eq:deloc-first-2} and \eqref{eq:deloc-first-1}, we obtain \[ \E^\infty N_n(\eta)\ge p^N\frac{c_{\mathrm{tune}}}{T_n}\cdot\frac{c_{\mathrm{seg}}Q_*\alpha\rho_{\mathrm{aux}}}{8}\,e^{-\iota n-sb_n}. \] Since $\log p-\iota-\widehat\gamma-\delta=\widehat\varphi(s)+s\widehat\gamma-\widehat\gamma-\delta=\widehat F-\delta$, we have $p^NT_n^{-1}e^{-\iota n}=p^ce^{(\widehat F-\delta)n}$, and the claim follows with $c_4=p^cc_{\mathrm{tune}}c_{\mathrm{seg}}Q_*\alpha\rho_{\mathrm{aux}}/8$. \end{proof} \section{The second moment}\label{sec:deloc-second} For pairs of resonant vertices, we use the backbone two-spine laws $\mathbb Q_{N,j}^\infty$ of \Cref{def:palm-spine-laws}, with endpoints $\mathsf u$ and $\mathsf v$ and fork $\mathsf v_j=\mathsf u_j$ at generation $j$. We apply the notation \eqref{eq:deloc-path-rows} to $x=\mathsf u$ and to $x=\mathsf v$, and we indicate the endpoint by a superscript: $t^{\mathsf u}_k$, $d^{\mathsf u}_k(z)$, $Z^{\mathsf u}_l(z)$, and so on. For $kj$, which we call the factors of the arm of $\mathsf u$; similarly for $\mathsf v$. \begin{definition}\label{def:deloc-envelope} Let $n\in m\N$ and $0\le j\le n-1$. For $\eta\in(0,1]$, set $z=E+\iu\eta$; for $\eta=0$, all quantities below are evaluated at the boundary values $E+\iu0$. The \emph{enlarged pair event} $\mathcal E_j(\eta)$ is the event that the following conditions hold under $\mathbb Q_{N,j}^\infty$. \begin{enumerate}[label=\textup{(E\arabic*)}] \item Conditions (R1) and (R2) hold for $\mathsf u$ and for $\mathsf v$. \item If $\lambda_j\ge1$, then the common blocks $1,\dots,\lambda_j$ are retained, and $Z_1(z)+\dots+Z_{\lambda_j}(z)\le\lambda_jm\widehat\gamma+b_n$. \item For $w\in\{\mathsf u,\mathsf v\}$, if $\lambda_j+2\le L$, then the blocks $\lambda_j+2,\dots,L$ of $w$ are retained, and $\sum_{l=\lambda_j+2}^LZ^w_l(z)\le(L-\lambda_j-1)m\widehat\gamma+2b_n$. \end{enumerate} The \emph{interaction event} $\mathcal E'_j(\eta)$ is the event that (E1) and (E2) hold. Further, we set \begin{equation}\label{eq:deloc-D} \mathfrak D_\eta=|z|\,\big(t^{\mathsf u}_Nt^{\mathsf v}_N\big)^{1/2}\,\big|G^{V\setminus\{\mathsf u,\mathsf v\}}_{\mathsf u_{N-1}\mathsf v_{N-1}}(z)\big|, \end{equation} where $G^{V\setminus\{\mathsf u,\mathsf v\}}$ denotes the resolvent of the tree from which $\mathsf u$ and $\mathsf v$ are deleted. \end{definition} We first estimate the enlarged pair event and the interaction event at the real axis. \begin{lemma}\label{lem:deloc-pair-real} There exists $C_5<\infty$, depending on $p$, $E$, and the constants (a)--(d), such that for every $n\in m\N$ and every $0\le j\le n-1$, with $\mathbb Q=\mathbb Q_{N,j}^\infty$, \begin{align} \mathbb Q\big(\mathcal E_j(0)\big)&\le Q_*^2\,e^{-\iota(2n-j)+2m\iota+5sb_n},\label{eq:deloc-pair-product}\\ \E_{\mathbb Q}\big[\one_{\mathcal E'_j(0)}\min\{1,\mathfrak D_0\}\big]&\le C_5\,Q_*^2\,e^{-\iota(j-m)+sb_n+2(n-j)(\varphi(s)-\log p)}.\label{eq:deloc-pair-interaction} \end{align} \end{lemma} \begin{proof} \proofstep{1}{Independence.} By \Cref{lem:palm-spine-structure}(iv), under $\mathbb Q$ the side sequences at the spine vertices other than the endpoints, the spine edge marks, and the two endpoint configurations are independent. The side sequences and edge marks have their unconditioned laws, and the endpoint configurations have law $\P^\infty$. Since $jj$ and $k0$, we compute its side denominators and weights in the sense of \Cref{def:res-rows}(ii). By \Cref{lem:tree-branch}(iii), the cavity variables of the branches hanging off the path are the same in $C$ as in the whole tree, since these branches contain neither $\mathsf u$ nor $\mathsf v$. The side denominator at $\mathsf u_{N-1}$ is $z$, because its only child $\mathsf u$ is deleted, and similarly at $\mathsf v_{N-1}$. At $\mathsf u_k$ and $\mathsf v_k$ with $j1$ and $\Gamma\le1$ on $[1,2]$. Further, $A_1=HV$, where $H$ is the product of the factors of $A_1$ at the vertices $\mathsf u_k$ of the single-child segment, $\max\{n,j+4\}\le k\le N-4$, whose denominators equal $E$, and $V$ is the product of the $(n-j-4)_+$ factors at $\mathsf u_k$, $j+4\le k\le n-1$, which are independent with law $\nu_{\mathrm{fac}}$. By \Cref{lem:res-arm} with $u=s$, with at most $c$ constant factors, and with $C_{\mathrm{arm}}\ge1$, and since $\varphi(s)<\log p$ by \Cref{prop:press-properties}(v), \[ \E_{\mathbb Q}\big[\|A_1\|^{-s}\bigm|\text{(E1)}\big]\le C_{\mathrm{arm}}^ce^{(n-j-4)_+(\varphi(s)-\log p)}\le C_{\mathrm{arm}}^ce^{-4(\varphi(s)-\log p)}e^{(n-j)(\varphi(s)-\log p)}, \] and the same bound holds for $A_2=V'H'$, where $V'$ is the product of the factors at $\mathsf v_k$, $j+4\le k\le n-1$, and $H'$ is the product of the factors at the vertices $\mathsf v_k$ of the single-child segment, $\max\{n,j+4\}\le k\le N-5$. Finally, $\mathbb Q(\mathcal E'_j(0))\le Q_*^2e^{-\iota m\lambda_j+sb_n}$ by Steps 1 and 2, and $m\lambda_j\ge j-m$. Combining these bounds, we obtain \eqref{eq:deloc-pair-interaction} with $C_5=\max\{1,E\}C_*^4C_{\mathrm{arm}}^{2c}e^{-8(\varphi(s)-\log p)}$. \end{proof} We now bound the second factorial moment of the number of resonant vertices. \begin{lemma}\label{lem:deloc-second} There exist $C_6<\infty$ and $n_2\ge n_1$, depending on $p$, $E$, and the constants (a)--(d), such that for every $n\in m\N$ with $n\ge n_2$ there exists $\eta_2(n)\in(0,\min\{1,E/8\}]$ with \[ \E^\infty\big[N_n(\eta)(N_n(\eta)-1)\big]\le C_6\,e^{2(\widehat F-\delta)n+5sb_n}\qquad\text{for every }\eta\in(0,\eta_2(n)]. \] \end{lemma} \begin{proof} Let $n_2=n_1$, fix $n\in m\N$ with $n\ge n_2$, let $\eta\in(0,\min\{1,E/8\}]$, and write $z=E+\iu\eta$ and $\mathbb Q_j=\mathbb Q^\infty_{N,j}$. \proofstep{1}{Many-to-two formula.} Resonant vertices belong to $\mathcal V^\infty_N$. By \eqref{eq:palm-two-backbone}, we have \[ \E^\infty\big[N_n(\eta)(N_n(\eta)-1)\big]=\sum_{j=0}^{N-1}\varrho_p\,p^{2N-j}\,\mathbb Q_j\big(\mathsf u\text{ and }\mathsf v\text{ are resonant at }E+\iu\eta\big). \] If $j\ge n$, then the fork $\mathsf v_j$ is a vertex of the single-child segment of $\mathsf v$ with the two children $\mathsf u_{j+1}\neq\mathsf v_{j+1}$, so (R1) fails for $\mathsf v$, and the corresponding term vanishes. Let $0\le j\le n-1$. \proofstep{2}{Inclusion in the enlarged pair event.} Suppose that $\mathsf u$ and $\mathsf v$ are resonant at $E+\iu\eta$. Then (E1) holds. The common blocks $1,\dots,\lambda_j$ are blocks of $\mathsf u$, so (R3) for $\mathsf u$ with $i=\lambda_j$ implies (E2). For $w\in\{\mathsf u,\mathsf v\}$, (R3) for $w$ with $i=L$ and $i=\lambda_j+1$ gives \[ \sum_{l=\lambda_j+2}^LZ^w_l(z)<\big(Lm\widehat\gamma+b_n\big)-\big((\lambda_j+1)m\widehat\gamma-b_n\big), \] and strictly retained blocks are retained; this is (E3). We conclude that the event in Step 1 is contained in $\mathcal E_j(\eta)\cap\{|G_{\mathsf u\mathsf u}(z)|\ge T_n,\ |G_{\mathsf v\mathsf v}(z)|\ge T_n\}$. \proofstep{3}{The two leaf weights.} Let $\mathcal A$ be the set of selected leaf children $\ell(w)$ of the endpoints $w\in\{\mathsf u,\mathsf v\}$ which have a leaf child; on (E1), we have $\mathcal A=\{\ell_{\mathsf u},\ell_{\mathsf v}\}$. By \Cref{lem:palm-decorations}(iv) with $X=\{\mathsf u,\mathsf v\}$, applied under $\mathbb Q_j$, the squared moduli of the marks of the edges from $\mathsf u$ to $\ell(\mathsf u)$ and from $\mathsf v$ to $\ell(\mathsf v)$ are fresh given the $\sigma$-field $\mathscr H_{\mathcal A}$ of \Cref{lem:palm-fresh-edges}(iii). We write $t_{\mathsf u}=|h_{\mathsf u\ell_{\mathsf u}}|^2$ and $t_{\mathsf v}=|h_{\mathsf v\ell_{\mathsf v}}|^2$ for the corresponding squared weights. The event $\mathcal E_j(\eta)$ and the number $\mathfrak D_\eta$ do not depend on the marks of these two edges: the factors in the retained common and arm blocks, which exclude the block containing the branching point, involve only the side subtrees of $\mathsf u_k$ and $\mathsf v_k$ and the spine edges, and $\mathfrak D_\eta$ is computed in the tree from which $\mathsf u$ and $\mathsf v$ are deleted. As in Step 2 of the proof of \Cref{lem:deloc-first}, they are therefore $\mathscr H_{\mathcal A}$-measurable. On (E1), the vertices $\mathsf u$ and $\mathsf v$ have the same generation and are therefore not adjacent, and we apply \Cref{lem:res-schur-two} to the whole tree with $x=\mathsf u$ and $y=\mathsf v$. The neighbors of $\mathsf u$ are $\mathsf u_{N-1}$, $\ell_{\mathsf u}$, and $y_{\mathsf u}$. In the tree with $\mathsf u$ and $\mathsf v$ deleted, the leaf $\ell_{\mathsf u}$ is an isolated vertex, with diagonal resolvent entry $-1/z$, the child $y_{\mathsf u}$ is the root of the component $V(\mathcal T_{y_{\mathsf u}})$, with diagonal resolvent entry $\Gamma_{y_{\mathsf u}}(z)$, and $\mathsf u_{N-1}$ lies in the component that contains the root. We obtain \[ S_{\mathsf u\mathsf u}(z)=-z+\frac{t_{\mathsf u}}z-\sigma_{\mathsf u}(z),\qquad\sigma_{\mathsf u}(z)=t^{\mathsf u}_N\,G^{V\setminus\{\mathsf u,\mathsf v\}}_{\mathsf u_{N-1}\mathsf u_{N-1}}(z)+|h_{\mathsf uy_{\mathsf u}}|^2\,\Gamma_{y_{\mathsf u}}(z), \] and similarly for $\mathsf v$. Let $\mathsf B$ be the matrix with the diagonal entries $z^2+z\sigma_{\mathsf u}(z)$ and $z^2+z\sigma_{\mathsf v}(z)$ and the off-diagonal entries $\mathsf B_{\mathsf u\mathsf v}=-zS_{\mathsf u\mathsf v}(z)$ and $\mathsf B_{\mathsf v\mathsf u}=-zS_{\mathsf v\mathsf u}(z)$. Then $zS(z)=\operatorname{diag}(t_{\mathsf u},t_{\mathsf v})-\mathsf B$, and \eqref{eq:res-schur-two} shows that the $2\times2$ block of the resolvent at $\{\mathsf u,\mathsf v\}$ equals $\mathsf R(t_{\mathsf u},t_{\mathsf v})=z(\operatorname{diag}(t_{\mathsf u},t_{\mathsf v})-\mathsf B)^{-1}$. The matrix $\mathsf B$ does not depend on the marks of the leaf edges, and it is $\mathscr H_{\mathcal A}$-measurable. Further, $\operatorname{diag}(t_1,t_2)-\mathsf B$ is invertible for all $t_1,t_2>0$. Indeed, let $S'(z)$ be the matrix of \Cref{lem:res-schur-two} for the tree from which the two leaves are deleted. Its entries are those of $S(z)$ without the terms $t_{\mathsf u}/z$ and $t_{\mathsf v}/z$, so that $\operatorname{diag}(t_1,t_2)-\mathsf B=z\big(S'(z)+\operatorname{diag}(t_1,t_2)/z\big)$. By \Cref{lem:res-schur-two}, $\Im S'(z)\le-\eta I$, and $\Im(\operatorname{diag}(t_1,t_2)/z)=-\eta|z|^{-2}\operatorname{diag}(t_1,t_2)\le0$. Thus $\Im\langle f,(S'(z)+\operatorname{diag}(t_1,t_2)/z)f\rangle\le-\eta\|f\|^2<0$ for $f\neq0$, and the matrix is invertible. By \eqref{eq:res-schur-offdiag}, applied with $x'=\mathsf u_{N-1}$ and $y'=\mathsf v_{N-1}$, we have $|\mathsf B_{\mathsf u\mathsf v}\mathsf B_{\mathsf v\mathsf u}|^{1/2}=|z|\,|S_{\mathsf u\mathsf v}(z)S_{\mathsf v\mathsf u}(z)|^{1/2}=\mathfrak D_\eta$. We now apply \Cref{lem:res-two-terminal} conditionally on $\mathscr H_{\mathcal A}$, using the substitution identity \eqref{eq:palm-freeze} (consequence (c) after \Cref{def:palm-fresh}) with the revealed variable $X=\mathsf B\in\C^4$ and the squared moduli of the two leaf marks as fresh variables. With $|z|\le E+1$, we find that, almost surely on (E1), \[ \mathbb Q_j\big(|G_{\mathsf u\mathsf u}(z)|\ge T_n,\ |G_{\mathsf v\mathsf v}(z)|\ge T_n\bigm|\mathscr H_{\mathcal A}\big)\le\frac{C_{\mathrm{tt}}}{T_n^2}+\frac{C_{\mathrm{tt}}}{T_n}\min\{1,\mathfrak D_\eta\},\qquad C_{\mathrm{tt}}=8p^2(E+1)^2. \] Combining this with Step 2, and using $\mathcal E_j(\eta)\subset\mathcal E'_j(\eta)$, we obtain \begin{equation}\label{eq:deloc-second-1} \mathbb Q_j\big(\mathsf u\text{ and }\mathsf v\text{ are resonant}\big)\le\frac{C_{\mathrm{tt}}}{T_n^2}\,\mathbb Q_j\big(\mathcal E_j(\eta)\big)+\frac{C_{\mathrm{tt}}}{T_n}\,\E_{\mathbb Q_j}\big[\one_{\mathcal E'_j(\eta)}\min\{1,\mathfrak D_\eta\}\big]. \end{equation} \proofstep{4}{Passage to the real axis.} By \Cref{prop:rde-fixed-depth}(i), $\mathbb Q_j$-almost surely, the factors of $\mathsf u$ and $\mathsf v$ at $E+\iu\eta'$ converge to their values at $\eta'=0$ as $\eta'\downarrow0$, and $\mathfrak D_{\eta'}\to\mathfrak D_0$, since $G^{V\setminus\{\mathsf u,\mathsf v\}}_{\mathsf u_{N-1}\mathsf v_{N-1}}(E+\iu\eta')$ is a coordinate of the depth-$(N+1)$ data. Let $V_{\eta'}$ be the vector of the factors of $\mathsf u$ and $\mathsf v$ with index $k0. \] Since there are finitely many $j$, there exists $\eta_2(n)\in(0,\min\{1,E/8\}]$ such that the left side of \eqref{eq:deloc-second-1} is at most $2B_j$ for every $\eta\in(0,\eta_2(n)]$ and every $0\le j\le n-1$. \proofstep{5}{Summation.} Let $\eta\in(0,\eta_2(n)]$, and set $\mathsf M_n=p^NT_n^{-1}Q_*e^{-\iota n}=p^cQ_*e^{(\widehat F-\delta)n}$, as in the proof of \Cref{lem:deloc-first}. Then $\mathsf M_n^2=p^{2N}T_n^{-2}Q_*^2e^{-2\iota n}$, and for $0\le j\le n-1$ we have \[ \begin{aligned} p^{2N-j}\,\frac{Q_*^2}{T_n^2}\,e^{-\iota(2n-j)+2m\iota+5sb_n}&=\mathsf M_n^2\,e^{2m\iota+5sb_n}\,e^{-(\log p-\iota)j},\\ p^{2N-j}\,\frac{Q_*^2}{T_n}\,e^{-\iota(j-m)+sb_n+2(n-j)(\varphi(s)-\log p)}&=\mathsf M_n^2\,e^{m\iota+sb_n}\,e^{\Psi_n(j)}, \end{aligned} \] where \[ \Psi_n(j)=-j\log p+(\widehat\gamma+\delta)n+\iota(2n-j)+2(n-j)\big(\varphi(s)-\log p\big). \] Indeed, the first identity follows from $p^{2N-j}e^{\iota j}=p^{2N}e^{-(\log p-\iota)j}$, and the second from $p^{2N-j}=p^{2N}e^{-j\log p}$, $T_n^{-1}=T_n^{-2}e^{(\widehat\gamma+\delta)n}$, and $e^{-\iota(j-m)}=e^{-2\iota n}e^{\iota(2n-j)}e^{m\iota}$. By Steps 1 and 4, we obtain \[ \begin{aligned} \E^\infty\big[N_n(\eta)(N_n(\eta)-1)\big]&\le\sum_{j=0}^{n-1}2\varrho_pp^{2N-j}B_j\\ &=2\varrho_pC_{\mathrm{tt}}\,\mathsf M_n^2\sum_{j=0}^{n-1}\Big(e^{2m\iota+5sb_n}e^{-(\log p-\iota)j}+C_5e^{m\iota+sb_n}e^{\Psi_n(j)}\Big). \end{aligned} \] Using $\iota=\log p-\widehat\varphi(s)-s\widehat\gamma$, we find that $\Psi_n$ is affine in $j$, \[ \begin{aligned} \Psi_n(j)&=\big(\widehat\gamma+\delta+2\log p-2\widehat\varphi(s)-2s\widehat\gamma+2\varphi(s)-2\log p\big)n\\ &\qquad+\big(-\log p-\iota-2\varphi(s)+2\log p\big)j\\ &=-\kappa_0n+\big(\widehat\varphi(s)+s\widehat\gamma-2\varphi(s)\big)j, \end{aligned} \] with $\kappa_0$ as in \eqref{eq:deloc-kappa}. Its values at the endpoints are $\Psi_n(0)=-\kappa_0n$ and $\Psi_n(n)=((1-s)\widehat\gamma-\widehat\varphi(s)+\delta)n=-(\widehat F-\delta)n$. Since an affine function attains its maximum over an interval at an endpoint, we have $\Psi_n(j)\le-\kappa_1n$ for $0\le j\le n$, where $\kappa_1=\min\{\kappa_0,\widehat F-\delta\}>0$. Further, $\log p-\iota>0$ by \Cref{lem:res-block-selection}(iii). Thus \[ \sum_{j=0}^{n-1}e^{-(\log p-\iota)j}\le\frac1{1-e^{-(\log p-\iota)}},\qquad\sum_{j=0}^{n-1}e^{\Psi_n(j)}\le ne^{-\kappa_1n}\le\frac1{e\kappa_1}. \] Inserting these bounds, and using $\mathsf M_n^2=p^{2c}Q_*^2e^{2(\widehat F-\delta)n}$, we obtain the claim. \end{proof} \section{Proof of the delocalization criterion}\label{sec:deloc-proof} Let $p\ge p_*$, let $E\in\mathcal E_p$ with $E>0$, and suppose that $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Suppose, for a contradiction, that $\varphi_{\pi_E}(1)>0$. We fix the constants (a)--(d) of \Cref{sec:deloc-candidates}, and the numbers $c_4$, $n_1$, $C_6$, and $n_2$ of \Cref{lem:deloc-first,lem:deloc-second}. For $n\in m\N$ with $n\ge n_2$, we set \[ \eta_n=\min\{\eta_{\mathrm{reg}},\eta_1(n),\eta_2(n)\}. \] Then $\eta_n\in(0,\min\{1,E/8\}]$, and \Cref{lem:deloc-first,lem:deloc-second} apply at $\eta=\eta_n$. The proof has three steps: we bound the probability that a resonant vertex exists from below, we amplify this probability, and we compare the result with the definition of the quantile. \proofstep{1}{A resonant vertex exists with subexponential probability.} Write $N_n=N_n(\eta_n)$, $\mu_n=c_4e^{(\widehat F-\delta)n-sb_n}$, and $\chi_n=C_6e^{2(\widehat F-\delta)n+5sb_n}$. Since $N_n\le|\mathcal V_N|$ and $\E^\infty|\mathcal V_N|\le\varrho_p^{-1}p^N$ by \eqref{eq:palm-survival-domination}, the expectation $\E^\infty N_n$ is finite. By the Cauchy--Schwarz inequality, we have \[ (\E^\infty N_n)^2=\big(\E^\infty[N_n\one\{N_n\ge1\}]\big)^2\le\E^\infty[N_n^2]\,\P^\infty(N_n\ge1); \] this is the Paley--Zygmund inequality \cite{PaleyZygmund1932note}. Since $\E^\infty[N_n^2]=\E^\infty N_n+\E^\infty[N_n(N_n-1)]\le\E^\infty N_n+\chi_n$ by \Cref{lem:deloc-second}, and since $t\mapsto t^2/(t+\chi_n)$ is increasing on $(0,\infty)$, \Cref{lem:deloc-first} gives \[ \P^\infty(N_n\ge1)\ge\frac{(\E^\infty N_n)^2}{\E^\infty N_n+\chi_n}\ge\frac{\mu_n^2}{\mu_n+\chi_n}. \] Since $\widehat F-\delta>0$ and $b_n=o(n)$, we have $\mu_n\ge1$ for all large $n$. Then $\mu_n+\chi_n\le\mu_n^2+\chi_n\le2\max\{\mu_n^2,\chi_n\}$, and \[ \frac{\mu_n^2}{\mu_n+\chi_n}\ge\min\Big\{\frac12,\frac{\mu_n^2}{2\chi_n}\Big\}=\min\Big\{\frac12,\frac{c_4^2}{2C_6}e^{-7sb_n}\Big\}\ge c_7e^{-7sb_n},\qquad c_7=\min\Big\{\frac12,\frac{c_4^2}{2C_6}\Big\}. \] \proofstep{2}{Amplification.} On the event $\{N_n\ge1\}$, there exists a vertex which is resonant at $E+\iu\eta_n$, and \Cref{lem:deloc-gain} gives $\Im\Gamma_o(E+\iu\eta_n)\ge y_n$, where $y_n=c_{\mathrm{gain}}e^{2\delta n-2b_n}\zeta_\alpha(\eta_n)$. By Step 1, for all large $n$, \[ \P^\infty\big(\Im\Gamma_o(E+\iu\eta_n)\ge y_n\big)\ge e^{-a_n},\qquad a_n=7sb_n+\log(1/c_7)\ge0. \] Since $\eta_n\le\eta_{\mathrm{reg}}$, \Cref{cor:deloc-amplify} implies that \[ \P^\infty\big(\Im\Gamma_o(E+\iu\eta_n)\ge c_3e^{-C_3(1+a_n)}y_n\big)\ge q_*>\alpha. \] \proofstep{3}{The quantile.} By \Cref{lem:res-quantile}(i), we have $\P^\infty(\Im\Gamma_o(E+\iu\eta_n)\ge t)<\alpha$ for every $t>\zeta_\alpha(\eta_n)$. By Step 2, this implies that $c_3e^{-C_3(1+a_n)}y_n\le\zeta_\alpha(\eta_n)$. Since $0<\zeta_\alpha(\eta_n)<\infty$, dividing by $\zeta_\alpha(\eta_n)$, we obtain \[ c_3c_{\mathrm{gain}}\exp\Big(2\delta n-2b_n-C_3\big(1+7sb_n+\log(1/c_7)\big)\Big)\le1 \] for every large $n\in m\N$. The constants $c_3$, $C_3$, $c_{\mathrm{gain}}$, $c_7$, $s$, and $\delta>0$ do not depend on $n$, and $b_n=n^{3/4}$. Therefore the left side tends to infinity as $n\to\infty$, which is a contradiction. We conclude that $\varphi_{\pi_E}(1)\le0$. The threshold $p_*$ was fixed in \eqref{eq:deloc-pstar} and does not depend on $E$. This completes the proof of \Cref{thm:deloc-criterion}. \part{The free energy at large degree}\label{part:largep} \chapter{The principal solution}\label{sec:stab} In this section, we construct the principal solution $\nu_{p,E}$ of the cavity equation at large degree and establish the properties of it that are used in the rest of the paper. We work in the normalization of \Cref{def:rde-normalizations}. For $E>2$ we write $\theta=\theta(E)=g(E)^2$, and we recall that $g(E)E=1+\theta$. If $D$ satisfies the real cavity equation $D\eqdist E-p^{-1}\sum_{j\le N}Y_j/D_j$, then multiplying by $g(E)$ shows that $B=g(E)D$ satisfies \begin{equation}\label{eq:stab-B-equation} B\eqdist 1+\theta-\frac{\theta}{p}\sum_{j=1}^{N}\frac{Y_j}{B_j}, \end{equation} where $N\sim\Pois(p)$, $Y_j\sim\Exp(1)$, and $B_j$ are copies of $B$, all independent. Conversely, every law of $B$ on $\R\setminus\{0\}$ satisfying \eqref{eq:stab-B-equation} with $\theta=\theta(E)$ is the law of $g(E)D$ for a real solution $D$ at $E$ in the sense of \Cref{def:rde-real}. Since $g(E)=2/(E+\sqrt{E^2-4})$, we have $E^{-2}\le\theta\le4E^{-2}$ for $E\ge2$; for a scaled energy $E=\xi\sqrt p$ with $\xi$ in a compact subset of $(0,\infty)$, the parameter $\theta$ is of order $p^{-1}$. The main results of this section are \Cref{thm:stab-window}, which constructs $\nu_{p,E}$ on compact scaled windows as the unique fixed point of \eqref{eq:stab-B-equation} in an explicit class of laws; \Cref{prop:stab-entry} and \Cref{cor:stab-halfline}, which treat large energies and the whole half-line $E\ge\xi_0\sqrt p$; and \Cref{prop:stab-selection}, which shows that on compact scaled windows every real solution, and in particular the real boundary law of \Cref{prop:rde-boundary}, coincides with $\nu_{p,E}$ once $p$ is large. We now describe the mechanism. When $p$ is large and $\theta\le C_0/p$, the sum in \eqref{eq:stab-B-equation} has about $p$ terms, and a term with $|B_j|$ of order one has size of order $p^{-2}$. Three effects then control the law of $B$. First, the children with $|B_j|$ bounded below contribute a sum that concentrates around $\theta$ on a scale much smaller than $r_p=p^{-1/4}$, so $B$ is close to $1$ with overwhelming probability. Second, for these children to bring $B$ close to zero, their sum must be of order one. Since the squared moduli $Y_j$ are exponential, an exponential tilt shows that this costs a factor $e^{-cp^2}$, at the level of densities. Third, a child with $B_j$ close to zero produces a jump $\theta Y_j/(pB_j)$ that may be of order one. The jumps produced in this way have an intensity bounded by $\theta H/y^2$, where $H$ bounds the density of $B_j$ near zero, and a one-large-jump argument shows that their contribution to the density of $B$ near zero is of order $\theta H$. Since $\theta\le C_0/p$, the density near zero contracts. These three effects define a class of laws that is invariant under the map defined by \eqref{eq:stab-B-equation} (\Cref{sec:stab-class}). To prove uniqueness, we construct a metric for which this map is a contraction. Differences of laws are measured in a Kantorovich norm near $1$, in the supremum norm of densities near $0$, and in a bounded--Lipschitz norm elsewhere, with exponentially large weights on the last two components (\Cref{sec:stab-mixed,sec:stab-columns}). The contraction is proved in \Cref{sec:stab-fixed}, the small-parameter regime, which contains the large-energy regime, is treated in \Cref{sec:stab-entry}, and real solutions are identified in \Cref{sec:stab-real}. Throughout this section, $Y$, $Y_j$, and $T$ denote standard exponential random variables, $N\sim\Pois(p)$, and these variables are independent of each other and of all other variables, unless stated otherwise. For a finite signed measure $\alpha$ on $\R$ and a bounded Borel function $\psi$, we write $\langle\alpha,\psi\rangle=\int\psi\dd\alpha$, $\|\alpha\|_{\TV}=|\alpha|(\R)$, and $\alpha^\vee$ for the image of $\alpha$ under $y\mapsto-y$. A constant is called numerical if it is absolute. \section{The law map}\label{sec:stab-lawmap} \begin{definition}\label{def:stab-lawmap} Let $p>1$ and $\theta>0$. For a probability measure $\nu$ on $\R$ with $\nu(\{0\})=0$, let $\mathcal R_{p,\theta}(\nu)$ be the law of \[ 1+\theta-\frac{\theta}{p}\sum_{j=1}^{N}\frac{Y_j}{B_j}, \] where $N\sim\Pois(p)$, $Y_j\sim\Exp(1)$, and $B_j\sim\nu$, all independent. We call $\mathcal R_{p,\theta}$ the law map. A fixed point of $\mathcal R_{p,\theta}$ is a probability measure $\nu$ on $\R$ with $\nu(\{0\})=0$ and $\mathcal R_{p,\theta}(\nu)=\nu$. \end{definition} By the discussion at the beginning of this section, for $E>2$ a probability measure $\pi$ on $\R\setminus\{0\}$ is a real solution at $E$ if and only if the law of $g(E)D$, $D\sim\pi$, is a fixed point of $\mathcal R_{p,\theta(E)}$. For $b\neq0$ and $\theta>0$, the variable $\theta Y/(pb)$ is exponential with rate $p|b|/\theta$ and has the sign of $b$. Its density is \begin{equation}\label{eq:stab-jump} j_\theta(b;y)=\frac{p|b|}{\theta}\,e^{-p|b||y|/\theta}\,\one_{\{by>0\}},\qquad y\in\R . \end{equation} For a finite signed measure $\sigma$ on $\R$ with $|\sigma|(\{0\})=0$, define the finite signed measure \[ \mathsf K_\theta\sigma(\mathrm dy)=\Big(\int j_\theta(b;y)\,\sigma(\mathrm db)\Big)\dd y . \] Since $\int j_\theta(b;y)\dd y=1$, we have $\|\mathsf K_\theta\sigma\|_{\TV}\le\|\sigma\|_{\TV}$ and $\mathsf K_\theta\sigma(\R)=\sigma(\R)$. For a finite signed measure $\alpha$, set \[ \mathsf P_p(\alpha)=e^{-p}\sum_{k=0}^{\infty}\frac{p^k}{k!}\,\alpha^{*k},\qquad\alpha^{*0}=\delta_0 . \] The series converges in total variation, since $\|\alpha^{*k}\|_{\TV}\le\|\alpha\|_{\TV}^k$. Finally, let $s_\theta(y)=1+\theta-y$, and write $(s_\theta)_\sharp\alpha$ for the image of $\alpha$ under $s_\theta$. Since $s_\theta(y_1+y_2)=s_\theta(y_1)-y_2$, we have \begin{equation}\label{eq:stab-image-convolution} (s_\theta)_\sharp(\alpha*\beta)=\big((s_\theta)_\sharp\alpha\big)*\beta^\vee . \end{equation} \begin{lemma}\label{lem:stab-lawmap} Let $p>1$, $\theta>0$, and let $\nu$ be a probability measure on $\R$ with $\nu(\{0\})=0$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\mathcal R_{p,\theta}(\nu)=(s_\theta)_\sharp\,\mathsf P_p(\mathsf K_\theta\nu)$. \item Let $A_1,\dots,A_k$ be disjoint Borel sets with union $\R\setminus\{0\}$, let $B'=1+\theta-(\theta/p)\sum_{j\le N}Y_j/B_j$ be as in \Cref{def:stab-lawmap}, and set \[ S_{A_i}=\frac{\theta}{p}\sum_{j\le N:\,B_j\in A_i}\frac{Y_j}{B_j}. \] Then $B'=1+\theta-\sum_{i}S_{A_i}$, and the variables $S_{A_1},\dots,S_{A_k}$ are independent. For each $i$, the points $\theta Y_j/(pB_j)$ with $B_j\in A_i$ form a Poisson process on $\R\setminus\{0\}$ with intensity measure $\ell_{A_i}(y)\dd y$, where \begin{equation}\label{eq:stab-levy-density} \ell_{A}(y)=p\int_{A}j_\theta(b;y)\,\nu(\mathrm db),\qquad\int\ell_A(y)\dd y=p\,\nu(A). \end{equation} In particular, the law of $S_{A}$ is $e^{-p\nu(A)}\sum_{m\ge0}\ell_A^{*m}/m!$, where $\ell_A^{*0}=\delta_0$. \item The measure $\mathcal R_{p,\theta}(\nu)$ is the sum of $e^{-p}\delta_{1+\theta}$ and an absolutely continuous measure. \end{enumerate} \end{lemma} \begin{proof} The pairs $(B_j,Y_j)$, $j\le N$, form a mixed binomial process with Poisson mixing distribution, which is a Poisson process on $(\R\setminus\{0\})\times(0,\infty)$ with intensity measure $p\,\nu\otimes\Exp(1)$. The restrictions of a Poisson process to disjoint sets are independent Poisson processes, and the image of a Poisson process under the map $(b,y)\mapsto\theta y/(pb)$ is a Poisson process whose intensity is the image of the original intensity \cite[Chapters~3 and~5]{LastPenrose2018lectures}. The image of $p\,\nu|_{A}\otimes\Exp(1)$ under this map is $\ell_A(y)\dd y$ by \eqref{eq:stab-jump}. This proves (ii); the formula for the law of $S_A$ follows by conditioning on the number of points. For (i), conditionally on $N=n$, the jumps $\theta Y_j/(pB_j)$, $j\le n$, are independent with law $\mathsf K_\theta\nu$, and the law of their sum is $(\mathsf K_\theta\nu)^{*n}$. Averaging over $N$ shows that the sum $\sum_{j\le N}\theta Y_j/(pB_j)$ has law $\mathsf P_p(\mathsf K_\theta\nu)$, and applying $s_\theta$ gives (i). For (iii), the event $N=0$ contributes $e^{-p}\delta_{1+\theta}$. Conditionally on $N=n\ge1$ and on $(B_j)_{j\le n}$, the sum is absolutely continuous because the summand $\theta Y_1/(pB_1)$ has a density, and the convolution of an absolutely continuous law with any law is absolutely continuous. \end{proof} \begin{lemma}\label{lem:stab-linearization} Let $p>1$ and $\theta>0$. For a probability measure $\lambda$ on $\R$ with $\lambda(\{0\})=0$ and a finite signed measure $\sigma$ with $|\sigma|(\{0\})=0$, define \begin{equation}\label{eq:stab-L-def} \mathcal L_{p,\theta;\lambda}\sigma=p\,\mathcal R_{p,\theta}(\lambda)*(\mathsf K_\theta\sigma)^\vee . \end{equation} Let $D_\lambda\sim\mathcal R_{p,\theta}(\lambda)$ and $T\sim\Exp(1)$ be independent, and for $b\neq0$ set $Z_b=D_\lambda-\theta T/(pb)$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every bounded Borel function $\psi$, \begin{equation}\label{eq:stab-L-dual} \langle\mathcal L_{p,\theta;\lambda}\sigma,\psi\rangle=p\int\E\big[\psi(Z_b)\big]\,\sigma(\mathrm db). \end{equation} \item For $b\neq0$, the law of $Z_b$ has the density $k^\lambda_b$ given, with $\mu_\lambda=\mathcal R_{p,\theta}(\lambda)$, by \begin{equation}\label{eq:stab-kernel} k^\lambda_b(x)= \begin{cases} \displaystyle\frac{pb}{\theta}\int_{(x,\infty)}e^{-pb(d-x)/\theta}\,\mu_\lambda(\mathrm dd), & b>0,\\[3mm] \displaystyle\frac{p|b|}{\theta}\int_{(-\infty,x)}e^{-p|b|(x-d)/\theta}\,\mu_\lambda(\mathrm dd), & b<0 . \end{cases} \end{equation} The measure $\mathcal L_{p,\theta;\lambda}\sigma$ is absolutely continuous with density $x\mapsto p\int k^\lambda_b(x)\,\sigma(\mathrm db)$. Moreover, $\|\mathcal L_{p,\theta;\lambda}\sigma\|_{\TV}\le p\|\sigma\|_{\TV}$ and $\mathcal L_{p,\theta;\lambda}\sigma(\R)=p\,\sigma(\R)$. \item Let $\nu,\mu$ be probability measures on $\R$ without atom at $0$, set $\sigma=\nu-\mu$ and $\lambda_t=(1-t)\mu+t\nu$ for $t\in[0,1]$. Then, for every bounded Borel function $\psi$, \begin{equation}\label{eq:stab-segment} \langle\mathcal R_{p,\theta}(\nu)-\mathcal R_{p,\theta}(\mu),\psi\rangle=\int_0^1\langle\mathcal L_{p,\theta;\lambda_t}\sigma,\psi\rangle\dd t, \end{equation} and \begin{equation}\label{eq:stab-second-order} \big\|\mathcal R_{p,\theta}(\nu)-\mathcal R_{p,\theta}(\mu)-\mathcal L_{p,\theta;\mu}\sigma\big\|_{\TV}\le\frac{p^2}{2}\|\sigma\|_{\TV}^2 . \end{equation} \end{enumerate} \end{lemma} The operator $\mathcal L_{p,\theta;\lambda}$ is the linearization of the law map at $\lambda$; by \eqref{eq:stab-L-dual}, it inserts one additional child with denominator distributed according to $\sigma$ into an independent output of the law map applied to $\lambda$. \begin{proof} By \eqref{eq:stab-L-def} and Fubini's theorem, we have \[ \langle\mathcal L_{p,\theta;\lambda}\sigma,\psi\rangle =p\int\!\!\int\psi(d-y)\,\mu_\lambda(\mathrm dd)\,\mathsf K_\theta\sigma(\mathrm dy) =p\int\sigma(\mathrm db)\int j_\theta(b;y)\,\E\big[\psi(D_\lambda-y)\big]\dd y , \] which is \eqref{eq:stab-L-dual}, since $j_\theta(b;\cdot)$ is the density of $\theta T/(pb)$. For (ii), let $b>0$. Then $\theta T/(pb)$ has density $j_\theta(b;\cdot)$ on $(0,\infty)$, and for a Borel set $A$, \[ \P(Z_b\in A)=\int\mu_\lambda(\mathrm dd)\int_{(0,\infty)}\one_A(d-y)\,j_\theta(b;y)\dd y=\int_A\Big(\int_{(x,\infty)}j_\theta(b;d-x)\,\mu_\lambda(\mathrm dd)\Big)\dd x , \] which is the first line of \eqref{eq:stab-kernel}. The case $b<0$ is the same, with $d-x<0$. Applying \eqref{eq:stab-L-dual} with $\psi=\one_A$ and using Fubini's theorem (which applies since $\int k^\lambda_b=1$ and $|\sigma|$ is finite), we find that $\mathcal L_{p,\theta;\lambda}\sigma(A)=\int_A p\int k^\lambda_b(x)\,\sigma(\mathrm db)\dd x$. The total variation bound and the mass identity follow from \eqref{eq:stab-L-dual} with $|\psi|\le1$ and $\psi=1$. For (iii), set $\alpha_t=\mathsf K_\theta\lambda_t=\mathsf K_\theta\mu+t\,\mathsf K_\theta\sigma$. Each $\alpha_t$ is a probability measure. Since convolution is bilinear and commutative, $t\mapsto\alpha_t^{*k}$ is a polynomial with values in the Banach space of finite signed measures with the total variation norm, with derivative $k\,\alpha_t^{*(k-1)}*\mathsf K_\theta\sigma$. The differentiated series $\sum_{k\ge1}e^{-p}p^kk\,\alpha_t^{*(k-1)}*\mathsf K_\theta\sigma/k!$ converges uniformly in $t\in[0,1]$, because its terms have norm at most $e^{-p}p^k\|\sigma\|_{\TV}/(k-1)!$. This implies that $t\mapsto\mathsf P_p(\alpha_t)$ is continuously differentiable with \[ \frac{\mathrm d}{\mathrm dt}\mathsf P_p(\alpha_t)=p\,\mathsf P_p(\alpha_t)*\mathsf K_\theta\sigma , \] and the fundamental theorem of calculus in this Banach space gives $\mathsf P_p(\alpha_1)-\mathsf P_p(\alpha_0)=\int_0^1p\,\mathsf P_p(\alpha_t)*\mathsf K_\theta\sigma\dd t$. Applying $(s_\theta)_\sharp$, which is a linear isometry, and using (i) of \Cref{lem:stab-lawmap} and \eqref{eq:stab-image-convolution}, we obtain $\mathcal R_{p,\theta}(\nu)-\mathcal R_{p,\theta}(\mu)=\int_0^1\mathcal L_{p,\theta;\lambda_t}\sigma\dd t$. Pairing with $\psi$, which is a bounded linear functional, gives \eqref{eq:stab-segment}. Further, since each $\mathsf P_p(\alpha_u)$ is a probability measure, the same computation shows that $\|\mathsf P_p(\alpha_t)-\mathsf P_p(\alpha_0)\|_{\TV}\le tp\|\sigma\|_{\TV}$, and therefore \[ \|\mathcal L_{p,\theta;\lambda_t}\sigma-\mathcal L_{p,\theta;\mu}\sigma\|_{\TV}=p\,\big\|(\mathsf P_p(\alpha_t)-\mathsf P_p(\alpha_0))*\mathsf K_\theta\sigma\big\|_{\TV}\le tp^2\|\sigma\|_{\TV}^2 . \] Integrating over $t\in[0,1]$ proves \eqref{eq:stab-second-order}. \end{proof} \section{Estimates for compound Poisson sums}\label{sec:stab-tools} In this subsection, we fix $p>1$ and $\theta>0$, and we consider a finite measure $\Lambda$ on $\R\setminus\{0\}$ with $\Lambda(\R)\le p$, a Poisson process $\{(b_j,Y_j)\}$ on $(\R\setminus\{0\})\times(0,\infty)$ with intensity measure $\Lambda\otimes\Exp(1)$, and the sum \begin{equation}\label{eq:stab-generic-sum} S=\sum_j\frac{\theta Y_j}{pb_j}. \end{equation} By \Cref{lem:stab-lawmap}, each sum $S_A$ in that lemma is of this form, with $\Lambda=p\,\nu|_A$. The first estimate is a Chernoff bound for the children whose denominators are bounded away from zero. \begin{lemma}\label{lem:stab-chernoff} Let $\delta\in(0,1]$ and suppose that $\Lambda$ is carried by $\{b:|b|\ge\delta\}$. Then $\E S=(\theta/p)\int b^{-1}\,\Lambda(\mathrm db)$, and for every $x>0$, \begin{align} \P\big(|S-\E S|>x\big)&\le2\exp\Big(-\min\Big\{\frac{p\delta^2x^2}{8\theta^2},\frac{p\delta x}{4\theta}\Big\}\Big),\label{eq:stab-chernoff}\\ \P\Big(\sum_j\frac{\theta Y_j}{p|b_j|}>x\Big)&\le\exp\Big(p-\frac{p\delta x}{2\theta}\Big).\label{eq:stab-chernoff-abs} \end{align} \end{lemma} \begin{proof} Conditioning on the number of points, which is Poisson with mean $\Lambda(\R)$, we have $\E e^{tS}=\exp\big(\int(\E e^{t\theta Y/(pb)}-1)\,\Lambda(\mathrm db)\big)$ whenever the integral converges. For $u<1$, we have $\E e^{uY}=(1-u)^{-1}$. Let $|t|\le p\delta/(2\theta)$ and $u_b=t\theta/(pb)$, so that $|u_b|\le1/2$ on the support of $\Lambda$. Differentiating at $t=0$ gives the formula for $\E S$, and we obtain \[ \log\E e^{t(S-\E S)}=\int\Big(\frac{1}{1-u_b}-1-u_b\Big)\Lambda(\mathrm db)=\int\frac{u_b^2}{1-u_b}\,\Lambda(\mathrm db)\le2\Lambda(\R)\frac{t^2\theta^2}{p^2\delta^2}\le\frac{2t^2\theta^2}{p\delta^2}. \] By Markov's inequality, $\P(S-\E S>x)\le\exp(-tx+2t^2\theta^2/(p\delta^2))$. If $2\theta\le\delta x$, we choose $t=p\delta/(2\theta)$; then $2t^2\theta^2/(p\delta^2)=t\theta/\delta\le tx/2$, and the exponent is at most $-p\delta x/(4\theta)$. If $2\theta>\delta x$, we choose $t=p\delta^2x/(4\theta^2)0$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\P(|V_1+\dots+V_m|\ge v)\le\sum_{i=1}^m\P(|V_i|\ge v/m)$. \item Suppose that, for each $i$, the restriction of the law of $V_i$ to $\{y:|y|\ge v/m\}$ is absolutely continuous with density at most $\varkappa_i$. Then the restriction of the law of $V_1+\dots+V_m$ to $\{u:|u|\ge v\}$ is absolutely continuous with density at most $\varkappa_1+\dots+\varkappa_m$. \item If $V_1,\dots,V_m$ are exponential with arbitrary rates $q_1,\dots,q_m>0$, then $V_1+\dots+V_m$ has a density $f$ with $f(u)\le m^2/(eu)$ for almost every $u>0$. The same holds if the rates are random and independent of the standard exponential variables $q_iV_i$. \end{enumerate} \end{lemma} \begin{proof} Let $\Sigma=V_1+\dots+V_m$ and $\Sigma^{(i)}=\Sigma-V_i$. On every outcome, some $i$ satisfies $|V_i|\ge|\Sigma|/m$. Part (i) follows. For (ii), let $A\subset\{u:|u|\ge v\}$ be a Borel set. Then \begin{align*} \P(\Sigma\in A)&\le\sum_{i=1}^m\P\big(\Sigma\in A,\ |V_i|\ge v/m\big)\\ &=\sum_{i=1}^m\E\Big[\P\big(V_i\in A-\Sigma^{(i)},\ |V_i|\ge v/m\,\big|\,\Sigma^{(i)}\big)\Big]\le\sum_{i=1}^m\varkappa_i|A| , \end{align*} where we used that $V_i$ is independent of $\Sigma^{(i)}$. For (iii), since $\sup_{q>0}qe^{-qy}=1/(ey)$ for $y>0$, the density of $V_i$ on $[v/m,\infty)$ is at most $m/(ev)$. By (ii), the density of $\Sigma$ on $[v,\infty)$ is at most $m^2/(ev)$. Applying this for all rational $v$ smaller than $u$ gives $f(u)\le m^2/(eu)$ for almost every $u>0$. For random rates, apply this conditionally on the rates and average. \end{proof} The third estimate is the fresh cancellation bound. It says that the children with denominators bounded away from zero can move the output by a distance of order one only with probability $e^{-cp/\theta}$, and that the same holds for densities. \begin{lemma}\label{lem:stab-fresh} Let $\delta\in(0,1]$, suppose that $\Lambda$ is carried by $\{b:|b|\ge\delta\}$, and let $c\in\R$. Then the restriction of the law of $c-S$ to $(-\infty,c-1/2]$ is absolutely continuous with density at most \begin{equation}\label{eq:stab-fresh-bound} F_{p,\theta}(\delta)=6p^2\exp\Big(p-\frac{\delta p}{4\theta}\Big). \end{equation} \end{lemma} \begin{proof} We write $S=S_+-S_-$, where $S_+$ is the sum over the points with $b_j>0$ and $S_-\ge0$ is minus the sum over the points with $b_j<0$. By the restriction property of Poisson processes, $S_+$ and $S_-$ are independent. If $c-S=x$ with $x\le c-1/2$, then $S_+=c-x+S_-\ge1/2$. Conditioning on $S_-$, we see that it suffices to show that the restriction of the law of $S_+$ to $[1/2,\infty)$ has density at most $F_{p,\theta}(\delta)$. Let $\Lambda_+=\Lambda([\delta,\infty))\le p$; if $\Lambda_+=0$ then $S_+=0$ and there is nothing to prove. The variable $S_+$ is a sum of $N_+\sim\Pois(\Lambda_+)$ independent jumps with common law $\pi_J(\mathrm dy)=\Lambda_+^{-1}\int_{[\delta,\infty)}j_\theta(b;y)\Lambda(\mathrm db)\dd y$. Let $f_n$ denote the density of the sum of $n$ independent jumps with law $\pi_J$. We use an exponential tilt (see \cite[Section~2.2]{DemboZeitouni1998large}). Let $\alpha=\delta/2$ and $t=\alpha p/\theta$. For $b\ge\delta$, we have \[ e^{ty}j_\theta(b;y)=\frac{b}{b-\alpha}\cdot\frac{p(b-\alpha)}{\theta}e^{-p(b-\alpha)y/\theta},\qquad y>0, \] and $1\le b/(b-\alpha)\le2$. Set $M=\int e^{ty}\pi_J(\mathrm dy)\in[1,2]$ and $\tilde\pi_J(\mathrm dy)=M^{-1}e^{ty}\pi_J(\mathrm dy)$. Then $\tilde\pi_J$ is a mixture of exponential laws with positive rates $p(b-\alpha)/\theta$, and $e^{tu}f_n(u)=M^n\tilde f_n(u)$, where $\tilde f_n$ is the density of the sum of $n$ independent jumps with law $\tilde\pi_J$. By part (iii) of \Cref{lem:stab-large-summand}, $\tilde f_n(u)\le n^2/(eu)$. We obtain, for almost every $u\ge1/2$, \[ \sum_{n\ge1}e^{-\Lambda_+}\frac{\Lambda_+^n}{n!}f_n(u) =e^{-tu}e^{\Lambda_+(M-1)}\sum_{n\ge1}e^{-\Lambda_+M}\frac{(\Lambda_+M)^n}{n!}\tilde f_n(u) \le e^{-tu}e^{p}\,\frac{2}{e}\,\E\big[\tilde N^2\big], \] where $\tilde N\sim\Pois(\Lambda_+M)$. In the inequality, we used $\Lambda_+(M-1)\le p$ and $1/(eu)\le2/e$. Since $\Lambda_+M\le2p$, we have $\E\tilde N^2\le2p+4p^2\le6p^2$, and $e^{-tu}\le e^{-\delta p/(4\theta)}$ for $u\ge1/2$. The left side is the density of the absolutely continuous part of the law of $S_+$, and the only atom of this law is at $0$. This proves the lemma. \end{proof} The last lemma concerns children whose denominators are close to zero. \begin{lemma}\label{lem:stab-envelope} \begin{enumerate}[label=\textup{(\roman*)}] \item Let $\rho'>0$, $H\ge0$, and let $\nu$ be a finite measure on $\R$ whose restriction to $(-\rho',\rho')$ is absolutely continuous with density at most $H$. Then $\ell(y)=p\int_{(-\rho',\rho')}j_\theta(b;y)\,\nu(\mathrm db)$ satisfies \begin{equation}\label{eq:stab-envelope} \ell(y)\le\min\Big\{\frac{\theta H}{y^2},\frac{p^2\rho'^2H}{2\theta}\Big\},\qquad\int\ell(y)\dd y=p\,\nu\big((-\rho',\rho')\big)\le2\rho'pH . \end{equation} \item Let $\ell\ge0$ be integrable with $\lambda=\int\ell\le1$ and $\ell(y)\le\varkappa/y^2$ for $y\neq0$, and let $S$ have law $e^{-\lambda}\sum_{m\ge0}\ell^{*m}/m!$. Then the law of $S$ is $e^{-\lambda}\delta_0+d_S(u)\dd u$ with $d_S(u)\le5\varkappa/u^2$ for almost every $u$, and $\P(|S|\ge u)\le4\varkappa/u$ for every $u>0$. \end{enumerate} \end{lemma} \begin{proof} For (i), we have $\ell(y)\le(p^2H/\theta)\int_0^{\rho'}be^{-pb|y|/\theta}\dd b$. Bounding the integral by $\int_0^\infty be^{-pb|y|/\theta}\dd b=\theta^2/(p^2y^2)$, and alternatively by $\rho'^2/2$, gives the two bounds. The identity for $\int\ell$ follows from $\int j_\theta(b;y)\dd y=1$. For (ii), we may assume $\lambda>0$. Let $m\ge1$. Then $\ell^{*m}=\lambda^m\,\Law(V_1+\dots+V_m)$, where $V_i$ are independent with density $\ell/\lambda$. On $\{|y|\ge v/m\}$ this density is at most $\varkappa m^2/(\lambda v^2)$. By part (ii) of \Cref{lem:stab-large-summand}, $\ell^{*m}$ has density at most $\varkappa m^3\lambda^{m-1}/v^2$ on $\{|u|\ge v\}$. This implies that $\ell^{*m}(u)\le\varkappa m^3\lambda^{m-1}/u^2$ for almost every $u$. Summing, we obtain \[ d_S(u)\le\frac{\varkappa}{u^2}e^{-\lambda}\sum_{m\ge1}\frac{m^3\lambda^{m-1}}{m!}=\frac{\varkappa}{u^2}e^{-\lambda}\sum_{k\ge0}\frac{(k+1)^2\lambda^k}{k!}=\frac{\varkappa}{u^2}(1+3\lambda+\lambda^2)\le\frac{5\varkappa}{u^2}. \] Similarly, by part (i) of \Cref{lem:stab-large-summand}, $\P(|V_1+\dots+V_m|\ge u)\le m\,\P(|V_1|\ge u/m)\le2\varkappa m^2/(\lambda u)$, and therefore \[ \P(|S|\ge u)\le\frac{2\varkappa}{u}e^{-\lambda}\sum_{m\ge1}\frac{m^2\lambda^{m-1}}{m!}=\frac{2\varkappa}{u}(1+\lambda)\le\frac{4\varkappa}{u}.\qedhere \] \end{proof} \section{The class \texorpdfstring{$\mathfrak S_{p,\theta}$}{S(p,theta)} and its invariance}\label{sec:stab-class} \begin{definition}\label{def:stab-class} A tuple $\mathbf q=(a,A,K',\rho)$ is admissible if $a>0$, $A\ge0$, $K'>0$, and $0<\rho\le1/16$. For $p>1$, set \[ r_p=p^{-1/4},\qquad\mathcal C_p=[1-r_p,1+r_p],\qquad\mathfrak h_p=p^Ae^{-ap}. \] The ambient class $\mathscr A_p(\mathbf q)$ is the set of probability measures $\nu$ on $\R$ such that \begin{enumerate}[label=\textup{(\Alph*)}] \item[\textup{(C)}] $\nu(\R\setminus\mathcal C_p)\le e^{-a\sqrt p}$; \item[\textup{(B)}] the restriction of $\nu$ to $(-\rho,\rho)$ is absolutely continuous with density at most $\mathfrak h_p$ almost everywhere; \item[\textup{(T)}] $\nu(\{b:|b|>L\})\le K'\mathfrak h_p/L+e^{-apL}$ for every $L\ge3$. \end{enumerate} For $\theta>0$, the class $\mathfrak S_{p,\theta}(\mathbf q)$ is the set of $\nu\in\mathscr A_p(\mathbf q)$ of the form $\nu=e^{-p}\delta_{1+\theta}+f(b)\dd b$ with $f\in L^1(\R)$. We call (C), (B), and (T) the core, boundary, and tail bounds. The canonical tuple is $\mathbf q_{\mathrm{can}}=(1,0,1,1/16)$, and we write $\mathscr A_p=\mathscr A_p(\mathbf q_{\mathrm{can}})$ and $\mathfrak S_{p,\theta}=\mathfrak S_{p,\theta}(\mathbf q_{\mathrm{can}})$. \end{definition} The ambient class is convex, it contains $\delta_1$, and it does not depend on $\theta$. Its members have no atom in $(-\rho,\rho)$, and in particular the law map is defined on $\mathscr A_p(\mathbf q)$. The members of $\mathfrak S_{p,\theta}(\mathbf q)$ have a single atom, of mass $e^{-p}$ at $1+\theta$, which is the atom forced by the event $N=0$ in \Cref{lem:stab-lawmap}. The ambient class is introduced because $\mathfrak S_{p,\theta}(\mathbf q)$ is not complete for the metric used below; see \Cref{rem:stab-not-complete}. \paragraph{Constants and the threshold.} We now fix the constants used in \Cref{sec:stab-class,sec:stab-mixed,sec:stab-columns,sec:stab-fixed}, and we make the order of choices explicit. First, an admissible tuple $\mathbf q=(a,A,K',\rho)$ is fixed, and with it \[ \rho_0=\rho/4,\qquad\rho_1=\rho/2 . \] Second, a number $C_0>0$ is fixed; the parameter $\theta$ will always satisfy $0<\theta\le C_0/p$. Third, with \begin{gather*} c_1=\frac{\rho^2}{128(1+C_0)^2},\qquad\omega_p=2e^{-c_1p^{7/4}}+2\rho p\mathfrak h_p,\qquad\delta_p=p^{A+3}e^{-ap},\\ t_p=\frac{\rho_0p^{7/4}}{2C_0},\qquad x_p=\frac{\rho\rho_0p^2}{C_0}, \end{gather*} we consider the following conditions on $p$. \begin{enumerate}[label=\textup{(S\arabic*)}] \item\label{it:stab-S1} $p\ge256$, $p\ge(2/\rho)^4$, and $p\ge12C_0$; in particular $r_p\le\min\{1/4,\rho/2\}$ and $\theta\le1/12$. \item\label{it:stab-S2} $(\rho^{-1}+2)e^{-a\sqrt p}\le r_p$. \item\label{it:stab-S3} $2e^{-c_1p^{7/4}}+(t_p+2)e^{-t_p}\le\rho p\mathfrak h_p$ and $4\rho p\mathfrak h_p\le e^{-a\sqrt p}/2$; in particular $(t_p+2)e^{-t_p}\le\omega_p\le3\rho p\mathfrak h_p\le e^{-a\sqrt p}/2$. \item\label{it:stab-S4} $2\rho p\mathfrak h_p\le1$ and $20C_0\le\rho^2p$. \item\label{it:stab-S5} $48p^2\exp(p-\rho p^2/(4C_0))\le\mathfrak h_p$. \item\label{it:stab-S6} $16C_0\le K'p$ and $\rho p^2/(18C_0)\ge(a+1)p+1$. \item\label{it:stab-S7} $x_p\ge2$ and $2(\rho\rho_0)^{-1}x_p^2e^{-x_p}\le\mathfrak h_p/2$. \item\label{it:stab-S8} $p^{3/4}\ge3100(1+C_0)\rho^{-2}$ and $160C_0\le\rho^2p^2$. \item\label{it:stab-S9} $20C_0/p+7pe^{-a\sqrt p/2}\le1/2$ and $2p^{A+3}e^{-ap/2}\le pe^{-a\sqrt p/2}$. \end{enumerate} Each condition holds for all sufficiently large $p$, because in each of them a quantity decaying like $e^{-cp}$ or faster is compared with a quantity decaying at most like $p^{-1}e^{-ap}$ or $e^{-a\sqrt p}$, or a polynomial is compared with a larger power of $p$. We let $p_*(\mathbf q,C_0)$ be the least integer such that \ref{it:stab-S1}--\ref{it:stab-S9} hold for every $p\ge p_*(\mathbf q,C_0)$. It depends only on $\mathbf q$ and $C_0$. The exponents $a$ and $A$ and the constant $K'$ of the class are never changed in the arguments below: the lemma that follows shows that the law map sends the ambient class into the class $\mathfrak S_{p,\theta}(\mathbf q)$ with the same tuple $\mathbf q$, and with the right sides of the three bounds halved. \begin{lemma}\label{lem:stab-invariance} Let $\mathbf q$ be admissible, $C_0>0$, $p\ge p_*(\mathbf q,C_0)$, and $0<\theta\le C_0/p$. For every $\nu\in\mathscr A_p(\mathbf q)$, the law $\mu=\mathcal R_{p,\theta}(\nu)$ satisfies the following. \begin{enumerate}[label=\textup{(\roman*)}] \item $\mu=e^{-p}\delta_{1+\theta}+h(b)\dd b$, where $h\le\mathfrak h_p/2$ almost everywhere on $[-2\rho,2\rho]$. \item $\mu(\{b:|b-1|>r_p/2\})\le\omega_p\le e^{-a\sqrt p}/2$. \item $\mu(\{b:|b|>L\})\le K'\mathfrak h_p/(2L)+e^{-apL}/2$ for every $L\ge3$. \end{enumerate} In particular, $\mathcal R_{p,\theta}$ maps $\mathscr A_p(\mathbf q)$ into $\mathfrak S_{p,\theta}(\mathbf q)$. \end{lemma} \begin{proof} We split the children according to whether $|B_j|\ge\rho$. By \Cref{lem:stab-lawmap}, a variable $B'$ with law $\mu$ can be written as \begin{equation}\label{eq:stab-split} B'=D-S_{\mathrm{bd}},\qquad D=1+\theta-S_{\mathrm{nb}},\qquad S_{\mathrm{nb}}=\frac{\theta}{p}\sum_{j\le N:\,|B_j|\ge\rho}\frac{Y_j}{B_j},\qquad S_{\mathrm{bd}}=\frac{\theta}{p}\sum_{j\le N:\,|B_j|<\rho}\frac{Y_j}{B_j}, \end{equation} with $S_{\mathrm{nb}}$ and $S_{\mathrm{bd}}$ independent. The sum $S_{\mathrm{nb}}$ is of the form \eqref{eq:stab-generic-sum} with $\Lambda=p\,\nu|_{\{|b|\ge\rho\}}$, and $S_{\mathrm{bd}}$ is of this form with $\Lambda=p\,\nu|_{(-\rho,\rho)}$. \proofstep{1}{Core concentration.} We have $\E S_{\mathrm{nb}}=\theta\int_{\{|b|\ge\rho\}}b^{-1}\nu(\mathrm db)$ by \Cref{lem:stab-chernoff}. Since $\int(b^{-1}-1)\one_{\{|b|\ge\rho\}}\dd\nu-\nu((-\rho,\rho))=\int_{\{|b|\ge\rho\}}b^{-1}\dd\nu-1$, and $|b^{-1}-1|\le r_p/(1-r_p)\le2r_p$ on $\mathcal C_p$, we have \begin{align*} \Big|\int_{\{|b|\ge\rho\}}\frac{\nu(\mathrm db)}{b}-1\Big|&\le2r_p+(\rho^{-1}+1)\nu(\R\setminus\mathcal C_p)+\nu((-\rho,\rho))\\ &\le2r_p+(\rho^{-1}+2)e^{-a\sqrt p}\le3r_p . \end{align*} In the second inequality, we used (C) and $(-\rho,\rho)\subset\R\setminus\mathcal C_p$, and in the last one \ref{it:stab-S2}. Since $\theta\le1/12$ by \ref{it:stab-S1}, we obtain $|\E S_{\mathrm{nb}}-\theta|\le3\theta r_p\le r_p/4$. Next, \eqref{eq:stab-chernoff} with $\delta=\rho$ and $x=r_p/4$, together with $\theta\le C_0/p$ and $r_p=p^{-1/4}$, gives \[ \P\big(|S_{\mathrm{nb}}-\E S_{\mathrm{nb}}|>r_p/4\big)\le2\exp\Big(-\min\Big\{\frac{\rho^2p^{5/2}}{128C_0^2},\frac{\rho p^{7/4}}{16C_0}\Big\}\Big)\le2e^{-c_1p^{7/4}} . \] Finally, $\P(S_{\mathrm{bd}}\neq0)=1-e^{-p\nu((-\rho,\rho))}\le2\rho p\mathfrak h_p$ by (B). Since $|B'-1|\le|\E S_{\mathrm{nb}}-\theta|+|S_{\mathrm{nb}}-\E S_{\mathrm{nb}}|$ on the event $S_{\mathrm{bd}}=0$, we conclude that $\P(|B'-1|>r_p/2)\le\omega_p$. The bound $\omega_p\le e^{-a\sqrt p}/2$ is \ref{it:stab-S3}. This proves (ii). \proofstep{2}{Fresh cancellation.} By \Cref{lem:stab-fresh} with $\delta=\rho$ and $c=1+\theta$, the restriction of the law of $D$ to $(-\infty,1/2]$ is absolutely continuous with density at most $F=F_{p,\theta}(\rho)$. Since $\theta\le C_0/p$, we have $F\le6p^2\exp(p-\rho p^2/(4C_0))\le\mathfrak h_p/8$ by \ref{it:stab-S5}. \proofstep{3}{Boundary renewal.} By part (i) of \Cref{lem:stab-envelope} with $\rho'=\rho$ and $H=\mathfrak h_p$, the intensity $\ell_{\mathrm{bd}}$ of the jumps of $S_{\mathrm{bd}}$ satisfies $\ell_{\mathrm{bd}}(y)\le\theta\mathfrak h_p/y^2$ and $\int\ell_{\mathrm{bd}}\le2\rho p\mathfrak h_p\le1$, by \ref{it:stab-S4}. By part (ii) of the same lemma, the law of $S_{\mathrm{bd}}$ is $e^{-\lambda}\delta_0+d_{\mathrm{bd}}(u)\dd u$ with $\lambda=\int\ell_{\mathrm{bd}}$ and $d_{\mathrm{bd}}(u)\le5\theta\mathfrak h_p/u^2$. Let $A\subset[-2\rho,2\rho]$ be a Borel set. Since $D$ and $S_{\mathrm{bd}}$ are independent, \[ \P(B'\in A)=e^{-\lambda}\P(D\in A)+\int\Big(\int\one_A(d-u)\,d_{\mathrm{bd}}(u)\dd u\Big)\P(D\in\mathrm dd). \] The first term is at most $F|A|$, because $A\subset(-\infty,1/2]$. We split the integral in the second term according to whether $|d|\le3\rho$. Since $3\rho\le1/2$, the part with $|d|\le3\rho$ equals $\int d_{\mathrm{bd}}(u)\P(D\in(A+u)\cap[-3\rho,3\rho])\dd u\le F|A|$. If $|d|>3\rho$ and $d-u\in A$, then $|u|\ge\rho$ and $d_{\mathrm{bd}}(u)\le5\theta\mathfrak h_p/\rho^2$, so the part with $|d|>3\rho$ is at most $5\theta\mathfrak h_p\rho^{-2}|A|$. We conclude that the restriction of $\mu$ to $[-2\rho,2\rho]$ has density at most \begin{equation}\label{eq:stab-renewal} 2F+\frac{5\theta}{\rho^2}\mathfrak h_p\le\frac{\mathfrak h_p}{4}+\frac{5C_0}{\rho^2p}\mathfrak h_p\le\frac{\mathfrak h_p}{2}, \end{equation} where we used Step 2 and \ref{it:stab-S4}. Together with part (iii) of \Cref{lem:stab-lawmap}, this proves (i). \proofstep{4}{Far tail.} Let $L\ge3$. If $|D|>L/2$, then $\sum_{|B_j|\ge\rho}\theta Y_j/(p|B_j|)\ge|S_{\mathrm{nb}}|>L/2-1-\theta\ge L/9$, where we used $\theta\le1/12$ and $L\ge3$. By \eqref{eq:stab-chernoff-abs} with $\delta=\rho$ and \ref{it:stab-S6}, \[ \P(|D|>L/2)\le\exp\Big(p-\frac{\rho p^2L}{18C_0}\Big)\le\exp\big(p-((a+1)p+1)L\big)\le\frac12e^{-apL}. \] Further, part (ii) of \Cref{lem:stab-envelope} gives $\P(|S_{\mathrm{bd}}|\ge L/2)\le8\theta\mathfrak h_p/L\le8C_0\mathfrak h_p/(pL)\le K'\mathfrak h_p/(2L)$, where we used \ref{it:stab-S6} in the last step. Since $|B'|>L$ implies $|D|>L/2$ or $|S_{\mathrm{bd}}|\ge L/2$, this proves (iii). Finally, (i)--(iii) show that $\mu$ satisfies (C), (B), and (T), since $\{b:|b-1|>r_p\}\subset\{b:|b-1|>r_p/2\}$ and $(-\rho,\rho)\subset[-2\rho,2\rho]$. We conclude that $\mu\in\mathfrak S_{p,\theta}(\mathbf q)$. \end{proof} \begin{remark}\label{rem:stab-sharper} The proof gives more than the bounds (C), (B), and (T). The output of the law map has density at most $2F_{p,\theta}(\rho)+5\theta\rho^{-2}\mathfrak h$ on $[-2\rho,2\rho]$ whenever the input has density at most $\mathfrak h$ on $(-\rho,\rho)$ and $2\rho p\mathfrak h\le1$; the first term is of order $e^{-cp^2}$. In particular, the boundary bound is not saturated at a fixed point, and any admissible choice of $a$ and $A$ can be used. This is why the tuple $\mathbf q$ in \Cref{def:stab-class} can be fixed once and for all, independently of the energy window; see also part (iii) of \Cref{thm:stab-fixed}. \end{remark} \section{The mixed metric}\label{sec:stab-mixed} We fix once and for all a $C^\infty$ function $\chi:\R\to[0,1]$ which is even, equal to $1$ on $[-1,1]$, equal to $0$ outside $(-2,2)$, and satisfies $|\chi'|\le2$. Its higher derivatives are then bounded by numerical constants. \begin{definition}\label{def:stab-mixed} Let $\mathbf q=(a,A,K',\rho)$ be admissible, let $p\ge\max\{256,(2/\rho)^4\}$, and recall $\rho_0=\rho/4$ and $\rho_1=\rho/2$. \begin{enumerate}[label=\textup{(\roman*)}] \item The cutoffs are $\chi_{\mathrm c}(b)=\chi((b-1)/r_p)$, $\chi_{\mathrm b}(b)=\chi(4b/\rho)$, and $\chi_{\mathrm i}=1-\chi_{\mathrm c}-\chi_{\mathrm b}$. \item The core region is $[1-2r_p,1+2r_p]$, the boundary region is $[-\rho_1,\rho_1]$, and the intermediate region is \[ I_p=(-\infty,-\rho_0]\cup[\rho_0,1-r_p]\cup\{1\}\cup[1+r_p,\infty). \] \item For a finite signed measure $\sigma$ on $\R$ with $\sigma(\R)=0$ and $\alpha\in\{\mathrm c,\mathrm b,\mathrm i\}$, the compensated projections are \[ \Pi_\alpha\sigma=\chi_\alpha\sigma-m_\alpha(\sigma)\,\delta_1,\qquad m_\alpha(\sigma)=\int\chi_\alpha\dd\sigma . \] \item For a finite signed measure $\tau$ with $\tau(\R)=0$, the component norms, with values in $[0,\infty]$, are \begin{align*} \|\tau\|_{\mathrm c}&=\sup\Big\{\Big|\int\varphi\dd\tau\Big|:\ \varphi(1)=0,\ \Lip(\varphi)\le1\Big\},\\ \|\tau\|_{\mathrm i}&=\sup\Big\{\Big|\int\varphi\dd\tau\Big|:\ \|\varphi\|_\infty\le1,\ \Lip(\varphi)\le1\Big\}, \end{align*} and $\|\tau\|_{\mathrm b}=\|g\|_{L^\infty}$ if $\tau=g(b)\dd b-(\int g)\delta_1$ for some $g\in L^\infty$ vanishing outside $[-\rho_1,\rho_1]$, and $\|\tau\|_{\mathrm b}=\infty$ otherwise. We write $n_\alpha(\sigma)=\|\Pi_\alpha\sigma\|_\alpha$ for $\sigma(\R)=0$. \item The weights are $W_{\mathrm c}=1$, $W_{\mathrm b}=e^{ap/2}$, and $W_{\mathrm i}=e^{a\sqrt p/2}$. The mixed norm of $\sigma$ with $\sigma(\R)=0$ is \[ N_p(\sigma)=n_{\mathrm c}(\sigma)+W_{\mathrm b}\,n_{\mathrm b}(\sigma)+W_{\mathrm i}\,n_{\mathrm i}(\sigma)\in[0,\infty], \] and the mixed metric on probability measures is $d_p(\nu,\mu)=N_p(\nu-\mu)$. \end{enumerate} \end{definition} The regions, the cutoffs, and the mixed metric depend on $p$ and on $\mathbf q$ through $\rho$ and $a$, but not on $\theta$. The norm $\|\cdot\|_{\mathrm c}$ is the Kantorovich--Rubinstein norm and $\|\cdot\|_{\mathrm i}$ is the dual bounded--Lipschitz norm. In the boundary norm, the decomposition $\tau=g\dd b-(\int g)\delta_1$ is unique when it exists, since the absolutely continuous part and the atom at $1$ of a measure are uniquely determined. \begin{lemma}\label{lem:stab-mixed-basic} Let $\mathbf q$ be admissible, $p\ge\max\{256,(2/\rho)^4\}$, and let $\sigma$ be a finite signed measure with $\sigma(\R)=0$. \begin{enumerate}[label=\textup{(\roman*)}] \item The supports of $\chi_{\mathrm c}$ and $\chi_{\mathrm b}$ are contained in the core and boundary regions, which are disjoint; $0\le\chi_{\mathrm i}\le1$; and $\chi_{\mathrm i}$ vanishes on $\mathcal C_p\cup[-\rho_0,\rho_0]$, so that $\{\chi_{\mathrm i}\neq0\}\subset I_p\setminus\{1\}$. We have $\sigma=\Pi_{\mathrm c}\sigma+\Pi_{\mathrm b}\sigma+\Pi_{\mathrm i}\sigma$, each $\Pi_\alpha\sigma$ has total mass zero, $\Pi_{\mathrm c}\sigma$ is carried by the core region, and $\Pi_{\mathrm i}\sigma$ is carried by $I_p$. \item For every Lipschitz function $\varphi$ with $\varphi(1)=0$, we have $\langle\Pi_{\mathrm c}\sigma,\varphi\rangle=\langle\sigma,\chi_{\mathrm c}\varphi\rangle$. For every bounded Lipschitz $\varphi$, we have $\langle\Pi_{\mathrm i}\sigma,\varphi\rangle=\langle\sigma,\chi_{\mathrm i}(\varphi-\varphi(1))\rangle$. \item If $\varphi(1)=0$ and $\Lip(\varphi)\le1$, then $\psi_{\mathrm c}=\chi_{\mathrm c}\varphi$ satisfies $\|\psi_{\mathrm c}\|_\infty\le2r_p$ and $\Lip(\psi_{\mathrm c})\le5$. If $\|\varphi\|_\infty\le1$ and $\Lip(\varphi)\le1$, then $\psi_{\mathrm i}=\chi_{\mathrm i}(\varphi-\varphi(1))$ satisfies $\|\psi_{\mathrm i}\|_\infty\le2$ and $\Lip(\psi_{\mathrm i})\le21/\rho$, and $\psi_{\mathrm i}$ vanishes on $\mathcal C_p\cup[-\rho_0,\rho_0]$. \item In the suprema defining $\|\cdot\|_{\mathrm c}$ and $\|\cdot\|_{\mathrm i}$, applied to a measure carried by a compact set in the first case, one may restrict to $C^\infty$ functions $\varphi$. \item Let $\tau$ be a finite signed measure with $\tau(\R)=0$. If $\tau$ is carried by an interval $I\ni1$ and $G:I\to\R$ is Lipschitz, then $|\int G\dd\tau|\le\Lip_I(G)\|\tau\|_{\mathrm c}$. If $\tau$ is carried by a closed set $I$ and $G:I\to\R$ is bounded and Lipschitz, then $|\int G\dd\tau|\le(\sup_I|G|+\Lip_I(G))\|\tau\|_{\mathrm i}$. If $G$ is continuously differentiable on each of the four pieces of $I_p$, with $|G'|\le M$ there and $|G|\le S$ on $I_p$, then $\Lip_{I_p}(G)\le M+2S/r_p$. \item $N_p$ is a norm on the space of finite signed measures $\sigma$ with $\sigma(\R)=0$ and $N_p(\sigma)<\infty$. For every $\nu,\mu\in\mathscr A_p(\mathbf q)$, $d_p(\nu,\mu)<\infty$, and $d_p$ is a metric on $\mathscr A_p(\mathbf q)$. \end{enumerate} \end{lemma} \begin{proof} For (i), the support of $\chi_{\mathrm c}$ is in $[1-2r_p,1+2r_p]\subset[1/2,3/2]$ and that of $\chi_{\mathrm b}$ is in $[-\rho/2,\rho/2]$, and these are disjoint. Since at each point at most one of $\chi_{\mathrm c},\chi_{\mathrm b}$ is nonzero, and both take values in $[0,1]$, we have $0\le\chi_{\mathrm i}\le1$. Since $\chi_{\mathrm c}=1$ on $\mathcal C_p$ and $\chi_{\mathrm b}=1$ on $[-\rho_0,\rho_0]$, the function $\chi_{\mathrm i}$ vanishes there. The identity $\sum_\alpha\Pi_\alpha\sigma=\sigma$ follows from $\sum_\alpha\chi_\alpha=1$ and $\sum_\alpha m_\alpha(\sigma)=\sigma(\R)=0$, and each $\Pi_\alpha\sigma$ has mass $m_\alpha(\sigma)-m_\alpha(\sigma)=0$. Part (ii) follows from the definition of $\Pi_\alpha$, since $\langle\chi_{\mathrm i}\sigma-m_{\mathrm i}(\sigma)\delta_1,\varphi\rangle=\langle\sigma,\chi_{\mathrm i}\varphi\rangle-\varphi(1)\langle\sigma,\chi_{\mathrm i}\rangle$. For (iii), on the support of $\chi_{\mathrm c}$ we have $|b-1|\le2r_p$ and $|\varphi(b)|\le2r_p$. Since $|\chi_{\mathrm c}'|\le2/r_p$, we obtain $|(\chi_{\mathrm c}\varphi)'|\le(2/r_p)(2r_p)+1=5$ almost everywhere. For $\psi_{\mathrm i}$, we have $\chi_{\mathrm i}'=-\chi_{\mathrm c}'-\chi_{\mathrm b}'$. On the support of $\chi_{\mathrm c}'$, $|\chi_{\mathrm c}'(\varphi-\varphi(1))|\le(2/r_p)(2r_p)=4$; and $|\chi_{\mathrm b}'(\varphi-\varphi(1))|\le(8/\rho)\cdot2$. Adding $\chi_{\mathrm i}|\varphi'|\le1$, we find $\Lip(\psi_{\mathrm i})\le5+16/\rho\le21/\rho$. The vanishing of $\psi_{\mathrm i}$ follows from (i). For (iv), let $\varphi$ be Lipschitz and let $\varphi_\eps$ be its convolution with a smooth probability density supported in $[-\eps,\eps]$. Then $\varphi_\eps$ is smooth, $\Lip(\varphi_\eps)\le\Lip(\varphi)$, $\|\varphi_\eps\|_\infty\le\|\varphi\|_\infty$, and $|\varphi_\eps-\varphi|\le\eps\Lip(\varphi)$. In the first case, $\varphi_\eps-\varphi_\eps(1)$ vanishes at $1$. Since $\tau$ is finite, $\int\varphi_\eps\dd\tau\to\int\varphi\dd\tau$, and (iv) follows. For (v), if $I$ is an interval, extend $G-G(1)$ to $\R$ by constants outside $I$. The extension $\varphi$ satisfies $\varphi(1)=0$ and $\Lip(\varphi)=\Lip_I(G)$, and $\int G\dd\tau=\int\varphi\dd\tau$ since $\tau$ has zero mass. If $I$ is closed, let $\Lambda=\Lip_I(G)$ and $S'=\sup_I|G|$. The McShane extension $\tilde G(x)=\inf_{y\in I}(G(y)+\Lambda|x-y|)$ agrees with $G$ on $I$ and has Lipschitz constant $\Lambda$, and truncating it at levels $\pm S'$ preserves both properties. The truncated extension divided by $\max\{S',\Lambda\}$ is one of the test functions in the definition of $\|\tau\|_{\mathrm i}$. For the last claim, note that $p\ge(2/\rho)^4$ gives $r_p\le\rho/2=2\rho_0$, so distinct pieces of $I_p$ are at distance at least $r_p$. For $b,b'$ in the same piece we have $|G(b)-G(b')|\le M|b-b'|$, and otherwise $|G(b)-G(b')|\le2S\le(2S/r_p)|b-b'|$. For (vi), suppose $N_p(\sigma)=0$. Then $\|\Pi_{\mathrm c}\sigma\|_{\mathrm c}=0$, and we have $\int\varphi\dd\Pi_{\mathrm c}\sigma=0$ for every Lipschitz $\varphi$ (subtract $\varphi(1)$ and use that $\Pi_{\mathrm c}\sigma$ has zero mass). Since bounded Lipschitz functions determine finite measures, $\Pi_{\mathrm c}\sigma=0$. Similarly $\Pi_{\mathrm b}\sigma=0$ and $\Pi_{\mathrm i}\sigma=0$, and (i) gives $\sigma=0$. Homogeneity and the triangle inequality are clear. If $\nu,\mu\in\mathscr A_p(\mathbf q)$ and $\sigma=\nu-\mu$, then $\|\Pi_{\mathrm c}\sigma\|_{\mathrm c}\le2r_p\|\Pi_{\mathrm c}\sigma\|_{\TV}\le8r_p$, because $\Pi_{\mathrm c}\sigma$ is carried by the core region, where the test functions in the definition of $\|\cdot\|_{\mathrm c}$ are bounded by $2r_p$; $\Pi_{\mathrm b}\sigma=\chi_{\mathrm b}(f_\nu-f_\mu)\dd b-m_{\mathrm b}(\sigma)\delta_1$, where $f_\nu,f_\mu$ are the densities on $(-\rho,\rho)$ given by (B), and $n_{\mathrm b}(\sigma)\le2\mathfrak h_p$; and $n_{\mathrm i}(\sigma)\le\|\Pi_{\mathrm i}\sigma\|_{\TV}\le4$. In particular, $d_p(\nu,\mu)<\infty$. \end{proof} \begin{lemma}\label{lem:stab-complete} Let $\mathbf q$ be admissible and $p\ge\max\{256,(2/\rho)^4\}$. \begin{enumerate}[label=\textup{(\roman*)}] \item The class $\mathscr A_p(\mathbf q)$ is tight and closed under weak convergence of probability measures. \item If $\nu_n,\nu\in\mathscr A_p(\mathbf q)$ and $\nu_n\Rightarrow\nu$, then $d_p(\mu,\nu)\le\liminf_{n}d_p(\mu,\nu_n)$ for every $\mu\in\mathscr A_p(\mathbf q)$. \item The metric space $(\mathscr A_p(\mathbf q),d_p)$ is complete, and convergence in $d_p$ implies weak convergence. \item Let $\theta_n\to\theta>0$, and let $\nu_n,\nu$ be probability measures on $\R$ without atom at $0$ with $\nu_n\Rightarrow\nu$. Then $\mathcal R_{p,\theta_n}(\nu_n)\Rightarrow\mathcal R_{p,\theta}(\nu)$. \end{enumerate} \end{lemma} \begin{proof} For (i), tightness follows from the uniform tail bound (T). Let $\nu_n\in\mathscr A_p(\mathbf q)$ converge weakly to $\nu$. Since the sets $\R\setminus\mathcal C_p$ and $\{|b|>L\}$ are open, the portmanteau theorem \cite[Theorem~2.1]{Billingsley1999convergence} shows that $\nu$ satisfies (C) and (T). For an open interval $J\subset(-\rho,\rho)$, we have $\nu(J)\le\liminf_n\nu_n(J)\le\mathfrak h_p|J|$ by the same theorem. Since every open subset of $(-\rho,\rho)$ is a countable disjoint union of open intervals, we obtain $\nu(U)\le\mathfrak h_p|U|$ for open $U\subset(-\rho,\rho)$, and by outer regularity of Lebesgue measure, $\nu(A)\le\mathfrak h_p|A|$ for every Borel set $A\subset(-\rho,\rho)$. By the Radon--Nikodym theorem, $\nu$ satisfies (B). For (ii), let $\mu\in\mathscr A_p(\mathbf q)$ and write $\sigma_n=\mu-\nu_n$ and $\sigma=\mu-\nu$. For each smooth $\varphi$ with $\varphi(1)=0$, the function $\chi_{\mathrm c}\varphi$ is bounded and continuous, and we have $\langle\Pi_{\mathrm c}\sigma_n,\varphi\rangle=\langle\sigma_n,\chi_{\mathrm c}\varphi\rangle\to\langle\sigma,\chi_{\mathrm c}\varphi\rangle=\langle\Pi_{\mathrm c}\sigma,\varphi\rangle$. By part (iv) of \Cref{lem:stab-mixed-basic}, $n_{\mathrm c}(\sigma)$ is a supremum of such limits, which implies that $n_{\mathrm c}(\sigma)\le\liminf_n n_{\mathrm c}(\sigma_n)$. The same argument applies to $n_{\mathrm i}$, using the bounded continuous functions $\chi_{\mathrm i}(\varphi-\varphi(1))$. For $n_{\mathrm b}$, let $f_n,f$ be the densities of $\nu_n,\nu$ on $(-\rho,\rho)$ and $f_\mu$ that of $\mu$. Since $L^\infty$ is the dual of $L^1$ and continuous functions with compact support are dense in $L^1$, we have \[ n_{\mathrm b}(\sigma)=\|\chi_{\mathrm b}(f_\mu-f)\|_{L^\infty}=\sup\Big\{\Big|\int\chi_{\mathrm b}(f_\mu-f)\,u\Big|:\ u\in C_c((-\rho,\rho)),\ \|u\|_{L^1}\le1\Big\}. \] For each such $u$, the function $\chi_{\mathrm b}u$ is bounded and continuous, and $\int\chi_{\mathrm b}f_nu=\langle\nu_n,\chi_{\mathrm b}u\rangle\to\langle\nu,\chi_{\mathrm b}u\rangle=\int\chi_{\mathrm b}fu$. We conclude that $n_{\mathrm b}(\sigma)\le\liminf_nn_{\mathrm b}(\sigma_n)$. Combining the three components proves (ii). For (iii), let $(\nu_n)$ be a Cauchy sequence for $d_p$ in $\mathscr A_p(\mathbf q)$. By (i) and Prohorov's theorem \cite[Theorem~5.1]{Billingsley1999convergence}, a subsequence $(\nu_{n_k})$ converges weakly to some $\nu\in\mathscr A_p(\mathbf q)$. By (ii), for each $n$, \[ d_p(\nu_n,\nu)\le\liminf_{k\to\infty}d_p(\nu_n,\nu_{n_k})\le\sup_{m\ge n}d_p(\nu_n,\nu_m), \] which tends to zero as $n\to\infty$. Hence $\nu_n\to\nu$ in $d_p$. Now suppose $d_p(\nu_n,\nu)\to0$ with $\nu_n,\nu\in\mathscr A_p(\mathbf q)$. By (i), every subsequence has a further subsequence converging weakly to some $\nu'\in\mathscr A_p(\mathbf q)$, and by (ii), $d_p(\nu',\nu)\le\liminf d_p(\nu_{n_k},\nu)=0$. Then $\nu'=\nu$, and $\nu_n\Rightarrow\nu$. For (iv), let $G_n$ and $G$ be the quantile functions of $\nu_n$ and $\nu$, and let $U_1,U_2,\dots$ be independent uniform variables on $(0,1)$, independent of $N$ and $(Y_j)$. Then $B_{n,j}=G_n(U_j)$ has law $\nu_n$, $B_j=G(U_j)$ has law $\nu$, and $B_{n,j}\to B_j$ almost surely for each $j$, because $G_n\to G$ at every continuity point of $G$ and $G$ has countably many discontinuities (this is the quantile form of Skorokhod's representation theorem; see \cite[Theorem~6.7]{Billingsley1999convergence}). Since $B_j\neq0$ almost surely, \[ 1+\theta_n-\frac{\theta_n}{p}\sum_{j\le N}\frac{Y_j}{B_{n,j}}\longrightarrow1+\theta-\frac{\theta}{p}\sum_{j\le N}\frac{Y_j}{B_j} \] almost surely, and almost sure convergence implies convergence in distribution. \end{proof} \begin{remark}\label{rem:stab-not-complete} The class $\mathfrak S_{p,\theta}(\mathbf q)$ is not complete for $d_p$. For example, let $\nu_n=e^{-p}\delta_{1+\theta}+(1-e^{-p})u_n$, where $u_n$ is the uniform law on $[1-1/n,1+1/n]$. If $\theta\le r_p$ and $n\ge1/r_p$, then $\nu_n\in\mathfrak S_{p,\theta}(\mathbf q)$, and $d_p(\nu_n,\nu_m)=n_{\mathrm c}(\nu_n-\nu_m)\le1/n+1/m$, because $\nu_n-\nu_m$ is carried by $\mathcal C_p$, where $\chi_{\mathrm c}=1$. The $d_p$-limit $e^{-p}\delta_{1+\theta}+(1-e^{-p})\delta_1$ has a second atom and is not in $\mathfrak S_{p,\theta}(\mathbf q)$. The core and intermediate norms metrize weak convergence on their regions, and weak limits of absolutely continuous laws may have atoms. The boundary norm is a supremum norm of densities, and its limits keep a density; this is why atoms are allowed in the ambient class exactly outside the open interval $(-\rho,\rho)$, which contains the boundary region. The law map sends $\mathscr A_p(\mathbf q)$ back into $\mathfrak S_{p,\theta}(\mathbf q)$ by \Cref{lem:stab-invariance}, and in particular fixed points in the ambient class belong to $\mathfrak S_{p,\theta}(\mathbf q)$. \end{remark} \section{Column estimates for the linearized map}\label{sec:stab-columns} Throughout this subsection, $\mathbf q=(a,A,K',\rho)$ is admissible, $C_0>0$, $p\ge p_*(\mathbf q,C_0)$, $0<\theta\le C_0/p$, and $\lambda\in\mathscr A_p(\mathbf q)$ is arbitrary. We use the notation of \Cref{lem:stab-linearization}: $\mu_\lambda=\mathcal R_{p,\theta}(\lambda)$, $D_\lambda\sim\mu_\lambda$, $T\sim\Exp(1)$ independent of $D_\lambda$, $Z_b=D_\lambda-\theta T/(pb)$, and $k^\lambda_b$ is the density of $Z_b$. By \Cref{lem:stab-invariance}, $\mu_\lambda$ has a density $h_\lambda\le\mathfrak h_p/2$ on $[-2\rho,2\rho]$ and satisfies $\mu_\lambda(\{d:|d-1|>r_p/2\})\le\omega_p$. In applications, $\lambda$ is a point of a segment joining two members of $\mathscr A_p(\mathbf q)$, or a fixed point of the law map; all estimates below are uniform in $\lambda\in\mathscr A_p(\mathbf q)$. \begin{lemma}\label{lem:stab-kernel} Under the assumptions above, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $b$ with $|b|\ge\rho_0$, we have $\P(|Z_b-1|\ge r_p)+\E\big[T\one_{\{|Z_b-1|\ge r_p\}}\big]\le3\omega_p$. \item For every $x\in[-\rho_1,\rho_1]$, the function $b\mapsto k^\lambda_b(x)$ is continuously differentiable on $(-\infty,-\rho_0]$ and on $[\rho_0,\infty)$, and there \[ 0\le k^\lambda_b(x)\le\mathfrak h_p,\qquad\big|\partial_bk^\lambda_b(x)\big|\le\mathfrak h_p/\rho_0 . \] \item For every $x\in[-\rho_1,\rho_1]$ and every $g\in L^\infty$ vanishing outside $[-\rho_1,\rho_1]$, \[ \Big|p\int k^\lambda_b(x)\,g(b)\dd b\Big|\le\Big(2\theta+\rho p\mathfrak h_p+\frac{2\theta\omega_p}{\rho^2}\Big)\|g\|_{L^\infty}. \] \end{enumerate} \end{lemma} \begin{proof} For (i), let $|b|\ge\rho_0$. On the event $|D_\lambda-1|\le r_p/2$, the inequality $|Z_b-1|\ge r_p$ forces $\theta T/(p|b|)\ge r_p/2$, that is, $T\ge p|b|r_p/(2\theta)\ge t_p$, where we used $|b|\ge\rho_0$ and $\theta\le C_0/p$. Since $T$ is independent of $D_\lambda$, we obtain \begin{align*} \P(|Z_b-1|\ge r_p)+\E\big[T\one_{\{|Z_b-1|\ge r_p\}}\big]&\le\omega_p+e^{-t_p}+\E[T]\,\omega_p+\E\big[T\one_{\{T\ge t_p\}}\big]\\ &=2\omega_p+(t_p+2)e^{-t_p}, \end{align*} which is at most $3\omega_p$ by \ref{it:stab-S3}. For (ii), let $e_s(u)=se^{-su}$ for $s>0$ and $u\ge0$. Let $b\ge\rho_0$; the case $b\le-\rho_0$ is identical after the reflection $d-x\mapsto x-d$ in \eqref{eq:stab-kernel}. By \eqref{eq:stab-kernel}, $k^\lambda_b(x)=\int_{(x,\infty)}e_{pb/\theta}(d-x)\,\mu_\lambda(\mathrm dd)$, and \[ \partial_b\,e_{pb/\theta}(u)=\frac p\theta\Big(1-\frac{pbu}{\theta}\Big)e^{-pbu/\theta}. \] This derivative is bounded by $p/\theta$ and continuous in $b$. By dominated convergence, $b\mapsto k^\lambda_b(x)$ is continuously differentiable with derivative $\int_{(x,\infty)}\partial_be_{pb/\theta}(d-x)\,\mu_\lambda(\mathrm dd)$. We split the integrals into $d\in(x,2\rho]$ and $d>2\rho$. On $(x,2\rho]\subset[-2\rho,2\rho]$, the measure $\mu_\lambda$ has density at most $\mathfrak h_p/2$, and this part of $k^\lambda_b(x)$ is at most $(\mathfrak h_p/2)\int_0^\infty e_{pb/\theta}=\mathfrak h_p/2$, and this part of the derivative is at most \[ \frac{\mathfrak h_p}{2}\int_0^\infty\frac p\theta\Big|1-\frac{pbu}{\theta}\Big|e^{-pbu/\theta}\dd u=\frac{\mathfrak h_p}{2b}\int_0^\infty|1-v|e^{-v}\dd v=\frac{\mathfrak h_p}{eb}\le\frac{\mathfrak h_p}{2\rho_0}. \] If $d>2\rho$, then $u=d-x\ge2\rho-\rho_1\ge\rho$. Let $S=p/\theta$, $w=Sbu$, and $x_0=\rho\rho_0S$. Since $S\ge p^2/C_0$, we have $x_0\ge x_p\ge2$ by \ref{it:stab-S7}, and since $b\ge\rho_0$ and $u\ge\rho$, we have $w\ge x_0$. The functions $we^{-w}$ and $(1+w)e^{-w}$ are decreasing on $[1,\infty)$, and $x^2e^{-x}$ is decreasing on $[2,\infty)$. Hence \begin{align*} e_{pb/\theta}(u)&=\frac wue^{-w}\le\frac{x_0e^{-x_0}}{\rho}\le\frac{2x_p^2e^{-x_p}}{\rho\rho_0},\\ \big|\partial_be_{pb/\theta}(u)\big|&\le S(1+w)e^{-w}\le\frac{x_0(1+x_0)e^{-x_0}}{\rho\rho_0}\le\frac{2x_p^2e^{-x_p}}{\rho\rho_0}, \end{align*} and the right sides are at most $\mathfrak h_p/2$ by \ref{it:stab-S7}. Since $\mu_\lambda$ is a probability measure, the parts of $k^\lambda_b(x)$ and of its derivative coming from $d>2\rho$ are at most $\mathfrak h_p/2$. Adding the two parts proves (ii). For (iii), consider first the contribution of $b\in(0,\rho_1]$. We split $(x,\infty)$ into $P_1=(x,2\rho]$, $P_2=(2\rho,\infty)\cap\{|d-1|>r_p/2\}$, and $P_3=[1-r_p/2,1+r_p/2]$; these sets cover $(x,\infty)$ because $1-r_p/2>2\rho$. As in the proof of (ii), the part of $p\,k^\lambda_b(x)$ coming from $P_1$ is at most $p\mathfrak h_p/2$, so the contribution of $P_1$ is at most $p\rho_1\mathfrak h_p\|g\|_{L^\infty}/2$. For $P_2\cup P_3$, we use Tonelli's theorem and \[ \int_0^\infty p\,e_{pb/\theta}(u)\dd b=\int_0^\infty\frac{p^2b}{\theta}e^{-pbu/\theta}\dd b=\frac{\theta}{u^2}, \] so that the contribution of $P_2\cup P_3$ is at most $\|g\|_{L^\infty}\int_{P_2\cup P_3}\theta(d-x)^{-2}\mu_\lambda(\mathrm dd)$. On $P_2$ we have $d-x\ge\rho$ and $\mu_\lambda(P_2)\le\omega_p$, and on $P_3$ we have $d-x\ge1-r_p/2-\rho_1\ge27/32$, where we used $r_p\le1/4$ and $\rho\le1/16$. Since $(32/27)^2\le2$, this contribution is at most $(\theta\omega_p/\rho^2+2\theta)\|g\|_{L^\infty}$. For $b\in[-\rho_1,0)$, the kernel integrates over $dr_p/2\}$. By the same arguments, the contribution of $b\in[-\rho_1,0)$ is at most $(p\rho_1\mathfrak h_p/2+\theta\omega_p/\rho^2)\|g\|_{L^\infty}$. Adding the contributions and using $\rho_1=\rho/2$ proves (iii). \end{proof} For $\alpha,\beta\in\{\mathrm c,\mathrm b,\mathrm i\}$, the quantity $n_\alpha(\mathcal L\Pi_\beta\sigma)$ below measures the $\alpha$ component of the output produced by the $\beta$ component of the input. Recall from \Cref{def:stab-mixed} that $n_\beta(\sigma)=\|\Pi_\beta\sigma\|_\beta$ is the norm of the measure $\Pi_\beta\sigma$ itself; the compensated projections are not idempotent, and we never project an input twice. \begin{lemma}\label{lem:stab-columns} Let $\mathbf q$ be admissible, $C_0>0$, $p\ge p_*(\mathbf q,C_0)$, $0<\theta\le C_0/p$, $\lambda\in\mathscr A_p(\mathbf q)$, and $\mathcal L=\mathcal L_{p,\theta;\lambda}$. Let $\mathscr D_p$ be the real vector space of finite signed measures $\sigma$ on $\R$ with $\sigma(\R)=0$ whose restriction to $(-\rho_1,\rho_1)$ is absolutely continuous with bounded density. Then $\Pi_\alpha\sigma\in\mathscr D_p$ and $\mathcal L\sigma\in\mathscr D_p$ for $\sigma\in\mathscr D_p$, and with $\delta_p=p^{A+3}e^{-ap}$ and \begin{equation}\label{eq:stab-matrix} M_{p,\theta}=\begin{pmatrix}M_{\mathrm{cc}}&M_{\mathrm{cb}}&M_{\mathrm{ci}}\\ M_{\mathrm{bc}}&M_{\mathrm{bb}}&M_{\mathrm{bi}}\\ M_{\mathrm{ic}}&M_{\mathrm{ib}}&M_{\mathrm{ii}}\end{pmatrix} =\begin{pmatrix}20\theta&p&6p\\ \delta_p&2\theta+\delta_p&\delta_p\\ \delta_p&p&\delta_p\end{pmatrix}, \end{equation} we have, for every $\sigma\in\mathscr D_p$ and all $\alpha,\beta\in\{\mathrm c,\mathrm b,\mathrm i\}$, \begin{equation}\label{eq:stab-columns} n_\alpha(\mathcal L\Pi_\beta\sigma)\le M_{\alpha\beta}\,n_\beta(\sigma), \qquad\text{and hence}\qquad n_\alpha(\mathcal L\sigma)\le\sum_{\beta}M_{\alpha\beta}\,n_\beta(\sigma). \end{equation} \end{lemma} \begin{proof} The proof is a direct computation with the dual formula \eqref{eq:stab-L-dual} for the core and intermediate outputs and the density formula of \Cref{lem:stab-linearization} for the boundary output, combined with \Cref{lem:stab-kernel}. We first record these formulas. \proofstep{1}{The three outputs.} Let $\tau\in\mathscr D_p$. Then $|\tau|(\{0\})=0$, and by \Cref{lem:stab-linearization}, $\mathcal L\tau$ has total mass zero and density $G_\tau(x)=p\int k^\lambda_b(x)\,\tau(\mathrm db)$. Then $\Pi_{\mathrm b}\mathcal L\tau=\chi_{\mathrm b}G_\tau\dd x-m_{\mathrm b}(\mathcal L\tau)\delta_1$ and \begin{equation}\label{eq:stab-out-b} n_{\mathrm b}(\mathcal L\tau)\le\sup_{|x|\le\rho_1}|G_\tau(x)|. \end{equation} For a $C^\infty$ function $\psi$ with bounded derivative, set $F_\psi(b)=p\,\E\psi(Z_b)$ for $b\neq0$. Since $\partial_bZ_b=\theta T/(pb^2)$, dominated convergence shows that $F_\psi$ is continuously differentiable on $(-\infty,0)$ and on $(0,\infty)$, with \begin{equation}\label{eq:stab-Fprime} F_\psi'(b)=\frac{\theta}{b^2}\E\big[T\,\psi'(Z_b)\big]. \end{equation} By parts (ii) and (iv) of \Cref{lem:stab-mixed-basic} and \eqref{eq:stab-L-dual}, we have \begin{equation}\label{eq:stab-out-ci} n_{\mathrm c}(\mathcal L\tau)=\sup_{\varphi}\Big|\int F_{\chi_{\mathrm c}\varphi}\dd\tau\Big|,\qquad n_{\mathrm i}(\mathcal L\tau)=\sup_{\varphi'}\Big|\int F_{\chi_{\mathrm i}(\varphi'-\varphi'(1))}\dd\tau\Big|, \end{equation} where $\varphi$ ranges over $C^\infty$ functions with $\varphi(1)=0$ and $\Lip(\varphi)\le1$, and $\varphi'$ over $C^\infty$ functions with $\|\varphi'\|_\infty\le1$ and $\Lip(\varphi')\le1$. We write $\psi_{\mathrm c}=\chi_{\mathrm c}\varphi$ and $\psi_{\mathrm i}=\chi_{\mathrm i}(\varphi'-\varphi'(1))$, and we recall the bounds of part (iii) of \Cref{lem:stab-mixed-basic}. Since $\psi_{\mathrm i}$ is smooth and vanishes on $\mathcal C_p$, its derivative vanishes on $\mathcal C_p$, so that \begin{equation}\label{eq:stab-psii} |\psi_{\mathrm i}(z)|\le2\cdot\one_{\{|z-1|\ge r_p\}},\qquad|\psi_{\mathrm i}'(z)|\le\frac{21}{\rho}\one_{\{|z-1|\ge r_p\}}. \end{equation} \proofstep{2}{Core inputs.} Let $\tau=\Pi_{\mathrm c}\sigma$, which is carried by $[1-2r_p,1+2r_p]\subset[1/2,3/2]$ and satisfies $\|\tau\|_{\mathrm c}=n_{\mathrm c}(\sigma)$. By \eqref{eq:stab-Fprime} and $\Lip(\psi_{\mathrm c})\le5$, we have $|F_{\psi_{\mathrm c}}'(b)|\le20\theta$ for $b\in[1/2,3/2]$. Part (v) of \Cref{lem:stab-mixed-basic} and \eqref{eq:stab-out-ci} give $n_{\mathrm c}(\mathcal L\tau)\le20\theta\,n_{\mathrm c}(\sigma)$. For $|x|\le\rho_1$, part (ii) of \Cref{lem:stab-kernel} shows that $b\mapsto p\,k^\lambda_b(x)$ is $p\mathfrak h_p/\rho_0$-Lipschitz on $[1/2,3/2]$. By part (v) of \Cref{lem:stab-mixed-basic}, we have $|G_\tau(x)|\le(p\mathfrak h_p/\rho_0)n_{\mathrm c}(\sigma)$, and \eqref{eq:stab-out-b} gives $n_{\mathrm b}(\mathcal L\tau)\le(p\mathfrak h_p/\rho_0)n_{\mathrm c}(\sigma)$. Finally, \eqref{eq:stab-Fprime}, \eqref{eq:stab-psii}, and part (i) of \Cref{lem:stab-kernel} give $|F_{\psi_{\mathrm i}}'(b)|\le4\theta(21/\rho)\cdot3\omega_p$ for $b\in[1/2,3/2]$, which implies that $n_{\mathrm i}(\mathcal L\tau)\le(252\theta\omega_p/\rho)\,n_{\mathrm c}(\sigma)$. \proofstep{3}{Boundary inputs.} Let $\tau=\Pi_{\mathrm b}\sigma=g\dd b-(\int g)\delta_1$, where $g=\chi_{\mathrm b}f_\sigma$ vanishes outside $[-\rho_1,\rho_1]$ and $\|g\|_{L^\infty}=n_{\mathrm b}(\sigma)$. Then $\|\tau\|_{\TV}\le2\|g\|_{L^1}\le2\rho\,n_{\mathrm b}(\sigma)$. Since $|F_{\psi_{\mathrm c}}|\le2pr_p$ and $|F_{\psi_{\mathrm i}}|\le2p$, \eqref{eq:stab-out-ci} gives $n_{\mathrm c}(\mathcal L\tau)\le4\rho pr_p\,n_{\mathrm b}(\sigma)\le p\,n_{\mathrm b}(\sigma)$ and $n_{\mathrm i}(\mathcal L\tau)\le4\rho p\,n_{\mathrm b}(\sigma)\le p\,n_{\mathrm b}(\sigma)$. For the boundary output, we have \[ G_\tau(x)=p\int k^\lambda_b(x)g(b)\dd b-\Big(\int g\Big)p\,k^\lambda_1(x). \] By part (iii) of \Cref{lem:stab-kernel}, the first term is at most $(2\theta+\rho p\mathfrak h_p+2\theta\omega_p/\rho^2)\|g\|_{L^\infty}$ in absolute value for $|x|\le\rho_1$. By part (ii) of the same lemma with $b=1$, and $|\int g|\le2\rho_1\|g\|_{L^\infty}=\rho\|g\|_{L^\infty}$, the second term is at most $\rho p\mathfrak h_p\|g\|_{L^\infty}$. Combining these bounds with \eqref{eq:stab-out-b}, we obtain \[ n_{\mathrm b}(\mathcal L\tau)\le\Big(2\theta+2\rho p\mathfrak h_p+\frac{2\theta\omega_p}{\rho^2}\Big)n_{\mathrm b}(\sigma). \] \proofstep{4}{Intermediate inputs.} Let $\tau=\Pi_{\mathrm i}\sigma$, which is carried by the closed set $I_p$ and satisfies $\|\tau\|_{\mathrm i}=n_{\mathrm i}(\sigma)$. Every $b\in I_p$ satisfies $|b|\ge\rho_0$. On $I_p$, we have $|F_{\psi_{\mathrm c}}|\le2pr_p$, and $|F_{\psi_{\mathrm c}}'|\le5\theta/\rho_0^2$ on each piece of $I_p$. By part (v) of \Cref{lem:stab-mixed-basic}, $\Lip_{I_p}(F_{\psi_{\mathrm c}})\le5\theta/\rho_0^2+4p$, and therefore \[ n_{\mathrm c}(\mathcal L\tau)\le\big(2pr_p+5\theta\rho_0^{-2}+4p\big)n_{\mathrm i}(\sigma)\le6p\,n_{\mathrm i}(\sigma), \] where we used $r_p\le1/4$ and $5\theta\rho_0^{-2}\le80C_0/(\rho^2p)\le p/2$, which follows from \ref{it:stab-S8}. For $|x|\le\rho_1$, part (ii) of \Cref{lem:stab-kernel} and part (v) of \Cref{lem:stab-mixed-basic} show that the bounded--Lipschitz norm of $b\mapsto p\,k^\lambda_b(x)$ on $I_p$ is at most $p\mathfrak h_p(1+\rho_0^{-1}+2r_p^{-1})$. Together with \eqref{eq:stab-out-b}, this implies that $n_{\mathrm b}(\mathcal L\tau)\le p\mathfrak h_p(1+\rho_0^{-1}+2r_p^{-1})\,n_{\mathrm i}(\sigma)$. Finally, \eqref{eq:stab-psii} and part (i) of \Cref{lem:stab-kernel} give $|F_{\psi_{\mathrm i}}|\le6p\omega_p$ on $I_p$ and $|F_{\psi_{\mathrm i}}'|\le63\theta\omega_p/(\rho\rho_0^2)$ on each piece, so that \[ n_{\mathrm i}(\mathcal L\tau)\le\Big(6p\omega_p(1+2r_p^{-1})+\frac{63\theta\omega_p}{\rho\rho_0^2}\Big)n_{\mathrm i}(\sigma). \] \proofstep{5}{The small entries.} By \ref{it:stab-S3}, $\omega_p\le3\rho p\mathfrak h_p$, and $\theta\le C_0/p$. Using these bounds, $\rho\le1$, $\rho_0=\rho/4$, and $r_p^{-1}=p^{1/4}$, each of the five coefficients \[ \frac{p\mathfrak h_p}{\rho_0},\quad\frac{252\theta\omega_p}{\rho},\quad2\rho p\mathfrak h_p+\frac{2\theta\omega_p}{\rho^2},\quad p\mathfrak h_p\Big(1+\frac1{\rho_0}+\frac2{r_p}\Big),\quad6p\omega_p\Big(1+\frac2{r_p}\Big)+\frac{63\theta\omega_p}{\rho\rho_0^2} \] obtained in Steps 2--4 is at most $3100(1+C_0)\rho^{-2}p^{9/4}\mathfrak h_p$. For instance, the last one is at most $54\rho p^{9/4}\mathfrak h_p+3024C_0\rho^{-2}\mathfrak h_p$. By \ref{it:stab-S8}, each of them is at most $p^3\mathfrak h_p=\delta_p$. This proves the first inequality in \eqref{eq:stab-columns}. \proofstep{6}{Conclusion.} Let $\sigma\in\mathscr D_p$. The projections $\Pi_\alpha\sigma$ have zero mass, and their restrictions to $(-\rho_1,\rho_1)$ are $0$, $\chi_{\mathrm b}f_\sigma\dd b$, and $\chi_{\mathrm i}f_\sigma\dd b$, so they belong to $\mathscr D_p$. The measure $\mathcal L\sigma$ has zero mass and is absolutely continuous, and the bounds on $\sup_{|x|\le\rho_1}|G_{\Pi_\beta\sigma}(x)|$ obtained in Steps 2--4 show that its density is bounded on $(-\rho_1,\rho_1)$, since $n_\beta(\sigma)<\infty$ for $\sigma\in\mathscr D_p$ (see the proof of part (vi) of \Cref{lem:stab-mixed-basic}). It follows that $\mathcal L\sigma\in\mathscr D_p$. Since $\sigma=\sum_\beta\Pi_\beta\sigma$ and $\Pi_\alpha\mathcal L$ is linear, we have $n_\alpha(\mathcal L\sigma)\le\sum_\beta n_\alpha(\mathcal L\Pi_\beta\sigma)$, and the second inequality in \eqref{eq:stab-columns} follows from the first. \end{proof} \begin{remark}\label{rem:stab-completion} The estimates of \Cref{lem:stab-columns} are linear and apply to arbitrary inputs in $\mathscr D_p$, not only to differences of probability measures; the side law $\lambda$ enters only through $\mu_\lambda=\mathcal R_{p,\theta}(\lambda)$. At a fixed point $\nu=\mathcal R_{p,\theta}(\nu)$ in $\mathscr A_p(\mathbf q)$, we have $\mathcal L_{p,\theta;\nu}\sigma=p\,\nu*(\mathsf K_\theta\sigma)^\vee$. By part (vi) of \Cref{lem:stab-mixed-basic}, $N_p$ is a norm on $\mathscr D_p$. Let $\mathscr Y_p$ be the completion of $(\mathscr D_p,N_p)$. Since $n_\alpha\le W_\alpha^{-1}N_p$, each component seminorm $n_\alpha$ extends uniquely to a continuous seminorm on $\mathscr Y_p$, and \Cref{lem:stab-contraction} below shows that $\mathcal L$ is bounded on $(\mathscr D_p,N_p)$. Then $\mathcal L$ extends uniquely to a bounded operator on $\mathscr Y_p$. By density and continuity, the inequalities \eqref{eq:stab-columns}, in their second form, and the bound $N_p(\mathcal L\sigma)\le\kappa_{p,\theta}N_p(\sigma)$ of \Cref{lem:stab-contraction}, continue to hold on $\mathscr Y_p$. \end{remark} \section{The principal fixed point}\label{sec:stab-fixed} \begin{lemma}\label{lem:stab-contraction} Let $\mathbf q$ be admissible, $C_0>0$, $p\ge p_*(\mathbf q,C_0)$, and $0<\theta\le C_0/p$. Set \[ \kappa_{p,\theta}=20\theta+7pe^{-a\sqrt p/2}. \] Then $\kappa_{p,\theta}\le20C_0/p+7pe^{-a\sqrt p/2}\le1/2$, and the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $\lambda\in\mathscr A_p(\mathbf q)$ and $\sigma\in\mathscr D_p$, we have $N_p(\mathcal L_{p,\theta;\lambda}\sigma)\le\kappa_{p,\theta}N_p(\sigma)$. \item For all $\nu,\mu\in\mathscr A_p(\mathbf q)$, we have $d_p(\mathcal R_{p,\theta}(\nu),\mathcal R_{p,\theta}(\mu))\le\kappa_{p,\theta}\,d_p(\nu,\mu)$. \end{enumerate} \end{lemma} \begin{proof} The bound on $\kappa_{p,\theta}$ is \ref{it:stab-S9}. For (i), by \Cref{lem:stab-columns}, we have \[ N_p(\mathcal L\sigma)=\sum_\alpha W_\alpha n_\alpha(\mathcal L\sigma)\le\sum_\alpha W_\alpha\sum_\beta M_{\alpha\beta}n_\beta(\sigma)=\sum_\beta c_\beta\,W_\beta n_\beta(\sigma),\qquad c_\beta=\sum_\alpha\frac{W_\alpha M_{\alpha\beta}}{W_\beta}. \] It suffices to show that $c_\beta\le\kappa_{p,\theta}$ for each $\beta$. By \ref{it:stab-S9}, $2p^{A+3}e^{-ap/2}\le pe^{-a\sqrt p/2}$; since $\delta_p\le p^{A+3}e^{-ap/2}$ and $\delta_pW_{\mathrm b}=p^{A+3}e^{-ap/2}$, each of $\delta_p$, $\delta_pW_{\mathrm b}$, and $\delta_pW_{\mathrm i}$ is at most $pe^{-a\sqrt p/2}/2$. Using \eqref{eq:stab-matrix}, we find \begin{align*} c_{\mathrm c}&=20\theta+(W_{\mathrm b}+W_{\mathrm i})\delta_p\le20\theta+pe^{-a\sqrt p/2},\\ c_{\mathrm b}&=pe^{-ap/2}+2\theta+\delta_p+pe^{-a(p-\sqrt p)/2}\le2\theta+3pe^{-a\sqrt p/2},\\ c_{\mathrm i}&=6pe^{-a\sqrt p/2}+(W_{\mathrm b}+W_{\mathrm i})W_{\mathrm i}^{-1}\delta_p\le7pe^{-a\sqrt p/2}, \end{align*} where in the second line we used $p-\sqrt p\ge\sqrt p$. This proves (i). For (ii), let $\sigma=\nu-\mu$ and $\lambda_t=(1-t)\mu+t\nu$. Then $\sigma\in\mathscr D_p$ by (B), and $\lambda_t\in\mathscr A_p(\mathbf q)$ by convexity. We write $\mathcal L_t=\mathcal L_{p,\theta;\lambda_t}$ and $\zeta=\mathcal R_{p,\theta}(\nu)-\mathcal R_{p,\theta}(\mu)$. For a smooth $\varphi$ with $\varphi(1)=0$ and $\Lip(\varphi)\le1$, \eqref{eq:stab-segment} and part (ii) of \Cref{lem:stab-mixed-basic} give \[ \langle\Pi_{\mathrm c}\zeta,\varphi\rangle=\langle\zeta,\chi_{\mathrm c}\varphi\rangle=\int_0^1\langle\mathcal L_t\sigma,\chi_{\mathrm c}\varphi\rangle\dd t=\int_0^1\langle\Pi_{\mathrm c}\mathcal L_t\sigma,\varphi\rangle\dd t, \] so that $n_{\mathrm c}(\zeta)\le\sup_{t}n_{\mathrm c}(\mathcal L_t\sigma)$. The same argument gives $n_{\mathrm i}(\zeta)\le\sup_tn_{\mathrm i}(\mathcal L_t\sigma)$. By \Cref{lem:stab-invariance}, $\zeta$ has a density on $(-\rho,\rho)$, and if $G_t$ denotes the density of $\mathcal L_t\sigma$, then \eqref{eq:stab-segment} with $\psi=\chi_{\mathrm b}u$, for bounded Borel $u$ supported in $(-\rho_1,\rho_1)$, gives $|\langle\zeta,\chi_{\mathrm b}u\rangle|\le\sup_t\|\chi_{\mathrm b}G_t\|_{L^\infty}\|u\|_{L^1}$. This implies that $n_{\mathrm b}(\zeta)\le\sup_tn_{\mathrm b}(\mathcal L_t\sigma)$. Since the bounds of \Cref{lem:stab-columns} are uniform in $t$, the computation in (i) gives $N_p(\zeta)\le\kappa_{p,\theta}N_p(\sigma)$. \end{proof} \begin{theorem}\label{thm:stab-fixed} Let $\mathbf q=(a,A,K',\rho)$ be admissible, $C_0>0$, and $p\ge p_*(\mathbf q,C_0)$. For every $\theta\in(0,C_0/p]$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item The law map $\mathcal R_{p,\theta}$ has exactly one fixed point in $\mathscr A_p(\mathbf q)$. This fixed point, denoted $\nu_p^\theta$, belongs to $\mathfrak S_{p,\theta}(\mathbf q)$; in particular, it is the unique fixed point in $\mathfrak S_{p,\theta}(\mathbf q)$. It has the properties (i)--(iii) of \Cref{lem:stab-invariance}. \item For every $\mu\in\mathscr A_p(\mathbf q)$, in particular for $\mu=\delta_1$, we have $d_p(\mathcal R_{p,\theta}^n(\mu),\nu^\theta_p)\le\kappa_{p,\theta}^n\,d_p(\mu,\nu_p^\theta)$ for $n\ge0$, and $\mathcal R_{p,\theta}^n(\mu)\Rightarrow\nu_p^\theta$. \item Let $\mathbf q'=(a',A',K'',\rho')$ be another admissible tuple, and let \[ \mathbf q\vee\mathbf q'=\big(\max\{a,a'\},\min\{A,A'\},\min\{K',K''\},\max\{\rho,\rho'\}\big). \] If $p\ge\max\{p_*(\mathbf q,C_0),p_*(\mathbf q',C_0),p_*(\mathbf q\vee\mathbf q',C_0)\}$, then the fixed points given by \textup{(i)} for $\mathbf q$ and for $\mathbf q'$ coincide. \item The map $\theta\mapsto\nu_p^\theta$ is weakly continuous on $(0,C_0/p]$. \item For every $s\in(0,1)$, we have \[ \int|b|^{-s}\,\nu_p^\theta(\mathrm db)\le\rho^{-s}+\frac{2\mathfrak h_p\rho^{1-s}}{1-s},\qquad\int|b|^{s}\,\nu_p^\theta(\mathrm db)\le4+\frac{K'\mathfrak h_p}{1-s}, \] and \[ \int|b-1|^s\,\nu_p^\theta(\mathrm db)\le\theta^s\Big(1+p^{1-s}\int|b|^{-s}\,\nu_p^\theta(\mathrm db)\Big). \] In particular, for fixed $p$, $\nu_p^\theta\Rightarrow\delta_1$ as $\theta\downarrow0$. \end{enumerate} \end{theorem} \begin{proof} By \Cref{lem:stab-complete}, $(\mathscr A_p(\mathbf q),d_p)$ is a complete metric space. By \Cref{lem:stab-invariance,lem:stab-contraction}, $\mathcal R_{p,\theta}$ maps it into $\mathfrak S_{p,\theta}(\mathbf q)\subset\mathscr A_p(\mathbf q)$ and is a contraction with factor $\kappa_{p,\theta}\le1/2$. The Banach fixed-point theorem gives a unique fixed point $\nu_p^\theta$ in $\mathscr A_p(\mathbf q)$, and $\nu_p^\theta=\mathcal R_{p,\theta}(\nu_p^\theta)$ belongs to $\mathfrak S_{p,\theta}(\mathbf q)$ and has the properties of \Cref{lem:stab-invariance}. A fixed point in $\mathfrak S_{p,\theta}(\mathbf q)$ lies in $\mathscr A_p(\mathbf q)$ and therefore equals $\nu_p^\theta$. For (ii), we have $d_p(\mathcal R^n_{p,\theta}(\mu),\nu_p^\theta)=d_p(\mathcal R^n_{p,\theta}(\mu),\mathcal R^n_{p,\theta}(\nu_p^\theta))\le\kappa_{p,\theta}^nd_p(\mu,\nu_p^\theta)$, which is finite by part (vi) of \Cref{lem:stab-mixed-basic}, and weak convergence follows from part (iii) of \Cref{lem:stab-complete}. Since $\delta_1\in\mathscr A_p(\mathbf q)$, this applies to $\mu=\delta_1$. For (iii), let $\mathbf q''=\mathbf q\vee\mathbf q'$. Comparing the bounds in \Cref{def:stab-class} term by term, we see that $\mathfrak S_{p,\theta}(\mathbf q'')\subset\mathfrak S_{p,\theta}(\mathbf q)\cap\mathfrak S_{p,\theta}(\mathbf q')$: a larger $a$ and smaller $A$ and $K'$ make the right sides of (C), (B), and (T) smaller, and a larger $\rho$ enlarges the interval in (B). Then the fixed point given by (i) for $\mathbf q''$ is a fixed point in $\mathfrak S_{p,\theta}(\mathbf q)$ and in $\mathfrak S_{p,\theta}(\mathbf q')$, and by uniqueness it equals both of the other fixed points. For (iv), let $\theta_n\to\theta$ in $(0,C_0/p]$ and $\nu_n=\nu_p^{\theta_n}$. By part (i) of \Cref{lem:stab-complete}, every subsequence of $(\nu_n)$ has a further subsequence converging weakly to some $\nu'\in\mathscr A_p(\mathbf q)$. Along it, part (iv) of \Cref{lem:stab-complete} gives $\nu_n=\mathcal R_{p,\theta_n}(\nu_n)\Rightarrow\mathcal R_{p,\theta}(\nu')$, so $\nu'=\mathcal R_{p,\theta}(\nu')$. By (i), $\nu'=\nu_p^\theta$. Since every subsequence has a further subsequence converging to $\nu_p^\theta$, we have $\nu_n\Rightarrow\nu_p^\theta$. For (v), the first bound follows by splitting at $|b|=\rho$ and using (B). For the second, we have $\int|b|^s\dd\nu=\int_0^\infty st^{s-1}\nu(|b|>t)\dd t\le3^s+\int_3^\infty st^{s-1}(K'\mathfrak h_p/t+e^{-apt})\dd t\le4+K'\mathfrak h_p/(1-s)$ by (T), where we used that $ap\ge1$, which follows from \ref{it:stab-S2}. For the third, let $B$ satisfy \eqref{eq:stab-B-equation} with $B_j\sim\nu_p^\theta$. Since $x\mapsto x^s$ is subadditive on $[0,\infty)$, we have \[ \E|B-1|^s\le\theta^s+\Big(\frac\theta p\Big)^s\E\sum_{j\le N}Y_j^s|B_j|^{-s}=\theta^s+\Big(\frac\theta p\Big)^sp\,\Gamma(1+s)\int|b|^{-s}\dd\nu_p^\theta, \] and $\Gamma(1+s)\le1$. The first bound in (v) does not depend on $\theta$, so $\E|B-1|^s\to0$ as $\theta\downarrow0$, which gives the last claim. \end{proof} The principal law is thus a fixed point which does not depend on the choice of the admissible tuple, and which is obtained by iterating the law map from $\delta_1$. We now show that functionals of finite-depth recursions built from it depend continuously on $\theta$. Let $L\ge0$. Let $\mathcal T$ be the marked $\PGW(p)$ tree of \Cref{def:tree-pgw}, let $\mathcal T_{\le L}$ be its restriction to the generations $0,\dots,L$, with vertex set $V_{\le L}$, squared moduli $Y_e$ on its edges, and numbers of children $N_v$. Given $\theta$, let $(B_v)_{|v|=L}$ be, conditionally on $\mathcal T_{\le L}$ and its marks, independent with law $\nu_p^\theta$, and define recursively, for $|v|0$, $p\ge p_*(\mathbf q,C_0)$, $J\subset(0,C_0/p]$ an interval, $L\ge0$, $s_0\in[0,1)$, $k\ge0$, and $C_\Psi\ge1$. For each $\theta\in J$, let $\Psi_\theta$ be a real Borel function of the depth-$L$ data $\omega=(t,y,b)$, where $t$ is a rooted tree of depth at most $L$, $y=(y_e)$ are positive edge marks, and $b=(b_v)\in(\R\setminus\{0\})^{V(t)}$. Assume: \begin{enumerate}[label=\textup{(\alph*)}] \item for every $\theta\in J$ and $\P_\theta$-almost every $\omega=(t,y,b)$, we have $\Psi_{\theta'}(t,y,b')\to\Psi_\theta(t,y,b)$ as $(\theta',b')\to(\theta,b)$ with $\theta'\in J$; \item for all $\theta\in J$ and $\omega$, \[ |\Psi_\theta(\omega)|\le C_\Psi^{|V(t)|}\prod_{e}(1+y_e)^k\prod_{v\in V(t)}(1+|b_v|^{-1})^{s_0}\prod_{|v|=L}(1+|b_v|)^{s_0}. \] \end{enumerate} Then the law of the depth-$L$ principal recursion is weakly continuous in $\theta\in(0,C_0/p]$, we have $\sup_{\theta\in J}\E_\theta|\Psi_\theta|^r<\infty$ with $r=s_0^{-1/2}$ (and $r=2$ if $s_0=0$), and $\theta\mapsto\E_\theta\Psi_\theta$ is finite and continuous on $J$. \end{lemma} Condition (b) allows, at every vertex, a total inverse power of $B_v$ at most $s_0<1$; positive powers are allowed only at generation $L$. Positive powers of $B_v$ at internal vertices can be reduced to this form using \eqref{eq:stab-depth-recursion}, at the cost of increasing the inverse powers of the children. Indicator functions of events whose boundaries are $\P_\theta$-null sets satisfy (a). \begin{proof} \proofstep{1}{Coupling.} Let $\theta_n\to\theta$ in $J$. We realize the depth-$L$ data at all parameters on one probability space. Let $\mathcal T$ and its marks be given, and let $(U_v)$ be independent uniform variables on $(0,1)$, indexed by the Ulam--Harris labels and independent of $\mathcal T$ and its marks. With $G_\vartheta$ the quantile function of $\nu^\vartheta_p$, set $B_v^{(n)}=G_{\theta_n}(U_v)$ and $B_v=G_\theta(U_v)$ for $|v|=L$, and define the internal values by \eqref{eq:stab-depth-recursion} with $\theta_n$ and $\theta$, respectively. By part (iv) of \Cref{thm:stab-fixed}, $G_{\theta_n}\to G_\theta$ at the continuity points of $G_\theta$, which are all but countably many points, so $B^{(n)}_v\to B_v$ almost surely for $|v|=L$. Since the limiting values are almost surely nonzero, induction over the generations shows that $B^{(n)}_v\to B_v$ almost surely for every $v\in V_{\le L}$. This proves the weak continuity of the law, and by (a), $\Psi_{\theta_n}(\omega^{(n)})\to\Psi_\theta(\omega)$ almost surely. \proofstep{2}{A uniform moment bound.} First suppose that $s_0=0$, so $r=2$. Let $M_k=\E(1+Y)^{2k}<\infty$ for $Y\sim\Exp(1)$. By condition (b) and independence of the edge marks, for every $\vartheta\in J$, \[ \E_\vartheta\big[|\Psi_\vartheta|^2\mid\text{shape}\big]\le C_\Psi^{2|V_{\le L}|}M_k^{|V_{\le L}|-1}\le(C_\Psi^2M_k)^{|V_{\le L}|}. \] The fixed-depth exponential moment calculation below therefore gives the required uniform $L^2$ bound. Now suppose that $01$, which implies that it is uniformly integrable. By Vitali's convergence theorem, $\E_{\theta_n}\Psi_{\theta_n}\to\E_\theta\Psi_\theta$. \end{proof} We now state the result on compact scaled windows. Recall that $\theta(E)=g(E)^2$ and that $E^{-1}\le g(E)\le2E^{-1}$ for $E\ge2$, since $g(E)=2/(E+\sqrt{E^2-4})$. \begin{theorem}\label{thm:stab-window} Let $\Xi\subset(0,\infty)$ be compact and $\xi_-=\min\Xi$. Let $C_\Xi=4/\xi_-^2$ and $p_{\mathrm{st}}(\Xi)=\max\{p_*(\mathbf q_{\mathrm{can}},C_\Xi),\lceil5/\xi_-^2\rceil\}$. For every $p\ge p_{\mathrm{st}}(\Xi)$, $\xi\in\Xi$, and $E=\xi\sqrt p$, with $\theta=\theta(E)$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $E>2$ and $\theta\le C_\Xi/p$. The law map $\mathcal R_{p,\theta}$ has exactly one fixed point in the class $\mathfrak S_{p,\theta}$, which is also its unique fixed point in $\mathscr A_p$. We denote it by $\nu_{p,E}=\nu_p^{\theta(E)}$, and we write $\nu^x_{p,E}$ for the law of $1/B$, $B\sim\nu_{p,E}$. \item For every $\mu\in\mathscr A_p$, in particular for $\mu=\delta_1$, the iterates $\mathcal R^n_{p,\theta}(\mu)$ converge to $\nu_{p,E}$ in $d_p$ and weakly, with $d_p(\mathcal R^n_{p,\theta}(\mu),\nu_{p,E})\le\kappa^n d_p(\mu,\nu_{p,E})$ and \[ \kappa=\kappa_{p,\theta}\le\frac{80}{\xi_-^2\,p}+7pe^{-\sqrt p/2}. \] \item The law $\nu_{p,E}$ satisfies, uniformly in $\xi\in\Xi$: $\nu_{p,E}=e^{-p}\delta_{1+\theta}+f(b)\dd b$ with $f\le e^{-p}/2$ almost everywhere on $[-1/8,1/8]$; $\nu_{p,E}(\{|b-1|>r_p/2\})\le\omega_p\le e^{-\sqrt p}/2$; $\nu_{p,E}(\{|b|>L\})\le e^{-p}/(2L)+e^{-pL}/2$ for $L\ge3$; and, for $s\in(0,1)$, $\int|b|^{-s}\dd\nu_{p,E}\le16+(1-s)^{-1}$ and $\int|b|^s\dd\nu_{p,E}\le4+(1-s)^{-1}$. In particular, if $B\sim\nu_{p,E}$, then $\sup_{\xi\in\Xi}\P(|B-1|>r_p)\to0$ as $p\to\infty$. \item The map $E\mapsto\nu_{p,E}$ is weakly continuous on $\{\xi\sqrt p:\xi\in\Xi\}$. For every family $(\Psi_\theta)$ as in \Cref{lem:stab-fixed-depth}, with $\mathbf q=\mathbf q_{\mathrm{can}}$ and $J\subset(0,C_\Xi/p]$ an interval containing $\{\theta(\xi\sqrt p):\xi\in\Xi\}$, the map $E\mapsto\E_{\theta(E)}\Psi_{\theta(E)}$ is continuous on $\{\xi\sqrt p:\xi\in\Xi\}$. \item If $B\sim\nu_{p,E}$, then the law of $D=B/g(E)$ is a real solution at $E$. \end{enumerate} \end{theorem} \begin{proof} Since $p\ge5/\xi_-^2$, we have $E\ge\xi_-\sqrt p>2$, and $\theta=g(E)^2\le4/E^2\le C_\Xi/p$. Parts (i)--(iii) follow from \Cref{thm:stab-fixed} with $\mathbf q=\mathbf q_{\mathrm{can}}$ and $C_0=C_\Xi$, from \Cref{lem:stab-invariance}, and from $\kappa_{p,\theta}\le20C_\Xi/p+7pe^{-\sqrt p/2}$; for the moment bounds we used $\rho=1/16$, $K'=1$, and $\mathfrak h_p=e^{-p}\le1$ in part (v) of \Cref{thm:stab-fixed}. For (iv), the function $E\mapsto\theta(E)$ is continuous on $(2,\infty)$, so (iv) follows from part (iv) of \Cref{thm:stab-fixed} and \Cref{lem:stab-fixed-depth}. Part (v) follows from the discussion at the beginning of this section, since $\nu_{p,E}(\{0\})=0$. \end{proof} The canonical tuple plays no special role. If $\mathbf q$ is any admissible tuple and $p\ge\max\{p_{\mathrm{st}}(\Xi),p_*(\mathbf q,C_\Xi),p_*(\mathbf q\vee\mathbf q_{\mathrm{can}},C_\Xi)\}$, then part (iii) of \Cref{thm:stab-fixed} shows that $\nu_{p,E}$ is also the unique fixed point of $\mathcal R_{p,\theta(E)}$ in $\mathfrak S_{p,\theta(E)}(\mathbf q)$, and \Cref{lem:stab-invariance} gives the bounds of (iii) with $e^{-p}$, $e^{-\sqrt p}$, and $1/16$ replaced by $p^Ae^{-ap}$, $e^{-a\sqrt p}$, and $\rho$. The same applies to the mixed metric of \Cref{def:stab-mixed} built with $\mathbf q$. \section{Small parameter and the half-line}\label{sec:stab-entry} The fixed point $\nu_p^\theta$ of \Cref{thm:stab-fixed} is characterized as the unique fixed point of $\mathcal R_{p,\theta}$ in $\mathscr A_p(\mathbf q)$, a class which does not involve $C_0$; the number $C_0$ only enters the threshold $p_*(\mathbf q,C_0)$. In particular, the construction covers all small values of $\theta$, which correspond to large energies, with a single class. \begin{proposition}\label{prop:stab-entry} Let $c_{\mathrm{ent}}=1$, let $\mathfrak S^{\mathrm{ent}}_{p,\theta}=\mathfrak S_{p,\theta}$ be the class of \Cref{def:stab-class} for the canonical tuple $\mathbf q_{\mathrm{can}}=(1,0,1,1/16)$, which we call the small-parameter class, and let $p_{\mathrm{ent}}=p_*(\mathbf q_{\mathrm{can}},c_{\mathrm{ent}})$. For every $p\ge p_{\mathrm{ent}}$ and $0<\theta\le c_{\mathrm{ent}}/p$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item The law map $\mathcal R_{p,\theta}$ has exactly one fixed point in $\mathfrak S^{\mathrm{ent}}_{p,\theta}$, namely $\nu_p^\theta$; it is also the unique fixed point in $\mathscr A_p$. The iterates $\mathcal R_{p,\theta}^n(\delta_1)$ converge to $\nu_p^\theta$ in $d_p$ and weakly, with contraction factor at most $20/p+7pe^{-\sqrt p/2}$. \item For every $s_0\in[0,1)$, $\sup_{0<\theta\le c_{\mathrm{ent}}/p}\int|b|^{-s_0}\,\nu_p^\theta(\mathrm db)\le16+(1-s_0)^{-1}$. \item The map $\theta\mapsto\nu_p^\theta$ is weakly continuous on $(0,c_{\mathrm{ent}}/p]$, and for every family $(\Psi_\theta)$ as in \Cref{lem:stab-fixed-depth} with $J\subset(0,c_{\mathrm{ent}}/p]$, the map $\theta\mapsto\E_\theta\Psi_\theta$ is continuous on $J$. \item As $\theta\downarrow0$, $\nu_p^\theta\Rightarrow\delta_1$, and the law of $1/B$, $B\sim\nu_p^\theta$, converges weakly to $\delta_1$. \item Let $\Xi\subset(0,\infty)$ be compact, $p\ge\max\{p_{\mathrm{ent}},p_{\mathrm{st}}(\Xi)\}$, $\xi\in\Xi$, and $E=\xi\sqrt p$. If $\theta(E)\le c_{\mathrm{ent}}/p$, then $\nu_p^{\theta(E)}=\nu_{p,E}$, the law of \Cref{thm:stab-window}. \end{enumerate} The same statements hold with $c_{\mathrm{ent}}$ replaced by any $c>0$ and $p_{\mathrm{ent}}$ by $p_*(\mathbf q_{\mathrm{can}},c)$. \end{proposition} \begin{proof} Parts (i), (iii), and (iv) follow from \Cref{thm:stab-fixed} and \Cref{lem:stab-fixed-depth} with $\mathbf q=\mathbf q_{\mathrm{can}}$ and $C_0=c_{\mathrm{ent}}$; for the reciprocal law in (iv), note that $B\to1$ in probability implies $1/B\to1$ in probability. For (ii), part (v) of \Cref{thm:stab-fixed} with $\rho=1/16$ and $\mathfrak h_p=e^{-p}\le1$ gives the bound $16^{s_0}+2e^{-p}16^{s_0-1}/(1-s_0)\le16+(1-s_0)^{-1}$. For (v), both $\nu_p^{\theta(E)}$ and $\nu_{p,E}$ are fixed points of $\mathcal R_{p,\theta(E)}$ in $\mathfrak S_{p,\theta(E)}(\mathbf q_{\mathrm{can}})$, which has at most one fixed point by part (i) of \Cref{thm:stab-fixed}. The last sentence follows by the same arguments with $C_0=c$. \end{proof} \begin{corollary}\label{cor:stab-halfline} Let $M_0=3$. Then $g(E)^2\le c_{\mathrm{ent}}/p$ for every $p>1$ and $E\ge M_0\sqrt p$. Fix $\xi_0>0$, and let \[ p_0(\xi_0)=\max\big\{p_*\big(\mathbf q_{\mathrm{can}},\max\{4\xi_0^{-2},1\}\big),\ \lceil5\xi_0^{-2}\rceil,\ p_{\mathrm{ent}}\big\}. \] For every $p\ge p_0(\xi_0)$ and $E\ge\xi_0\sqrt p$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $E>2$, and the iterates $\mathcal R^n_{p,\theta(E)}(\delta_1)$ converge weakly, and in $d_p$, to the unique fixed point $\nu_{p,E}=\nu_p^{\theta(E)}$ of $\mathcal R_{p,\theta(E)}$ in $\mathfrak S_{p,\theta(E)}$. The law $\nu_{p,E}$ has the properties (i)--(iii) of \Cref{lem:stab-invariance} with $\mathbf q=\mathbf q_{\mathrm{can}}$ and $C_0=\max\{4\xi_0^{-2},1\}$. In particular, $\nu_{p,E}(\{b:|b|>L\})\le e^{-p}/(2L)+e^{-pL}/2$ for $L\ge3$, and $\int|b|^{-s}\dd\nu_{p,E}\le16+(1-s)^{-1}$ and $\int|b|^{s}\dd\nu_{p,E}\le4+(1-s)^{-1}$ for $s\in(0,1)$. \item Let $\Xi\subset[\xi_0,\infty)$ be compact with $E/\sqrt p\in\Xi$. Then $p\ge p_{\mathrm{st}}(\Xi)$, and $\nu_{p,E}$ is the law given by \Cref{thm:stab-window} for the window $\Xi$; for example, this applies with $\Xi=[\xi_0,M_0+1]$ if $E\le(M_0+1)\sqrt p$. If $E\ge M_0\sqrt p$, then $\nu_{p,E}$ is the law given by \Cref{prop:stab-entry} with $\theta=\theta(E)$. \item The map $E\mapsto\nu_{p,E}$ is weakly continuous on $[\xi_0\sqrt p,\infty)$. \item If $B\sim\nu_{p,E}$, then the law of $D=B/g(E)$ is a real solution at $E$. \end{enumerate} \end{corollary} \begin{proof} For $E\ge3\sqrt p$ we have $E>2$ and $g(E)^2\le4/E^2\le4/(9p)\le c_{\mathrm{ent}}/p$. Let $C_0=\max\{4\xi_0^{-2},1\}$. For $p\ge p_0(\xi_0)$ and $E\ge\xi_0\sqrt p$, we have $E\ge\sqrt5>2$ and $\theta(E)\le4/E^2\le C_0/p$. Part (i) then follows from parts (i), (ii), and (v) of \Cref{thm:stab-fixed} with $\mathbf q=\mathbf q_{\mathrm{can}}$, as in the proof of \Cref{thm:stab-window}. For (ii), let $\xi_-=\min\Xi\ge\xi_0$. Then $p_{\mathrm{st}}(\Xi)=\max\{p_*(\mathbf q_{\mathrm{can}},4\xi_-^{-2}),\lceil5\xi_-^{-2}\rceil\}\le p_0(\xi_0)$, because $4\xi_-^{-2}\le\max\{4\xi_0^{-2},1\}$ and $p_*(\mathbf q,C_0)$ is nondecreasing in $C_0$ (the conditions \ref{it:stab-S1}--\ref{it:stab-S9} become stronger as $C_0$ grows). The laws in \Cref{thm:stab-window} and \Cref{prop:stab-entry} are fixed points of $\mathcal R_{p,\theta(E)}$ in $\mathfrak S_{p,\theta(E)}$, so they coincide with $\nu_{p,E}$ by uniqueness. Part (iii) follows from part (iv) of \Cref{thm:stab-fixed} and the continuity of $E\mapsto\theta(E)$, and part (iv) from the discussion at the beginning of this section. \end{proof} \begin{remark}\label{rem:stab-M0} If a later argument requires a smaller value of $c_{\mathrm{ent}}$ or a larger value of $M_0$, nothing changes: for $M\ge3$, the energies in the large-energy regime $E\ge M\sqrt p$ satisfy $\theta(E)\le4/(M^2p)$, and \Cref{prop:stab-entry} holds with $c_{\mathrm{ent}}$ replaced by $4/M^2$ and $p_{\mathrm{ent}}$ by $p_*(\mathbf q_{\mathrm{can}},4/M^2)\le p_{\mathrm{ent}}$. The principal law $\nu_{p,E}$ itself does not depend on these choices. \end{remark} \section{Real solutions and the real boundary law}\label{sec:stab-real} In this subsection, we show that on compact scaled windows and for large $p$, every real solution of the cavity equation is the principal solution. We begin with an a priori bound for the physical cavity variables and with the observation that real limit points of their laws are real solutions. \begin{lemma}\label{lem:stab-apriori} Let $p>1$, $E>2$, $\theta=\theta(E)$, and for $\eta>0$ set $X_\eta=-\Gamma_o(E+\iu\eta)/g(E)$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $r\in(0,1)$ and $\eta>0$, we have $\E|X_\eta|^r\le C_{r,1}(p+1)^{r/2}\le2C_{r,1}p^{r/2}$ and $\E^\infty|X_\eta|^r\le2\varrho_p^{-1}C_{r,1}p^{r/2}$, where $C_{r,1}$ is the constant of \Cref{lem:palm-affine} with $d=1$. \item Let $\eta_k\downarrow0$, and suppose that the laws of $X_{\eta_k}$ under $\P$ converge weakly to a probability measure carried by $\R$. Let $X$ have this law. Then $\P(X=0)=0$, and the law of $B=1/X$ is a fixed point of $\mathcal R_{p,\theta}$; equivalently, the law of $B/g(E)$ is a real solution at $E$. \end{enumerate} \end{lemma} \begin{proof} By \Cref{prop:tree-schur}, $\Gamma_o\eqdist-(z+p^{-1}\sum_{j\le N}Y_j\Gamma_j)^{-1}$ with $z=E+\iu\eta$, where $N\sim\Pois(p)$, $Y_j\sim\Exp(1)$, and the $\Gamma_j$ are copies of $\Gamma_o(z)$, all independent. Dividing by $-g(E)$ and using $g(E)E=1+\theta$, we obtain \begin{equation}\label{eq:stab-complex-recursion} X_\eta\eqdist\Big(1+\theta+\iu g(E)\eta-\frac\theta p\sum_{j\le N}Y_jX_{\eta,j}\Big)^{-1},\qquad X_{\eta,j}=-\Gamma_j/g(E). \end{equation} Conditionally on $N=m$ and on $(X_{\eta,j})_{j\le m}$, the denominator is $c+\sum_{j\le m}a_jY_j$ with $c=1+\theta+\iu g(E)\eta$, $|c|\ge1$, and complex coefficients $a_j=-(\theta/p)X_{\eta,j}$ that are measurable with respect to the conditioning, while $Y_1,\dots,Y_m$ are standard exponential variables which are independent of each other and of the conditioning. The conditional version of \Cref{lem:palm-affine} gives $\E[|X_\eta|^r\mid N,(X_{\eta,j})]\le C_{r,1}(N+1)^{r/2}$. Averaging and using Jensen's inequality, we find $\E|X_\eta|^r\le C_{r,1}(p+1)^{r/2}$. The bound under $\P^\infty$ follows from $\E^\infty Z\le\varrho_p^{-1}\E Z$ for $Z\ge0$ (\Cref{lem:palm-survival}). For (ii), write $X_k=X_{\eta_k}$, let $(X_j)_{j\ge1}$ be independent copies of $X$, independent of $N$ and $(Y_j)$, and let $X_{k,j}$ be independent copies of $X_k$, independent of $N$ and $(Y_j)$. Set \[ D_k=1+\theta+\iu g(E)\eta_k-\frac\theta p\sum_{j\le N}Y_jX_{k,j},\qquad D=1+\theta-\frac\theta p\sum_{j\le N}Y_jX_j . \] Fix $m\ge0$. Since the components are independent and converge weakly, the vector $(Y_1,\dots,Y_m,X_{k,1},\dots,X_{k,m})$ converges weakly to $(Y_1,\dots,Y_m,X_1,\dots,X_m)$ \cite[Theorem~2.8]{Billingsley1999convergence}. Since $\eta_k\to0$, the continuous mapping theorem \cite[Theorem~2.7]{Billingsley1999convergence} shows that the conditional law of $D_k$ given $N=m$ converges weakly to that of $D$ given $N=m$. Averaging over $m$ with dominated convergence, we obtain $D_k\Rightarrow D$. By \eqref{eq:stab-complex-recursion}, $1/D_k$ has the law of $X_k$. The laws of $X_k$ converge, and in particular they are tight. For every $R>0$, the portmanteau theorem applied to the open set $\{|w|R)\le\sup_k\P(|X_k|>R), \] which tends to zero as $R\to\infty$. Hence $\P(D=0)=0$, and the continuous mapping theorem gives $1/D_k\Rightarrow1/D$. We obtain $X\eqdist1/D$. Since $D$ is real and finite, $X$ is real and almost surely nonzero, and $1/X\eqdist D$. By the definition of $D$, this says that the law of $1/X$ is a fixed point of $\mathcal R_{p,\theta}$. The equivalence with real solutions was noted after \Cref{def:stab-lawmap}. \end{proof} The next lemma is the main step. It shows that every real fixed point on a scaled window satisfies the bounds defining the class $\mathfrak S_{p,\theta}$. The only input is the fixed-point equation: we start from the bound $\sqrt p$ on the density near zero given by \Cref{lem:palm-affine}, and we improve it by a one-big-jump argument. \begin{lemma}\label{lem:stab-bootstrap} Let $00$. \proofstep{2}{Decomposition.} We split the children according to whether $|B_j|\ge2\rho$, and write $B=D-S$ with $D=1+\theta-(\theta/p)\sum_{|B_j|\ge2\rho}Y_j/B_j$ and $S=(\theta/p)\sum_{|B_j|<2\rho}Y_j/B_j$, which are independent by \Cref{lem:stab-lawmap}. By \Cref{lem:stab-fresh} with $\delta=2\rho$ and $c=1+\theta$, the law of $D$ restricted to $(-\infty,1/2]$ has density at most $F_{p,\theta}(2\rho)\le F_p$. By part (i) of \Cref{lem:stab-envelope} with $\rho'=2\rho$, the jumps of $S$ form a Poisson process with intensity $\ell(y)\dd y$, where \begin{equation}\label{eq:stab-boot-envelope} \ell(y)\le\min\Big\{\frac{c_*}{y^2},M\Big\},\qquad c_*=\theta H\le c^{(0)},\qquad M=\frac{2p^2\rho^2H}{\theta}\le M_p . \end{equation} Let $a_*=c_*^{1/4}\le1$, and write $S=S_<+S_>$, where $S_<$ and $S_>$ are the sums of the jumps with $|y|\le a_*$ and $|y|>a_*$. These are independent compound Poisson variables with intensities $\ell_<=\ell\one_{\{|y|\le a_*\}}$ and $\ell_>=\ell\one_{\{|y|>a_*\}}$. By \eqref{eq:stab-boot-envelope}, \begin{equation}\label{eq:stab-boot-large} \lambda_>=\int\ell_>\le2\int_{a_*}^\infty\frac{c_*}{y^2}\dd y=2c_*^{3/4}\le1,\qquad\|\ell_>\|_\infty\le\frac{c_*}{a_*^2}=c_*^{1/2}. \end{equation} \proofstep{3}{The small jumps.} We claim that the law of $S_<$ restricted to $\{|s|\ge\rho\}$ has density at most $e^2Me^{-\rho/a_*}$. Let $m_<=\int y\,\ell_<(y)\dd y$, $v_<=\int y^2\ell_<(y)\dd y$, and $y_0=(c_*/M)^{1/2}=\theta/(\sqrt2p\rho)$. By \eqref{eq:stab-boot-envelope}, if $y_0$, and the law of $S_>$ is $e^{-\lambda_>}\sum_{m\ge0}\ell_>^{*m}/m!$. Let $A\subset[-2\rho,2\rho]$ be a Borel set. The term $m=0$ contributes at most $G|A|$ to $\P(B\in A)$. The term $m=1$ contributes at most \[ \int_A\Big(\int\ell_>(w-x)\,\P(W\in\mathrm dw)\Big)\dd x\le\Big(G\lambda_>+\frac{c_*}{\rho^2}\Big)|A| , \] where we split the inner integral according to whether $|w-x|<\rho$, in which case $w\in[-3\rho,3\rho]$ and the density of $W$ is at most $G$, or $|w-x|\ge\rho$, in which case $\ell_>(w-x)\le c_*/\rho^2$ by \eqref{eq:stab-boot-envelope}. The terms $m\ge2$ contribute at most $\sum_{m\ge2}\|\ell_>^{*m}\|_\infty|A|/m!\le\|\ell_>\|_\infty\lambda_>|A|\le2c_*^{5/4}|A|$, by \eqref{eq:stab-boot-large}. Since $\lambda_>\le1$, we conclude that \begin{equation}\label{eq:stab-one-big-jump} H\le2G+\frac{\theta H}{\rho^2}+2(\theta H)^{5/4}. \end{equation} \proofstep{5}{The bootstrap.} By \ref{it:stab-B2}, $\theta/\rho^2\le C_0/(p\rho^2)\le1/4$. Further, $2(\theta H)^{5/4}\le H/4$ is equivalent to $H\le8^{-4}\theta^{-5}$, which holds since $H\le h_0\le8^{-4}(p/C_0)^5$ by \ref{it:stab-B2}. Inserting these bounds into \eqref{eq:stab-one-big-jump}, we obtain $H\le4G$. Since $M\le M_p$ and $a_*=(\theta H)^{1/4}$, this reads $H\le\phi(H)$, where \[ \phi(h)=4F_p+4e^2M_p\exp\big(-\rho(\theta h)^{-1/4}\big),\qquad h>0, \] is nondecreasing. We now iterate this inequality. Since $H\le h_0$ and $\theta h_0\le c^{(0)}\le p^{-1/3}$ by \ref{it:stab-B1}, we have $H\le\phi(h_0)\le h_1$. Next, $\theta h_1\le C_0h_1/p\le\rho^4p^{-8}$ by \ref{it:stab-B3}, so that $\rho(\theta h_1)^{-1/4}\ge p^2$ and $H\le\phi(h_1)\le4F_p+4e^2M_pe^{-p^2}\le e^{-2p}$, again by \ref{it:stab-B3}. This proves the density bound in the lemma, and in particular $\nu_B$ satisfies the boundary bound (B) with $\mathfrak h_p=e^{-p}$. \proofstep{6}{Core and tail bounds.} By \Cref{lem:stab-chernoff} with $\delta=2\rho$, $\E(1+\theta-D)=\theta\int_{|b|\ge2\rho}b^{-1}\nu_B(\mathrm db)$, which has absolute value at most $\theta/(2\rho)=8\theta$. This gives $|\E D-1|\le9\theta\le9C_0/p\le r_p/4$, by \ref{it:stab-B4}. Applying \eqref{eq:stab-chernoff} with $\delta=2\rho$ and $x=r_p/4$ as in Step 1 of the proof of \Cref{lem:stab-invariance}, we obtain $\P(|D-\E D|>r_p/4)\le2e^{-c_1p^{7/4}}$. Further, $\P(S\neq0)\le p\,\nu_B((-2\rho,2\rho))\le4\rho pH\le4\rho pe^{-2p}$. By \ref{it:stab-B4}, we obtain \[ \P(|B-1|>r_p)\le\P(|B-1|>r_p/2)\le2e^{-c_1p^{7/4}}+4\rho pe^{-2p}\le\frac12e^{-\sqrt p}, \] which is the core bound (C). For $L\ge3$, as in Step 4 of the proof of \Cref{lem:stab-invariance}, \eqref{eq:stab-chernoff-abs} with $\delta=2\rho$ gives $\P(|D|>L/2)\le\exp(p-\rho p^2L/(9C_0))\le e^{-pL}$ by \ref{it:stab-B4}. By part (ii) of \Cref{lem:stab-envelope}, which applies since $\int\ell\le4\rho pH\le1$ and $\ell(y)\le\theta H/y^2$, we have $\P(|S|\ge L/2)\le8\theta H/L\le8C_0e^{-2p}/(pL)\le e^{-p}/L$, by \ref{it:stab-B4}. We obtain $\nu_B(\{|b|>L\})\le e^{-p}/L+e^{-pL}$, which is the tail bound (T) for the canonical tuple. Finally, $\nu_B=\mathcal R_{p,\theta}(\nu_B)$ is of the form $e^{-p}\delta_{1+\theta}+f\dd b$ by part (iii) of \Cref{lem:stab-lawmap}. Therefore $\nu_B\in\mathfrak S_{p,\theta}$, and $\nu_B=\nu_p^\theta$ by part (i) of \Cref{thm:stab-fixed}. \end{proof} \begin{proposition}\label{prop:stab-selection} Let $\Xi\subset(0,\infty)$ be compact, $\xi_-=\min\Xi$, $\xi_+=\max\Xi$, and \[ p_{\mathrm{phys}}(\Xi)=\max\big\{p_{\mathrm{boot}}(\xi_+^{-2},4\xi_-^{-2}),\ p_{\mathrm{st}}(\Xi)\big\}. \] For every $p\ge p_{\mathrm{phys}}(\Xi)$, $\xi\in\Xi$, and $E=\xi\sqrt p$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $r\in(0,1)$, we have $\sup_{\eta>0}\E|\Gamma_o(E+\iu\eta)/g(E)|^r\le C_rp^{r/2}$ and $\sup_{\eta>0}\E^\infty|\Gamma_o(E+\iu\eta)/g(E)|^r\le\varrho_p^{-1}C_rp^{r/2}$, where $C_r$ depends only on $r$. \item For every real solution $\pi$ at $E$, the law of $g(E)D$, $D\sim\pi$, is $\nu_{p,E}$. \item If $\eta_k\downarrow0$ and the laws under $\P$ of $-\Gamma_o(E+\iu\eta_k)/g(E)$ converge weakly to a probability measure carried by $\R$, then this limit is the law of $1/B$ with $B\sim\nu_{p,E}$. In particular, if $\Im\Gamma_o(E+\iu\eta)\to0$ in $\P$-probability as $\eta\downarrow0$, then the law of $-\Gamma_o(E+\iu\eta)/g(E)$ under $\P$ converges weakly to the law of $1/B$. \item Suppose that $E\in\mathcal E_p$ and $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Then the real boundary law at $E$ of \Cref{prop:rde-boundary}, in the $B$ normalization, equals $\nu_{p,E}$. Moreover, $-\Gamma_o(E+\iu\eta)/g(E)$ converges $\P$-almost surely as $\eta\downarrow0$ to a random variable with the law of $1/B$, $B\sim\nu_{p,E}$. \end{enumerate} \end{proposition} \begin{proof} Part (i) is part (i) of \Cref{lem:stab-apriori} with $C_r=2C_{r,1}$. For (ii), since $p\ge p_{\mathrm{st}}(\Xi)$ we have $E>2$, and $E^{-1}\le g(E)\le2E^{-1}$ gives $\xi_+^{-2}/p\le\theta(E)\le4\xi_-^{-2}/p$. By the discussion after \Cref{def:stab-lawmap}, the law of $g(E)D$ is a fixed point of $\mathcal R_{p,\theta(E)}$, and \Cref{lem:stab-bootstrap} with $c_0=\xi_+^{-2}$ and $C_0=4\xi_-^{-2}$ shows that it equals $\nu_p^{\theta(E)}=\nu_{p,E}$. The first claim in (iii) follows from part (ii) of \Cref{lem:stab-apriori} and (ii). For the second claim, let $X_\eta=-\Gamma_o(E+\iu\eta)/g(E)$ and let $\eta_k\downarrow0$. By (i) and Markov's inequality, the laws of $X_{\eta_k}$ are tight, so every subsequence has a further subsequence along which $X_{\eta_k}$ converges in law to some $X$. Since $\Im X_{\eta_k}\to0$ in probability and $\Im$ is continuous, $\Im X=0$ almost surely, and the first claim identifies the law of $X$. This shows that the laws of $X_\eta$ converge to the law of $1/B$ along every sequence $\eta_k\downarrow0$. For (iv), since $E\in\mathcal E_p$, the boundary value $\Gamma_o(E+\iu0)=\lim_{\eta\downarrow0}\Gamma_o(E+\iu\eta)$ exists almost surely (\Cref{def:rde-regular}). By part (ii) of \Cref{prop:rde-boundary}, $\Gamma_o(E+\iu0)$ is $\P$-almost surely finite, real, and nonzero, and the law of $D=-1/\Gamma_o(E+\iu0)$ under $\P$ is a real solution at $E$. By (ii), the law of $B=g(E)D$ is $\nu_{p,E}$, and $-\Gamma_o(E+\iu\eta)/g(E)\to-\Gamma_o(E+\iu0)/g(E)=1/B$ almost surely. In other words, under the no-ac hypothesis the laws of $-\Gamma_o(E+\iu\eta)/g(E)$ have a unique limit point as $\eta\downarrow0$, this limit point is carried by $\R$, and (iii) identifies it with $\nu^x_{p,E}$. \end{proof} \begin{remark}\label{rem:stab-selection} Part (ii) of \Cref{prop:stab-selection} does not use the a priori bound (i): the one-big-jump argument of \Cref{lem:stab-bootstrap} starts from the density bound $\sqrt{p+1}$ near zero, which holds for every real fixed point by \Cref{lem:palm-affine}. The bound (i) is used only for tightness, in the second claim of (iii). Under survival conditioning, the corresponding boundary law is not $\nu_{p,E}$, since a surviving vertex has at least one child; it is dominated by $\varrho_p^{-1}$ times the unconditioned law, as in \Cref{prop:palm-endpoints}, and therefore inherits the upper bounds of \Cref{thm:stab-window} with an additional factor $\varrho_p^{-1}$. \end{remark} \chapter{Analyticity of the principal solution and bounds on its derivatives}\label{sec:jets} In this section, we study how the principal solution $\nu_{p,E}$ of \Cref{thm:stab-window} depends on the scaled energy $\xi=E/\sqrt p$. We prove three results. The first, \Cref{thm:jets-analytic}, states that for each fixed large $p$, the law of the principal denominator, written in the scaling $g=1/E$, is a real-analytic function of the energy with values in a Banach space $X_q$ of signed measures. The second, \Cref{prop:jets-critical-analytic}, deduces from this that the critical growth rate $\lambda_p(\xi)$ is real-analytic in $\xi$. The third, \Cref{prop:jets-quantitative} and its consequence \Cref{cor:jets-tests}, bounds the first four $\xi$-derivatives of the law with constants that are explicit in $p$: polynomial for the weighted total variation, exponentially small for the parts of the derivatives near zero and away from one, and of order $p^{-1}$ for the parts near one that are seen by smooth test functions. These bounds are used in \Cref{sec:uniq}. The analyticity statements in \Cref{sec:jets-analytic-sec,sec:jets-critical} are qualitative: they hold at every fixed $p$ above a threshold, and the constants in their proofs depend on $p$. The quantitative bounds in \Cref{sec:jets-quant} are proved separately. There we differentiate the fixed point equation and estimate the derivatives with the component norms of \Cref{def:stab-mixed}; analyticity is used only to know that the derivatives exist and satisfy the differentiated equations. \Cref{sec:jets-fixed} introduces the scaling $g=1/E$ and collects the properties of the principal law that we use. \Cref{sec:jets-space} defines the space $X_q$ and proves that the law map is holomorphic on it. \Cref{sec:jets-invert} shows that the linearization of the law map at the principal law is a contraction for the mixed norm of \Cref{sec:stab} and deduces that $I$ minus the linearization is invertible on $X_q$. \Cref{sec:jets-analytic-sec} proves \Cref{thm:jets-analytic}, and \Cref{sec:jets-critical} proves \Cref{prop:jets-critical-analytic}. \Cref{sec:jets-quant} proves \Cref{prop:jets-quantitative,cor:jets-tests}. \section{The scaling \texorpdfstring{$g=1/E$}{g=1/E}}\label{sec:jets-fixed} \subsection{Conventions}\label{sec:jets-conventions} Throughout this section, $\Xi=[\xi_-,\xi_+]\subset(0,\infty)$ is a compact interval. Every compact set $\Xi\subset(0,\infty)$ is contained in its convex hull, which is a compact interval in $(0,\infty)$, and the results below for intervals imply those for compact sets. We set \begin{equation}\label{eq:jets-Xiprime} \Xi'=[\xi_-/2,\,2\xi_+], \end{equation} so that $\Xi$ is contained in the interior of $\Xi'$, and we apply the results of \Cref{sec:stab} on the window $\Xi'$. Let $p_{\mathrm{st}}(\Xi')$ be the threshold of \Cref{thm:stab-window} for this window. By \Cref{thm:stab-window}, for $p\ge p_{\mathrm{st}}(\Xi')$ and $E=\xi\sqrt p$ with $\xi\in\Xi'$, the law $\nu_{p,E}$ is the weak limit of the iteration of the cavity map in the $B$-normalization started from $\delta_1$. In particular, $\nu_{p,E}$ does not depend on the window used to construct it. For $\xi\in\Xi'$ and $p\ge p_{\mathrm{st}}(\Xi')$, we write $E=\xi\sqrt p$, $g=g(E)$, $\theta=\theta(\xi)=g(E)^2$, and \begin{equation}\label{eq:jets-tau} \tau=\tau(\xi)=\frac{1}{pE^2}=\frac{1}{p^2\xi^2}. \end{equation} Recall that $Eg(E)=1+\theta$, so that $\theta/(1+\theta)^2=g^2/(Eg)^2=E^{-2}$ and \begin{equation}\label{eq:jets-tau-theta} \tau=\frac{\theta}{p(1+\theta)^2}. \end{equation} Constants $c,C>0$ in this section may depend on $\Xi$, on the exponent $q\in(0,1)$ when one is fixed, and on the constants of \Cref{sec:stab} for the window $\Xi'$ (which depend only on $\Xi$); they never depend on $p$ or $\xi$ unless this is indicated, and they may change from line to line. We write $Y,T,Y_1,Y_2,\dots$ for independent $\Exp(1)$ variables and $N\sim\Pois(p)$, all independent of each other and of the other variables in the same formula. We first record the size of the parameters and of their derivatives. \begin{lemma}\label{lem:jets-parameters} There exist $p_0(\Xi)\ge p_{\mathrm{st}}(\Xi')$ and $C>0$ such that for every $p\ge p_0(\Xi)$, the functions $\xi\mapsto\theta(\xi)$ and $\xi\mapsto\tau(\xi)$ are real-analytic on a neighborhood of $\Xi'$, and for all $\xi\in\Xi'$ and $k\in\{0,1,\dots,4\}$, \begin{equation}\label{eq:jets-parameters} \frac{1}{Cp}\le\theta(\xi)\le\frac{C}{p},\qquad \bigl|\partial_\xi^k\theta(\xi)\bigr|\le\frac{C}{p},\qquad \bigl|\partial_\xi^k\tau(\xi)\bigr|\le\frac{C}{p^2}. \end{equation} In particular, $\theta(\xi)\le 1/24$ and $1\le 1+\theta(\xi)\le 9/8$ for $\xi\in\Xi'$. \end{lemma} \begin{proof} The bounds for $\tau$ follow from \eqref{eq:jets-tau}, since $\partial_\xi^k\xi^{-2}$ is bounded on $\Xi'$. For $\theta$, recall from \Cref{def:rde-normalizations} that $\sqrt p\,g(\xi\sqrt p)=a_p(\xi)^{-1}$, where $a_p(\xi)=(\xi+\sqrt{\xi^2-4/p})/2$. Then \[ \theta(\xi)=\frac{1}{p\,a_p(\xi)^2}. \] Let $p\ge128/\xi_-^2$. For $\xi\ge\xi_-/4$ we have $8/p\le\xi_-^2/16\le\xi^2$, that is, $\xi^2-4/p\ge\xi^2/2$. On the half-line $(\xi_-/4,\infty)$, which contains a neighborhood of $\Xi'$, the function $a_p$ is real-analytic and satisfies $\xi/2\le a_p(\xi)\le\xi$, and on $\Xi'$ its derivatives of order at most four are bounded by a constant depending only on $\xi_-$ and $\xi_+$, uniformly in $p\ge128/\xi_-^2$. This gives the bounds for $\theta$. The last assertion holds once $C/p\le1/24$. \end{proof} \subsection{The principal law in the scaling \texorpdfstring{$g=1/E$}{g=1/E}}\label{sec:jets-fixedlaw} Let $p\ge p_0(\Xi)$, $\xi\in\Xi'$, and let $B\sim\nu_{p,E}$. Recall from \Cref{thm:stab-window} that $\nu_{p,E}$ solves \begin{equation}\label{eq:jets-B-equation} B\eqdist 1+\theta-\frac{\theta}{p}\sum_{j=1}^N\frac{Y_j}{B_j}, \end{equation} where $B_1,B_2,\dots$ are independent with law $\nu_{p,E}$. Set $\widetilde B=B/(1+\theta)=D/E$, where $D=B/g$ is the corresponding real solution (\Cref{cor:stab-halfline}). Dividing \eqref{eq:jets-B-equation} by $1+\theta$ and using \eqref{eq:jets-tau-theta}, we obtain \begin{equation}\label{eq:jets-fixed-equation} \widetilde B\eqdist 1-\tau\sum_{j=1}^N\frac{Y_j}{\widetilde B_j}, \end{equation} where $\widetilde B_1,\widetilde B_2,\dots$ are independent copies of $\widetilde B$. The event $N=0$ gives $\widetilde B$ an atom of mass at least $e^{-p}$ at $1$, whose location does not depend on $\xi$. \begin{definition}\label{def:jets-fixed-law} For $p\ge p_0(\Xi)$ and $E=\xi\sqrt p$ with $\xi\in\Xi'$, the \emph{principal law in the scaling $g=1/E$} is the law $\widetilde\nu_{p,E}$ of $\widetilde B=B/(1+\theta)$, where $B\sim\nu_{p,E}$. When $p$ is fixed, we abbreviate $\widetilde\nu_\xi=\widetilde\nu_{p,\xi\sqrt p}$. \end{definition} The reason for this scaling is the following. The law $\nu_{p,E}$ of $B$ has the atom $e^{-p}\delta_{1+\theta(\xi)}$, and $\xi\mapsto\theta(\xi)$ is strictly monotone. For $\xi\neq\xi'$, the measure $\nu_{p,\xi\sqrt p}-\nu_{p,\xi'\sqrt p}$ gives mass $e^{-p}$ to the point $1+\theta(\xi)$, since the second law has no atom there. Its total variation is at least $e^{-p}$, and $\xi\mapsto\nu_{p,\xi\sqrt p}$ is not continuous in total variation, let alone differentiable. After the dilation by $(1+\theta)^{-1}$, the compulsory atom sits at $1$ for every $\xi$, and we show that $\xi\mapsto\widetilde\nu_\xi$ is real-analytic in a norm that dominates the weighted total variation. Smooth test functions of $B$ are recovered from $\widetilde\nu_\xi$ by the chain rule (see \Cref{cor:jets-tests}). \subsection{Inputs from the construction of the principal law}\label{sec:jets-inputs} We use the following facts from \Cref{sec:stab}, applied on the window $\Xi'$. We state them in the $B$-coordinate $b$, as in \Cref{sec:stab}, and write $r_p=p^{-1/4}$ and $\mathcal C_p=[1-r_p,1+r_p]$. Let $C_{\Xi'}=16/\xi_-^2$, which is the constant $C_\Xi$ of \Cref{thm:stab-window} for the window $\Xi'$, so that $\theta\le C_{\Xi'}/p$ for $\xi\in\Xi'$ and $p\ge p_{\mathrm{st}}(\Xi')$. \begin{enumerate}[label=\textup{(S\arabic*)}] \item\label{item:jets-S1} (\Cref{def:stab-class,thm:stab-window}.) There exist constants $a_{\mathrm{st}},K_{\mathrm{st}}>0$, $A_{\mathrm{st}}\ge0$, and $\rho\in(0,1/16]$ such that for all $p\ge p_{\mathrm{st}}(\Xi')$ and $\xi\in\Xi'$, we have $\nu_{p,E}=e^{-p}\delta_{1+\theta}+f(b)\dd b$ with \begin{equation}\label{eq:jets-class} \begin{gathered} \nu_{p,E}(\mathcal C_p^c)\le e^{-a_{\mathrm{st}}\sqrt p},\qquad \sup_{|b|<\rho}f(b)\le p^{A_{\mathrm{st}}}e^{-a_{\mathrm{st}}p},\\ \nu_{p,E}(|b|>L)\le\frac{K_{\mathrm{st}}p^{A_{\mathrm{st}}}e^{-a_{\mathrm{st}}p}}{L}+e^{-a_{\mathrm{st}}pL} \end{gathered} \end{equation} for $L\ge3$. Indeed, $\nu_{p,E}$ belongs to the class $\mathfrak S_{p,\theta}$ of \Cref{def:stab-class} with the canonical tuple, and we may take $(a_{\mathrm{st}},A_{\mathrm{st}},K_{\mathrm{st}},\rho)=(1,0,1,1/16)$; we keep general constants to indicate how they enter. Further, $\xi\mapsto\nu_{p,\xi\sqrt p}$ is weakly continuous on $\Xi'$, and $\nu_{p,E}$ is a fixed point of the law map $\mathcal R_{p,\theta}$ of \Cref{def:stab-lawmap}, that is, it satisfies \eqref{eq:jets-B-equation}. \item\label{item:jets-S2} (\Cref{def:stab-mixed,lem:stab-mixed-basic}.) The cutoffs $\chi_{\mathrm c},\chi_{\mathrm b},\chi_{\mathrm i}\colon\R\to[0,1]$ satisfy $\chi_{\mathrm c}+\chi_{\mathrm b}+\chi_{\mathrm i}=1$, $\chi_{\mathrm c}=1$ on $\mathcal C_p$, $\supp\chi_{\mathrm c}\subset\{|b-1|\le2r_p\}$, $|\chi_{\mathrm c}'|\le 2r_p^{-1}$, $\chi_{\mathrm b}=1$ on $[-\rho/4,\rho/4]$, and $\supp\chi_{\mathrm b}\subset[-\rho/2,\rho/2]$. For a finite real measure $\sigma$ on $\R$ with $\sigma(\R)=0$, we have $m_\alpha(\sigma)=\int\chi_\alpha\dd\sigma$ and $\Pi_\alpha\sigma=\chi_\alpha\sigma-m_\alpha(\sigma)\delta_1$ for $\alpha\in\{\mathrm c,\mathrm b,\mathrm i\}$, and the component norms are \begin{align*} \|\Pi_{\mathrm c}\sigma\|_{\mathrm c}&=\sup\Bigl\{\Bigl|\int\varphi\dd\Pi_{\mathrm c}\sigma\Bigr|:\varphi(1)=0,\ \Lip(\varphi)\le1\Bigr\},\\ \|\Pi_{\mathrm b}\sigma\|_{\mathrm b}&=\|h\|_{L^\infty([-\rho/2,\rho/2])}\quad\text{when }\chi_{\mathrm b}\sigma=h(b)\dd b,\\ \|\Pi_{\mathrm i}\sigma\|_{\mathrm i}&=\sup\Bigl\{\Bigl|\int\varphi\dd\Pi_{\mathrm i}\sigma\Bigr|:\|\varphi\|_\infty\le1,\ \Lip(\varphi)\le1\Bigr\}. \end{align*} The second norm is finite when $\chi_{\mathrm b}\sigma$ is absolutely continuous with bounded density. In the notation of \Cref{def:stab-mixed}, $\|\Pi_\alpha\sigma\|_\alpha=n_\alpha(\sigma)$. \item\label{item:jets-S3} (\Cref{lem:stab-linearization}.) The linearization of the law map at a law $\lambda$ is the operator $\mathcal L_{p,\theta;\lambda}$, given on finite real measures $\sigma$ with $|\sigma|(\{0\})=0$ by \begin{equation}\label{eq:jets-dual-L} \int\varphi\dd(\mathcal L_{p,\theta;\lambda}\sigma)=p\int\E\Bigl[\varphi\Bigl(D_\lambda-\frac{\theta T}{pb}\Bigr)\Bigr]\sigma(\dd b) \end{equation} for bounded Borel $\varphi$, where $D_\lambda\sim\mathcal R_{p,\theta}(\lambda)$ and $T\sim\Exp(1)$ are independent. We use only the case $\lambda=\nu_{p,E}$, for which $D_\lambda\sim\nu_{p,E}$. \item\label{item:jets-S4} (\Cref{lem:stab-columns}.) Let $\mathcal L=\mathcal L_{p,\theta;\nu_{p,E}}$, and let $\sigma$ be a finite real measure with $\sigma(\R)=0$ whose restriction to $(-\rho/2,\rho/2)$ is absolutely continuous with bounded density. Then the vectors $v(\sigma)=(\|\Pi_{\mathrm c}\sigma\|_{\mathrm c},\|\Pi_{\mathrm b}\sigma\|_{\mathrm b},\|\Pi_{\mathrm i}\sigma\|_{\mathrm i})$ satisfy $v(\mathcal L\sigma)\le M_{p,\theta}v(\sigma)$ entrywise, with the matrix $M_{p,\theta}$ of \eqref{eq:stab-matrix}; this applies since $\nu_{p,E}$ belongs to the ambient class of \Cref{def:stab-class}. We use this inequality only through a cruder form. Let \begin{equation}\label{eq:jets-matrix-entries} P_p=p^{A_{\mathrm{col}}},\qquad \delta_{\mathrm b,p}=p^{A_{\mathrm{col}}}e^{-c_{\mathrm b}p},\qquad \delta_{\mathrm i,p}=p^{A_{\mathrm{col}}}e^{-c_{\mathrm i}\sqrt p},\qquad \varepsilon_p=\frac{C_{\mathrm{col}}}{p}+\delta_{\mathrm i,p}, \end{equation} with $A_{\mathrm{col}}=A_{\mathrm{st}}+3$, $c_{\mathrm b}=c_{\mathrm i}=a_{\mathrm{st}}$, and $C_{\mathrm{col}}=20C_{\Xi'}$. Since $\theta\le C_{\Xi'}/p$ and $e^{-a_{\mathrm{st}}p}\le e^{-a_{\mathrm{st}}\sqrt p}$, every entry of $M_{p,\theta}$ is at most the corresponding entry of \begin{equation}\label{eq:jets-matrix} M_p=\begin{pmatrix} \varepsilon_p&P_p&P_p\\ \delta_{\mathrm b,p}&\varepsilon_p&\delta_{\mathrm b,p}\\ \delta_{\mathrm i,p}&P_p&\delta_{\mathrm i,p}+\delta_{\mathrm b,p} \end{pmatrix} \end{equation} once $p\ge6$, and therefore $v(\mathcal L\sigma)\le M_pv(\sigma)$. The proofs below use only this form, with the structure of polynomial and exponentially small entries; any constants $A_{\mathrm{col}},C_{\mathrm{col}},c_{\mathrm b},c_{\mathrm i}>0$ depending only on $\Xi$ would do. \end{enumerate} We write $\mathsf D_t\mu$ for the pushforward of a measure $\mu$ on $\R$ under $x\mapsto tx$ ($t>0$), and $\mathsf R\mu$ for its pushforward under $x\mapsto-x$ (denoted $\mu^\vee$ in \Cref{sec:stab}). In the scaling $g=1/E$, we denote the variable by $x$, so that $b=(1+\theta)x$. The following sets are used throughout: \begin{equation}\label{eq:jets-sets} \begin{gathered} \mathcal Z=(-\rho/5,\rho/5),\qquad J_0=[1/4,7/4],\\ \mathcal S_\xi=\Bigl\{x:\bigl|(1+\theta)x-1\bigr|\le\frac{r_p}{4}\Bigr\},\qquad \mathcal O_\xi=\Bigl\{x:\bigl|(1+\theta)x-1\bigr|>\frac{r_p}{2}\Bigr\}. \end{gathered} \end{equation} The set $\mathcal S_\xi$ is a strict core, the set $\mathcal O_\xi$ is the complement of a core of half radius, and $\mathcal Z$ is an interval around zero on which the boundary cutoff equals one. We also set $x_\xi=(1+\theta)^{-1}$, the point corresponding to the anchor $b=1$. \begin{lemma}\label{lem:jets-law} There exists $p_1(\Xi)\ge p_0(\Xi)$ such that the following holds for all $p\ge p_1(\Xi)$ and $\xi\in\Xi'$. Set $H_p=2p^{A_{\mathrm{st}}}e^{-a_{\mathrm{st}}p}$. \begin{enumerate}[label=\textup{(\roman*)}] \item The law $\widetilde\nu_\xi$ satisfies \eqref{eq:jets-fixed-equation} and has the form $\widetilde\nu_\xi=e^{-p}\delta_1+\widetilde f_\xi(x)\dd x$ with $\widetilde f_\xi\le H_p$ on $(-3\rho/4,3\rho/4)$. \item For every $q'\in(0,1)$, there exists $C_{q'}$ depending only on $q'$ and $\Xi$ such that $\int(1+|x|^{q'})\,\widetilde\nu_\xi(\dd x)\le C_{q'}$. \item We have $\widetilde\nu_\xi(\mathcal S_\xi^c)\le e^{-a_{\mathrm{st}}\sqrt p/2}$. \item The map $\xi\mapsto\widetilde\nu_\xi$ is weakly continuous on $\Xi'$. \end{enumerate} \end{lemma} \begin{proof} We derived \eqref{eq:jets-fixed-equation} above. The law $\widetilde\nu_\xi=\mathsf D_{1/(1+\theta)}\nu_{p,E}$ has the atom $e^{-p}\delta_1$ and the density $\widetilde f_\xi(x)=(1+\theta)f((1+\theta)x)$. For $|x|<3\rho/4$, we have $|(1+\theta)x|<(9/8)(3\rho/4)<\rho$ by \Cref{lem:jets-parameters}, and \eqref{eq:jets-class} implies that $\widetilde f_\xi(x)\le(9/8)p^{A_{\mathrm{st}}}e^{-a_{\mathrm{st}}p}\le H_p$. This proves (i). For (ii), we have $\widetilde\nu_\xi(|x|>L)\le\nu_{p,E}(|b|>L)\le C/L$ for $L\ge3$ by \eqref{eq:jets-class}, since $1+\theta\ge1$. Then \[ \int|x|^{q'}\,\widetilde\nu_\xi(\dd x)\le 3^{q'}+\int_3^\infty q'L^{q'-1}\,\widetilde\nu_\xi(|x|>L)\dd L \le3^{q'}+Cq'\int_3^\infty L^{q'-2}\dd L, \] which is finite and depends only on $q'$ and $\Xi$. For (iii), we work in the $B$-coordinate, where $\mathcal S_\xi^c$ corresponds to $\{|b-1|>r_p/4\}$. Let $B'$ denote the right side of \eqref{eq:jets-B-equation}, with $B_j\sim\nu_{p,E}$, so that $B'\sim\nu_{p,E}$. We split the children $j\le N$ according to whether $B_j\in\mathcal C_p$. By the thinning property of Poisson processes \cite[Chapter~5]{LastPenrose2018lectures}, the number $N_2$ of children with $B_j\notin\mathcal C_p$ is $\Pois(p\,\nu_{p,E}(\mathcal C_p^c))$, so \eqref{eq:jets-class} gives \[ \P(N_2\ge1)\le p\,\nu_{p,E}(\mathcal C_p^c)\le pe^{-a_{\mathrm{st}}\sqrt p}. \] The sum $S=(\theta/p)\sum_{j\le N:\,B_j\in\mathcal C_p}Y_j/B_j$ is compound Poisson with intensity $p_1=p\,\nu_{p,E}(\mathcal C_p)\le p$ and jumps distributed as $Z=\theta Y/(p\hat B)$, where $\hat B$ has law $\nu_{p,E}(\cdot\,|\,\mathcal C_p)$ and is independent of $Y$. On the event $\{N_2=0\}$ we have $B'=1+\theta-S$. Since $r_p\le1/2$, we have $1/\hat B\in[1-2r_p,1+2r_p]$, so \[ |\E S-\theta|=\bigl|\theta\,\nu_{p,E}(\mathcal C_p)\E[\hat B^{-1}]-\theta\bigr|\le\theta\bigl(\nu_{p,E}(\mathcal C_p^c)+2r_p\bigr)\le3\theta r_p \] once $e^{-a_{\mathrm{st}}\sqrt p}\le r_p$. Let $\lambda\in\R$ with $|\lambda|\le p/(4\theta)$. Then $|\lambda Z|\le Y/2$, and the inequality $e^w-1-w\le w^2e^{|w|}$ gives \[ \log\E e^{\lambda(S-\E S)}=p_1\E\bigl[e^{\lambda Z}-1-\lambda Z\bigr]\le p\lambda^2\Bigl(\frac{2\theta}{p}\Bigr)^2\E\bigl[Y^2e^{Y/2}\bigr]=\frac{64\lambda^2\theta^2}{p}. \] In the last equality we used that $\E[Y^2e^{Y/2}]=16$. Taking $\lambda=\pm p/(4\theta)$ in the exponential Chebyshev inequality, we obtain \[ \P\bigl(|S-\E S|\ge r_p/8\bigr)\le2\exp\Bigl(-\frac{pr_p}{32\theta}+4p\Bigr)\le2\exp\bigl(-c\,p^{7/4}+4p\bigr), \] where we used $\theta\le C/p$. On the event $\{N_2=0\}\cap\{|S-\E S|r_p/4\bigr)\le pe^{-a_{\mathrm{st}}\sqrt p}+2e^{-cp^{7/4}+4p}\le e^{-a_{\mathrm{st}}\sqrt p/2} \] for $p$ large, which proves (iii). For (iv), let $\xi_n\to\xi$ in $\Xi'$. By \ref{item:jets-S1}, $B_n\sim\nu_{p,\xi_n\sqrt p}$ converges in law to $B\sim\nu_{p,\xi\sqrt p}$, and $1+\theta(\xi_n)\to1+\theta(\xi)$. By Slutsky's lemma, $B_n/(1+\theta(\xi_n))$ converges in law to $B/(1+\theta(\xi))$. \end{proof} \subsection{Component norms in the scaling \texorpdfstring{$g=1/E$}{g=1/E}}\label{sec:jets-components} We transport the component norms of \ref{item:jets-S2} to the scaling $g=1/E$ by the dilation $b=(1+\theta)x$. \begin{definition}\label{def:jets-components} Let $p\ge p_1(\Xi)$ and $\xi\in\Xi'$. Let $\sigma$ be a finite real measure on $\R$ with $\sigma(\R)=0$ whose restriction to $[-\rho/2,\rho/2]$ is absolutely continuous with bounded density. Set $\sigma^B=\mathsf D_{1+\theta}\sigma$, and for $\alpha\in\{\mathrm c,\mathrm b,\mathrm i\}$ set $\Pi^\xi_\alpha\sigma=\mathsf D_{1/(1+\theta)}\Pi_\alpha\sigma^B$ and \[ \|\sigma\|_{\mathrm c,\xi}=\|\Pi_{\mathrm c}\sigma^B\|_{\mathrm c},\qquad \|\sigma\|_{\mathrm b,\xi}=\|\Pi_{\mathrm b}\sigma^B\|_{\mathrm b},\qquad \|\sigma\|_{\mathrm i,\xi}=\|\Pi_{\mathrm i}\sigma^B\|_{\mathrm i}. \] For $\gamma>0$ we define the mixed seminorm \[ N_{\xi,\gamma}(\sigma)=\|\sigma\|_{\mathrm c,\xi}+e^{\gamma p}\|\sigma\|_{\mathrm b,\xi}+e^{\gamma\sqrt p}\|\sigma\|_{\mathrm i,\xi}. \] For a complex measure $\sigma$ whose real and imaginary parts satisfy these assumptions, each of the four quantities is defined as its value at $\Re\sigma$ plus its value at $\Im\sigma$. \end{definition} The measure $\sigma^B$ is absolutely continuous on $[-\rho/2,\rho/2]$, because $\sigma$ is absolutely continuous on $[-\rho/(2(1+\theta)),\rho/(2(1+\theta))]$. In particular, all four quantities are finite. We record the properties of these norms that we use. \begin{lemma}\label{lem:jets-components} Let $p\ge p_1(\Xi)$, $\xi\in\Xi'$, and let $\sigma$ be as in \Cref{def:jets-components}. Set $S_{\mathrm c}=\{x:|(1+\theta)x-1|\le2r_p\}$ and $S_{\mathrm i}=\{x:|(1+\theta)x|\ge\rho/4,\ |(1+\theta)x-1|\ge r_p\}\cup\{x_\xi\}$. \begin{enumerate}[label=\textup{(\roman*)}] \item We have $\sigma=\Pi^\xi_{\mathrm c}\sigma+\Pi^\xi_{\mathrm b}\sigma+\Pi^\xi_{\mathrm i}\sigma$, and each summand has total mass zero. The measure $\Pi^\xi_{\mathrm c}\sigma$ is carried by $S_{\mathrm c}\subset J_0$, the measure $\Pi^\xi_{\mathrm i}\sigma$ is carried by $S_{\mathrm i}\subset\{|x|\ge2\rho/9\}$, and $\Pi^\xi_{\mathrm b}\sigma=g(x)\dd x-(\int g)\delta_{x_\xi}$ with $g$ supported in $[-\rho/2,\rho/2]$ and $|g|\le2\|\sigma\|_{\mathrm b,\xi}$. \item The restriction of $\sigma$ to $\mathcal Z$ is absolutely continuous with density bounded by $2\|\sigma\|_{\mathrm b,\xi}$. \item We have \[ \|\sigma\|_{\mathrm c,\xi}\le2r_p\|\sigma\|_\TV,\qquad \|\sigma\|_{\mathrm b,\xi}\le\sup_{|x|\le\rho/2}\Bigl|\frac{\dd\sigma}{\dd x}(x)\Bigr|,\qquad \|\sigma\|_{\mathrm i,\xi}\le2|\sigma|(\mathcal O_\xi). \] \item Let $V$ be a real Banach space. If $f\colon S_{\mathrm c}\to V$ is Lipschitz, then \[ \Bigl\|\int f\dd\Pi^\xi_{\mathrm c}\sigma\Bigr\|_V\le\Lip_{S_{\mathrm c}}(f)\,\|\sigma\|_{\mathrm c,\xi}. \] If $f\colon S_{\mathrm i}\to V$ is bounded and Lipschitz, then \[ \Bigl\|\int f\dd\Pi^\xi_{\mathrm i}\sigma\Bigr\|_V\le\Bigl(\sup_{S_{\mathrm i}}\|f\|_V+\Lip_{S_{\mathrm i}}(f)\Bigr)\|\sigma\|_{\mathrm i,\xi}. \] \end{enumerate} \end{lemma} \begin{proof} Part (i) follows from \ref{item:jets-S2} after the change of variables $b=(1+\theta)x$: the measure $\Pi_{\mathrm c}\sigma^B$ is carried by $\supp\chi_{\mathrm c}\cup\{1\}\subset\{|b-1|\le2r_p\}$, and the measure $\Pi_{\mathrm i}\sigma^B$ is carried by $\supp\chi_{\mathrm i}\cup\{1\}$, where $\supp\chi_{\mathrm i}\subset\{|b|\ge\rho/4,\ |b-1|\ge r_p\}$ because $\chi_{\mathrm i}$ vanishes on $\mathcal C_p\cup[-\rho/4,\rho/4]$. If $\Pi_{\mathrm b}\sigma^B=h(b)\dd b-(\int h)\delta_1$, then $g(x)=(1+\theta)h((1+\theta)x)$, which gives $|g|\le(9/8)\|\sigma\|_{\mathrm b,\xi}$ by \Cref{lem:jets-parameters}. The inclusions $S_{\mathrm c}\subset J_0$ and $S_{\mathrm i}\subset\{|x|\ge2\rho/9\}$ follow from $r_p\le1/4$ and $1\le1+\theta\le9/8$. For (ii), if $x\in\mathcal Z$ then $|(1+\theta)x|<\rho/4$, so $\chi_{\mathrm b}((1+\theta)x)=1$ and $\chi_{\mathrm c}((1+\theta)x)=\chi_{\mathrm i}((1+\theta)x)=0$; further, $x_\xi\notin\mathcal Z$. The restriction of $\sigma$ to $\mathcal Z$ is therefore the restriction of $g(x)\dd x$, and (i) gives the bound. For (iii), let $\varphi$ be one of the test functions in the definition of $\|\cdot\|_{\mathrm c}$. Since $\varphi(1)=0$ and $\Lip(\varphi)\le1$, we have $|\chi_{\mathrm c}\varphi|\le2r_p$, and $\int\varphi\dd\Pi_{\mathrm c}\sigma^B=\int\chi_{\mathrm c}\varphi\dd\sigma^B$ is bounded by $2r_p\|\sigma\|_\TV$. The second bound holds because the density of $\sigma^B$ at $b$ is $(1+\theta)^{-1}(\dd\sigma/\dd x)(b/(1+\theta))$ and $\chi_{\mathrm b}\le1$. For the third, if $\|\varphi\|_\infty\le1$, then \[ \Bigl|\int\varphi\dd\Pi_{\mathrm i}\sigma^B\Bigr|\le\int\chi_{\mathrm i}\dd|\sigma^B|+|m_{\mathrm i}(\sigma^B)|\le2\int\chi_{\mathrm i}\dd|\sigma^B|\le2|\sigma^B|(\mathcal C_p^c), \] and $|\sigma^B|(\mathcal C_p^c)=|\sigma|(\{|(1+\theta)x-1|>r_p\})\le|\sigma|(\mathcal O_\xi)$. For (iv), let $\lambda\in V^*$ with $\|\lambda\|\le1$, and set $F=\lambda\circ f$ and $F^B(b)=F(b/(1+\theta))$ on $(1+\theta)S_{\mathrm c}$. Then $\Lip(F^B)\le\Lip_{S_{\mathrm c}}(f)$. Let $\widetilde F$ be the McShane extension of $F^B$ to $\R$, which has the same Lipschitz constant, and set $\varphi=\widetilde F-\widetilde F(1)$. Since $\Pi_{\mathrm c}\sigma^B$ has mass zero and is carried by $(1+\theta)S_{\mathrm c}$, we obtain \[ \lambda\Bigl(\int f\dd\Pi^\xi_{\mathrm c}\sigma\Bigr)=\int F^B\dd\Pi_{\mathrm c}\sigma^B=\int\varphi\dd\Pi_{\mathrm c}\sigma^B, \] and the right side is bounded by $\Lip_{S_{\mathrm c}}(f)\|\sigma\|_{\mathrm c,\xi}$. The first identity holds because Bochner integrals commute with bounded functionals; the integral exists because $f$ is continuous and bounded on the compact set $S_{\mathrm c}$. Taking the supremum over $\lambda$ and using the Hahn--Banach theorem, we obtain the first inequality. For the second, we argue in the same way, with $\widetilde F$ replaced by the McShane extension truncated at $\pm s$, where $s=\sup_{S_{\mathrm i}}|F|$. Since the truncation does not increase the Lipschitz constant, $\widetilde F/\max\{s,\Lip(\widetilde F)\}$ is one of the test functions in the definition of $\|\cdot\|_{\mathrm i}$, and the bound follows since $\max\{s,\Lip\}\le s+\Lip$. \end{proof} \section{An analytic Banach space for the law map}\label{sec:jets-space} \subsection{The spaces}\label{sec:jets-spaces} Fix $q\in(0,1)$, and set \begin{equation}\label{eq:jets-R0} R_0=\frac{5\rho}{8}. \end{equation} Then $\rho/20\}$. For $\tau\in\Omega$, $x\in\R\setminus\{0\}$, and $y\in\R$, define \begin{equation}\label{eq:jets-kernel} k_\tau(x;y)=\frac{|x|}{\tau}\exp\Bigl(-\frac{|x||y|}{\tau}\Bigr)\one_{\{xy>0\}}, \end{equation} and for $\mu\in X_q$ let $\mathcal K_\tau\mu$ be the measure with density $y\mapsto\int k_\tau(x;y)\,\mu(\dd x)$. This is the analogue in the scaling $g=1/E$ of the kernel $j_\theta$ and the operator $\mathsf K_\theta$ of \Cref{sec:stab-lawmap}. For real $\tau>0$, $k_\tau(x;\cdot)$ is the density of $\tau Y/x$. If $\mu$ is a probability measure with $\mu(\{0\})=0$, then $\mathcal K_\tau\mu$ is the law of the jump $\tau Y/X$, where $X\sim\mu$ and $Y\sim\Exp(1)$ are independent. The parameter $\tau$ changes the density of the jump but not its support. \begin{lemma}\label{lem:jets-kernel} \begin{enumerate}[label=\textup{(\roman*)}] \item For all $\tau\in\Omega$ and $x\neq0$, we have $\int k_\tau(x;y)\dd y=1$. In particular $(\mathcal K_\tau\mu)(\R)=\mu(\R)$ for $\mu\in X_q$. \item The map $\tau\mapsto\mathcal K_\tau$ is holomorphic from $\Omega$ to the space of bounded operators from $X_q$ to $Z_q$, with the operator norm. For $j\ge1$, its $j$-th derivative is the integral operator with kernel $\partial_\tau^jk_\tau(x;y)$. For every compact $\Omega'\subset\Omega$ and $j\ge0$, $\sup_{\tau\in\Omega'}\|\partial_\tau^j\mathcal K_\tau\|_{X_q\to Z_q}$ is bounded by a constant depending only on $\Omega'$, $j$, $q$, and $\rho$. \end{enumerate} \end{lemma} \begin{proof} We write $\kappa=1/\tau$, which maps $\Omega$ biholomorphically onto itself, so that $k_\tau(x;y)=|x|\kappa e^{-\kappa s}\one_{\{xy>0\}}$ with $s=|x||y|$. Part (i) follows from $\int_0^\infty|x|\kappa e^{-\kappa|x|y}\dd y=1$ for $\Re\kappa>0$. For (ii), fix $\kappa_0\in\Omega$ and let $\mathbb D$ be the closed disc of radius $\alpha=\Re\kappa_0/2$ around $\kappa_0$, so that $\Re\kappa\ge\alpha$ and $|\kappa|\le2|\kappa_0|$ for $\kappa\in\mathbb D$. By the Leibniz rule, \[ \partial_\kappa^j\bigl(|x|\kappa e^{-\kappa s}\bigr)=|x|\bigl((-s)^j\kappa+j(-s)^{j-1}\bigr)e^{-\kappa s}, \] so for $\kappa\in\mathbb D$, with the substitution $u=|x||y|$, \begin{align*} \int(1+|y|^q)\bigl|\partial_\kappa^jk\bigr|\dd y&\le\int_0^\infty\bigl(2|\kappa_0|u^j+ju^{j-1}\bigr)e^{-\alpha u}\Bigl(1+\frac{u^q}{|x|^q}\Bigr)\dd u\le C_j\bigl(1+|x|^{-q}\bigr),\\ |y|\,\bigl|\partial_\kappa^jk\bigr|&\le u\bigl(2|\kappa_0|u^j+ju^{j-1}\bigr)e^{-\alpha u}\le C_j, \end{align*} where $C_j$ depends only on $\kappa_0$, $j$, and $q$. Integrating against $|\mu|$ and using \eqref{eq:jets-negative-moment}, we find that the integral operator with kernel $\partial_\kappa^jk$ is bounded from $X_q$ to $Z_q$, with norm at most $2C_jC_{q,\rho}$, uniformly for $\kappa\in\mathbb D$. For $\kappa,\kappa+h\in\mathbb D$ we have pointwise \[ k_{\kappa+h}-k_\kappa-h\,\partial_\kappa k_\kappa=h^2\int_0^1(1-t)\,\partial_\kappa^2k_{\kappa+th}\dd t, \] and the previous bound with $j=2$ gives that the corresponding operator has norm at most $2C_2C_{q,\rho}|h|^2$. Then $\kappa\mapsto\mathcal K$ is complex differentiable in operator norm, with derivative the integral operator with kernel $\partial_\kappa k$. Applying the same argument to $\partial_\kappa^jk$ in place of $k$ and inducting on $j$, we obtain the statement about higher derivatives. Composing with $\tau\mapsto1/\tau$ and covering a compact $\Omega'$ by finitely many discs proves (ii). \end{proof} \subsection{Convolution powers}\label{sec:jets-convolution} The local density of the law map is controlled by the following estimate: if a sum of $k$ jumps lands at distance $|z|$ from zero, then one of the jumps has modulus at least $|z|/k$. \begin{lemma}\label{lem:jets-onejump} Let $k\ge1$ and let $\ell_1,\dots,\ell_k\in L^1(\R)$ be complex functions with $|\ell_i(y)|\le M_i/|y|$ for almost every $y$. Then for almost every $z\neq0$, \begin{equation}\label{eq:jets-onejump} \bigl(|\ell_1|*\cdots*|\ell_k|\bigr)(z)\le\frac{k}{|z|}\sum_{i=1}^kM_i\prod_{j\neq i}\|\ell_j\|_{L^1}. \end{equation} In particular, $|(\ell_1*\cdots*\ell_k)(z)|\le k^2|z|^{-1}\prod_{i=1}^k\max\{M_i,\|\ell_i\|_{L^1}\}$ for almost every $z\ne0$. \end{lemma} \begin{proof} For $k=1$, the conclusion is the assumed almost-everywhere bound. Suppose $k\ge2$, and let $H_z=\{(y_1,\dots,y_k)\in\R^k:y_1+\dots+y_k=z\}$. The left side of \eqref{eq:jets-onejump} is the integral of $\prod_j|\ell_j(y_j)|$ over $H_z$ with respect to the Lebesgue measure in any $k-1$ of the coordinates; all these parametrizations give the same measure on $H_z$, since the changes of variables between them have Jacobian $\pm1$. On $H_z$ we have $\max_i|y_i|\ge|z|/k$. Then $H_z$ is covered by the sets $A_i=H_z\cap\{|y_i|\ge|z|/k\}$. On $A_i$ we bound $|\ell_i(y_i)|\le kM_i/|z|$; this holds outside a set of $y_i$ of measure zero, whose preimage in $H_z$ is null in the parametrization by $(y_j)_{j\ne i}$. Integrating the other $k-1$ factors over $(y_j)_{j\ne i}\in\R^{k-1}$, we obtain \[ \int_{A_i}\prod_{j=1}^k|\ell_j(y_j)|\le\frac{kM_i}{|z|}\prod_{j\ne i}\|\ell_j\|_{L^1}. \] Summing over $i$ proves \eqref{eq:jets-onejump}, and the second bound follows. \end{proof} For $\alpha\in Z_q$, define the compound Poisson measure (as in \Cref{sec:stab-lawmap}) \begin{equation}\label{eq:jets-CP} \mathsf P_p(\alpha)=e^{-p}\sum_{k=0}^\infty\frac{p^k}{k!}\alpha^{*k},\qquad\alpha^{*0}=\delta_0. \end{equation} For a probability density $\alpha$, this is the law of a sum of $N\sim\Pois(p)$ independent variables with law $\alpha$. \begin{lemma}\label{lem:jets-CP} For $k\ge1$ and $\alpha_1,\dots,\alpha_k\in Z_q$, the measure $\alpha_1*\dots*\alpha_k$ is absolutely continuous and \begin{equation}\label{eq:jets-multilinear} \|\alpha_1*\dots*\alpha_k\|_{W_q}\le3k^2\prod_{i=1}^k\|\alpha_i\|_{Z_q}. \end{equation} The series \eqref{eq:jets-CP} converges in $W_q$, and $\mathsf P_p\colon Z_q\to W_q$ is holomorphic. Its mass is $\mathsf P_p(\alpha)(\R)=\exp(p(\alpha(\R)-1))$, and for $m\ge1$ and $\beta_1,\dots,\beta_m\in Z_q$, \begin{equation}\label{eq:jets-CP-derivative} D^m\mathsf P_p(\alpha)[\beta_1,\dots,\beta_m]=p^m\,\mathsf P_p(\alpha)*\beta_1*\dots*\beta_m, \end{equation} where the right side is the series $e^{-p}\sum_{k\ge0}p^{k+m}(k!)^{-1}\alpha^{*k}*\beta_1*\dots*\beta_m$, which converges in $W_q$. \end{lemma} \begin{proof} Let $\alpha_i=\ell_i(y)\dd y$. The weighted total variation of $\alpha_1*\dots*\alpha_k$ is at most $\prod_i\|\alpha_i\|_{q,\TV}$ by \Cref{lem:jets-complete}. The convolution is absolutely continuous with density $\ell_1*\dots*\ell_k$. For $z\in[1-R_0,1+R_0]$ we have $|z|\ge1-R_0\ge1/2$, and \Cref{lem:jets-onejump} with $M_i=\|\alpha_i\|_{Z_q}\ge\|\ell_i\|_{L^1}$ implies that $|(\ell_1*\dots*\ell_k)(z)|\le2k^2\prod_i\|\alpha_i\|_{Z_q}$ for almost every such $z$. This proves \eqref{eq:jets-multilinear}. The term $k=0$ is $\delta_0$, which does not charge $[1-R_0,1+R_0]$ and has norm one in $W_q$. Since $\sum_kp^k3k^2r^k/k!<\infty$ for every $r>0$, the series \eqref{eq:jets-CP} and the series in \eqref{eq:jets-CP-derivative} converge in $W_q$, uniformly for $\alpha$ and $\beta_i$ in bounded sets. Let $A_k(\alpha_1,\dots,\alpha_k)=\alpha_1*\dots*\alpha_k$, a bounded symmetric $k$-linear map by \eqref{eq:jets-multilinear} and commutativity. For $\alpha,\beta\in Z_q$ and $k\ge2$, we have \[ \alpha_+^{*k}-\alpha^{*k}-k\,A_k(\alpha,\dots,\alpha,\beta)=\sum_{j=2}^k\binom kjA_k(\underbrace{\alpha,\dots,\alpha}_{k-j},\underbrace{\beta,\dots,\beta}_{j}),\qquad\alpha_+=\alpha+\beta. \] Since $\binom kj\le k^2\binom{k-2}{j-2}$ for $j\ge2$, the norm of the right side is at most $3k^4\|\beta\|_{Z_q}^2(\|\alpha\|_{Z_q}+\|\beta\|_{Z_q})^{k-2}$. Multiplying by $e^{-p}p^k/k!$ and summing over $k$, we find that $\mathsf P_p$ is complex differentiable at $\alpha$ with derivative $\beta\mapsto e^{-p}\sum_{k\ge1}p^kk(k!)^{-1}\alpha^{*(k-1)}*\beta=p\,\mathsf P_p(\alpha)*\beta$. The $m$-th derivative is computed in the same way, by induction on $m$: at each stage the derivative is a series of the same form in $\alpha$, multilinear in $\beta_1,\dots,\beta_m$, and it converges in $W_q$ uniformly on bounded sets by \eqref{eq:jets-multilinear}. This gives \eqref{eq:jets-CP-derivative}. The mass formula holds because the mass of a convolution is the product of the masses. \end{proof} \subsection{The law map}\label{sec:jets-lawmap} For $\tau\in\Omega$ and $\nu\in X_q$, define \begin{equation}\label{eq:jets-lawmap} \widetilde\Phi_\tau(\nu)=\delta_1*\mathsf R\,\mathsf P_p(\mathcal K_\tau\nu). \end{equation} For real $\tau>0$ and a probability measure $\nu\in X_q$, the measure $\mathcal K_\tau\nu$ is the law of $\tau Y/X$ with $X\sim\nu$, and $\widetilde\Phi_\tau(\nu)$ is the law of $1-\tau\sum_{j\le N}Y_j/X_j$ with $X_j$ independent with law $\nu$. This is the law map of the rescaled equation \eqref{eq:jets-fixed-equation}. \begin{lemma}\label{lem:jets-holomorphic} \begin{enumerate}[label=\textup{(\roman*)}] \item The map $(\tau,\nu)\mapsto\widetilde\Phi_\tau(\nu)$ is holomorphic from $\Omega\times X_q$ to $X_q$. \item For $\tau\in\Omega$ and $\nu\in\mathcal A_q$, we have $\widetilde\Phi_\tau(\nu)\in\mathcal A_q$, and $\widetilde\Phi_\tau(\nu)-e^{-p}\delta_1$ is absolutely continuous. \item For $\tau\in\Omega$, $\nu\in X_q$, $m\ge1$, and $\sigma_1,\dots,\sigma_m\in X_q$, \begin{equation}\label{eq:jets-Dm-lawmap} D^m_\nu\widetilde\Phi_\tau(\nu)[\sigma_1,\dots,\sigma_m]=p^m\,\widetilde\Phi_\tau(\nu)*\mathsf R\mathcal K_\tau\sigma_1*\dots*\mathsf R\mathcal K_\tau\sigma_m. \end{equation} \item Let $p\ge p_1(\Xi)$ and $\xi\in\Xi'$. Then $\widetilde\nu_\xi\in\mathcal A_q$ and $\widetilde\Phi_{\tau(\xi)}(\widetilde\nu_\xi)=\widetilde\nu_\xi$. \end{enumerate} \end{lemma} \begin{proof} The map $\mu\mapsto\delta_1*\mathsf R\mu$ is a bounded linear map from $W_q$ to $X_q$: it is the pushforward under $z\mapsto1-z$, which maps $[1-R_0,1+R_0]$ onto $[-R_0,R_0]$ and preserves the size of densities, and it at most doubles the weighted total variation by \Cref{lem:jets-complete}. The map $(\tau,\nu)\mapsto\mathcal K_\tau\nu$ is holomorphic from $\Omega\times X_q$ to $Z_q$: by \Cref{lem:jets-kernel}, for $h\in\C$ and $\mu\in X_q$, \begin{multline*} \bigl\|\mathcal K_{\tau+h}(\nu+\mu)-\mathcal K_\tau\nu-h(\partial_\tau\mathcal K_\tau)\nu-\mathcal K_\tau\mu\bigr\|_{Z_q}\\ \le\|\mathcal K_{\tau+h}-\mathcal K_\tau-h\partial_\tau\mathcal K_\tau\|\,\|\nu\|_{X_q}+\|\mathcal K_{\tau+h}-\mathcal K_\tau\|\,\|\mu\|_{X_q}, \end{multline*} which is $o(|h|+\|\mu\|_{X_q})$. Composing with the holomorphic map $\mathsf P_p$ of \Cref{lem:jets-CP} and with $\delta_1*\mathsf R$ proves (i). For (ii), we have $(\mathcal K_\tau\nu)(\R)=1$ by \Cref{lem:jets-kernel}(i), and \Cref{lem:jets-CP} implies that $\mathsf P_p(\mathcal K_\tau\nu)$ has mass one. The term $k=0$ of \eqref{eq:jets-CP} gives the atom $e^{-p}\delta_1$ of $\widetilde\Phi_\tau(\nu)$, and the terms $k\ge1$ are absolutely continuous. Part (iii) follows from \eqref{eq:jets-CP-derivative}, the linearity of $\nu\mapsto\mathcal K_\tau\nu$, and the identity $\delta_1*\mathsf R(\mu*\beta)=(\delta_1*\mathsf R\mu)*\mathsf R\beta$. For (iv), \Cref{lem:jets-law}(i) shows that $\widetilde\nu_\xi$ is a probability measure with a density bounded by $H_p$ on $[-R_0,R_0]\subset(-3\rho/4,3\rho/4)$, and \Cref{lem:jets-law}(ii) shows that its weighted total variation is finite. Then $\widetilde\nu_\xi\in\mathcal A_q$. The fixed point identity is \eqref{eq:jets-fixed-equation}, by the description of $\widetilde\Phi_\tau$ for real $\tau$ given before the lemma. \end{proof} \section{Invertibility of the linearization}\label{sec:jets-invert} \subsection{Score polynomials}\label{sec:jets-scores} For real parameters, derivatives of the jump kernel in the input $x$ and in the energy $\xi$ are polynomials in the normalized jump $u=|x||y|/\tau$ times the kernel itself. We use this in the rest of the section. \begin{lemma}\label{lem:jets-score} For all integers $m,r\ge0$ there exists a polynomial $P_{m,r}$ of degree at most $m+r$ with integer coefficients such that the following holds. Let $p>1$, $\xi>0$, $\tau=\tau(\xi)=(p\xi)^{-2}$, and let $x,y\in\R$ with $xy>0$. Then, with $u=|x||y|/\tau$, \begin{equation}\label{eq:jets-score} \partial_x^m\partial_\xi^r\,k_{\tau(\xi)}(x;y)=x^{-m}\xi^{-r}P_{m,r}(u)\,k_{\tau(\xi)}(x;y). \end{equation} \end{lemma} \begin{proof} For fixed $y\neq0$ and $x$ of the sign of $y$, we have $k_{\tau(\xi)}(x;y)=G(u)/|y|$ with $G(u)=ue^{-u}$ and $u=|x||y|p^2\xi^2$. Then $\partial_\xi u=2u/\xi$, $\partial_xu=u/x$, and $G'(u)=(1-u)G(u)/u$. Suppose that $\partial_x^{m}\partial_\xi^{r}k=x^{-m}\xi^{-r}P(u)G(u)/|y|$ for a polynomial $P$. Then \begin{align*} \partial_\xi\bigl(x^{-m}\xi^{-r}P(u)G(u)\bigr)&=x^{-m}\xi^{-r-1}G(u)\bigl(-rP(u)+2uP'(u)+2(1-u)P(u)\bigr),\\ \partial_x\bigl(x^{-m}\xi^{-r}P(u)G(u)\bigr)&=x^{-m-1}\xi^{-r}G(u)\bigl(-mP(u)+uP'(u)+(1-u)P(u)\bigr). \end{align*} Both brackets are polynomials with integer coefficients whose degree exceeds that of $P$ by at most one. Starting from $P_{0,0}=1$ and using that $\partial_x$ and $\partial_\xi$ commute, we obtain \eqref{eq:jets-score} by induction on $m+r$. \end{proof} \begin{lemma}\label{lem:jets-kernel-bounds} Let $q\in(0,1)$ and let $m,r\ge0$ be integers with $m+r\le5$. There exists $C>0$, depending only on $m$, $r$, $q$, and $\Xi$, such that for all $p>1$, $\xi\in\Xi'$, $\tau=\tau(\xi)$, and $x\neq0$: \begin{enumerate}[label=\textup{(\alph*)}] \item $\int(1+|y|^q)\,|\partial_x^m\partial_\xi^rk_\tau(x;y)|\dd y\le C|x|^{-m}\bigl(1+(\tau/|x|)^q\bigr)$; \item for $t>0$, $\int_{|y|\ge t}(1+|y|^q)\,|\partial_x^m\partial_\xi^rk_\tau(x;y)|\dd y\le C|x|^{-m}\bigl(1+(\tau/|x|)^q\bigr)e^{-|x|t/(2\tau)}$; \item for $y\neq0$, $|\partial_x^m\partial_\xi^rk_\tau(x;y)|\le C|x|^{-m}|y|^{-1}e^{-|x||y|/(2\tau)}$; \item if $m=0$, then $\int_\R|\partial_\xi^rk_\tau(x;y)|\dd x\le C\tau y^{-2}$ for $y\neq0$. \end{enumerate} \end{lemma} \begin{proof} On the half-line $\{xy>0\}$ we substitute $u=|x||y|/\tau$, so that $k_\tau(x;y)\dd y=e^{-u}\dd u$, $|y|^q=(\tau u/|x|)^q$, and $|y|\,k_\tau(x;y)=ue^{-u}$. By \Cref{lem:jets-score} and $\xi^{-r}\le(2/\xi_-)^r$, parts (a) and (b) follow from \[ \int_{u_0}^\infty|P_{m,r}(u)|\Bigl(1+\Bigl(\frac{\tau u}{|x|}\Bigr)^q\Bigr)e^{-u}\dd u\le\Bigl(1+\Bigl(\frac{\tau}{|x|}\Bigr)^q\Bigr)e^{-u_0/2}\int_0^\infty|P_{m,r}(u)|(1+u^q)e^{-u/2}\dd u, \] with $u_0=0$ and $u_0=|x|t/\tau$, respectively. Part (c) follows from $|P_{m,r}(u)|k_\tau(x;y)=|y|^{-1}|P_{m,r}(u)|ue^{-u}\le C|y|^{-1}e^{-u/2}$. For (d), we substitute $x=\tau u/y$ for $x$ of the sign of $y$ and obtain $\int|P_{0,r}(u)|k_\tau(x;y)\dd x=\tau y^{-2}\int_0^\infty|P_{0,r}(u)|ue^{-u}\dd u$. \end{proof} \subsection{The linearization at the principal law}\label{sec:jets-linearization} For $p\ge p_1(\Xi)$ and $\xi\in\Xi'$, let \begin{equation}\label{eq:jets-L} \widetilde{\mathcal L}_\xi=D_\nu\widetilde\Phi_{\tau(\xi)}(\widetilde\nu_\xi),\qquad\text{so that}\qquad\widetilde{\mathcal L}_\xi\sigma=p\,\widetilde\nu_\xi*\mathsf R\mathcal K_{\tau(\xi)}\sigma \end{equation} by \Cref{lem:jets-holomorphic}(iii),(iv). By \Cref{lem:jets-kernel}(i), $\widetilde{\mathcal L}_\xi$ maps $X_q^0$ into itself. Let $\widetilde D\sim\widetilde\nu_\xi$ and $T\sim\Exp(1)$ be independent. For $\sigma\in X_q$ and bounded Borel $\varphi$, by Fubini's theorem we have \begin{equation}\label{eq:jets-L-dual} \int\varphi\dd\widetilde{\mathcal L}_\xi\sigma=p\int\E\Bigl[\varphi\Bigl(\widetilde D-\frac{\tau T}{x}\Bigr)\Bigr]\sigma(\dd x). \end{equation} Further, $\widetilde{\mathcal L}_\xi\sigma$ is absolutely continuous, with density \begin{equation}\label{eq:jets-L-density} \frac{\dd\widetilde{\mathcal L}_\xi\sigma}{\dd z}(z)=p\int\bar k_{\xi,x}(z)\,\sigma(\dd x),\qquad \bar k_{\xi,x}(z)=\int k_{\tau(\xi)}(x;d-z)\,\widetilde\nu_\xi(\dd d). \end{equation} Here $\bar k_{\xi,x}$ is the density of $\widetilde D-\tau T/x$. \begin{lemma}\label{lem:jets-conjugate} Let $p\ge p_1(\Xi)$ and $\xi\in\Xi'$, and let $\mathcal L=\mathcal L_{p,\theta;\nu_{p,E}}$. For every real $\sigma\in X_q^0$, we have $\mathsf D_{1+\theta}\widetilde{\mathcal L}_\xi\sigma=\mathcal L\,\mathsf D_{1+\theta}\sigma$. Consequently, for every $\sigma\in X_q^0$, real or complex, the vector $v_\xi(\sigma)=(\|\sigma\|_{\mathrm c,\xi},\|\sigma\|_{\mathrm b,\xi},\|\sigma\|_{\mathrm i,\xi})$ satisfies $v_\xi(\widetilde{\mathcal L}_\xi\sigma)\le M_pv_\xi(\sigma)$ entrywise, with $M_p$ as in \eqref{eq:jets-matrix}. \end{lemma} \begin{proof} Let $\varphi$ be bounded and Borel, and set $\sigma^B=\mathsf D_{1+\theta}\sigma$. Let $D=(1+\theta)\widetilde D$, which has law $\nu_{p,E}$. Using \eqref{eq:jets-L-dual} with the test $x\mapsto\varphi((1+\theta)x)$ and substituting $b=(1+\theta)x$, we find \[ \int\varphi\dd\mathsf D_{1+\theta}\widetilde{\mathcal L}_\xi\sigma=p\int\E\Bigl[\varphi\Bigl(D-\frac{(1+\theta)^2\tau T}{b}\Bigr)\Bigr]\sigma^B(\dd b)=p\int\E\Bigl[\varphi\Bigl(D-\frac{\theta T}{pb}\Bigr)\Bigr]\sigma^B(\dd b), \] where we used \eqref{eq:jets-tau-theta} in the second equality. By \eqref{eq:jets-dual-L} the right side is $\int\varphi\dd\mathcal L\sigma^B$; the measure $\sigma^B$ is real, has mass zero, does not charge zero, and $\chi_{\mathrm b}\sigma^B$ has a bounded density, as noted after \Cref{def:jets-components}. This proves the identity. Since the components of $\widetilde{\mathcal L}_\xi\sigma$ in the sense of \Cref{def:jets-components} are those of $\mathcal L\sigma^B$, the matrix inequality for real $\sigma$ follows from \eqref{eq:jets-matrix}. Since $\widetilde{\mathcal L}_\xi$ maps real measures to real measures, the complex case follows by applying the real case to $\Re\sigma$ and $\Im\sigma$ and adding. \end{proof} In the next lemma and below we write $L^1_q(G)$, for a Borel set $G\subset\R$, for the space of real functions on $G$ with norm $\int_G(1+|z|^q)|F(z)|\dd z$. We also set \begin{equation}\label{eq:jets-a1} q'=\frac{1+q}{2},\qquad a_1=\frac{(1-q)\,a_{\mathrm{st}}}{2(1+q)}. \end{equation} By H\"older's inequality with exponents $q'/q$ and $q'/(q'-q)$, \Cref{lem:jets-law}(ii),(iii), and $(1+|d|^q)^{q'/q}\le2^{q'/q}(1+|d|^{q'})$, we have \begin{equation}\label{eq:jets-holder} \int_{\mathcal S_\xi^c}(1+|d|^q)\,\widetilde\nu_\xi(\dd d)\le C\,\widetilde\nu_\xi(\mathcal S_\xi^c)^{1-q/q'}\le Ce^{-a_1\sqrt p}, \end{equation} since $1-q/q'=(1-q)/(1+q)$. \begin{lemma}\label{lem:jets-L-bounds} There exists $C>0$, depending only on $q$ and $\Xi$, such that for all $p\ge p_1(\Xi)$, $\xi\in\Xi'$, and $\sigma\in X_q^0$, writing $\|\sigma\|_{\mathrm{cbi}}=\|\sigma\|_{\mathrm c,\xi}+\|\sigma\|_{\mathrm b,\xi}+\|\sigma\|_{\mathrm i,\xi}$: \begin{enumerate}[label=\textup{(\roman*)}] \item $\|\widetilde{\mathcal L}_\xi\sigma\|_{q,\TV}\le Cp\,\|\sigma\|_{\mathrm{cbi}}$; \item the density of $\widetilde{\mathcal L}_\xi\sigma$ is bounded by $Cp\,\|\sigma\|_{\mathrm{cbi}}$ on $[-R_0,R_0]$; \item $\int_{\mathcal O_\xi}(1+|z|^q)\,|\widetilde{\mathcal L}_\xi\sigma|(\dd z)\le Cpe^{-a_1\sqrt p}\bigl(\|\sigma\|_{\mathrm c,\xi}+\|\sigma\|_{\mathrm i,\xi}\bigr)+Cp\|\sigma\|_{\mathrm b,\xi}$; \item for every $\gamma>0$, $N_{\xi,\gamma}(\widetilde{\mathcal L}_\xi\sigma)\le Cpe^{\gamma p}\|\sigma\|_{X_q}$. \end{enumerate} \end{lemma} \begin{proof} It suffices to consider real $\sigma$. We decompose $\sigma=\Pi^\xi_{\mathrm c}\sigma+\Pi^\xi_{\mathrm b}\sigma+\Pi^\xi_{\mathrm i}\sigma$ as in \Cref{lem:jets-components}, write $\Pi^\xi_{\mathrm b}\sigma=g\dd x-(\int g)\delta_{x_\xi}$, and treat the three components separately. Throughout, $\tau=\tau(\xi)\le4/(p\xi_-)^2$. We use the following consequence of \Cref{lem:jets-kernel-bounds}(a) for $x\ne0$ and $s\in\R$: \begin{equation}\label{eq:jets-Lip-of-kernel} \int(1+|y|^q)\bigl|k_\tau(x;y)-k_\tau(s;y)\bigr|\dd y\le|x-s|\sup_{w\in[x,s]}\int(1+|y|^q)|\partial_xk_\tau(w;y)|\dd y, \end{equation} valid when the segment $[x,s]$ does not contain zero, which follows from the fundamental theorem of calculus in $x$ and Tonelli's theorem. The same inequality holds with the weight and the domain of integration restricted to any Borel set of $y$, and after integrating in $d$ against $\widetilde\nu_\xi$ with the shift $y=d-z$. \proofstep{1}{Weighted total variation.} Let $f(x)=k_\tau(x;\cdot)\in L^1_q(\R)$. By \Cref{lem:jets-kernel-bounds}(a), $\|f(x)\|\le C$ and $\|\partial_xf(x)\|\le C$ for $|x|\ge2\rho/9$. By \eqref{eq:jets-Lip-of-kernel}, $f$ is Lipschitz with constant at most $C$ on $S_{\mathrm c}\subset J_0$ and on each of $S_{\mathrm i}\cap(0,\infty)$ and $S_{\mathrm i}\cap(-\infty,0)$. If $x,s\in S_{\mathrm i}$ have opposite signs, then $|x-s|\ge4\rho/9$, and therefore $\|f(x)-f(s)\|\le2C\le(9C/(2\rho))|x-s|$. We conclude that $\Lip_{S_{\mathrm i}}(f)\le C$. The density of $\mathcal K_\tau\Pi^\xi_\alpha\sigma$ is the Bochner integral $\int f\dd\Pi^\xi_\alpha\sigma$, and \Cref{lem:jets-components}(iv) implies that $\|\mathcal K_\tau\Pi^\xi_{\mathrm c}\sigma\|_{q,\TV}\le C\|\sigma\|_{\mathrm c,\xi}$ and $\|\mathcal K_\tau\Pi^\xi_{\mathrm i}\sigma\|_{q,\TV}\le C\|\sigma\|_{\mathrm i,\xi}$. For the boundary component, \Cref{lem:jets-kernel-bounds}(a) gives \[ \|\mathcal K_\tau\Pi^\xi_{\mathrm b}\sigma\|_{q,\TV}\le\int_{-\rho/2}^{\rho/2}|g(x)|\,C\bigl(1+|x|^{-q}\bigr)\dd x+C\Bigl|\int g\Bigr|\le C\|\sigma\|_{\mathrm b,\xi}, \] since $q<1$ and $|g|\le2\|\sigma\|_{\mathrm b,\xi}$. By \Cref{lem:jets-complete} and \Cref{lem:jets-law}(ii), $\|\widetilde{\mathcal L}_\xi\sigma\|_{q,\TV}\le p\,\|\widetilde\nu_\xi\|_{q,\TV}\|\mathcal K_\tau\sigma\|_{q,\TV}$, which proves (i). \proofstep{2}{Local density.} Fix $|z|\le R_0$ and split the integral defining $\bar k_{\xi,x}(z)$ into $d\in(-3\rho/4,3\rho/4)$, where $\widetilde\nu_\xi$ has density at most $H_p$, and the complement, where $|d-z|\ge3\rho/4-R_0=\rho/8$. For $|x|\ge2\rho/9$, \Cref{lem:jets-kernel}(i) and \Cref{lem:jets-kernel-bounds}(a),(c) give \begin{align*} |\bar k_{\xi,x}(z)|+|\partial_x\bar k_{\xi,x}(z)|&\le H_p\int\bigl(k_\tau(x;y)+|\partial_xk_\tau(x;y)|\bigr)\dd y+\sup_{|y|\ge\rho/8}\bigl(k_\tau(x;y)+|\partial_xk_\tau(x;y)|\bigr)\\ &\le C\bigl(H_p+e^{-cp^2}\bigr). \end{align*} As in Step 1, the scalar function $x\mapsto\bar k_{\xi,x}(z)$ is then Lipschitz on $S_{\mathrm c}$ and bounded and Lipschitz on $S_{\mathrm i}$, with constants at most $C$. By \eqref{eq:jets-L-density} and \Cref{lem:jets-components}(iv), the core and intermediate components contribute at most $Cp(\|\sigma\|_{\mathrm c,\xi}+\|\sigma\|_{\mathrm i,\xi})$ to the density at $z$. For the boundary component, the same splitting and \Cref{lem:jets-kernel-bounds}(d) give \[ \int_{-\rho/2}^{\rho/2}|g(x)|\,\bar k_{\xi,x}(z)\dd x\le\|g\|_\infty\Bigl(\rho H_p+\sup_{|y|\ge\rho/8}\int_\R k_\tau(x;y)\dd x\Bigr)\le\|g\|_\infty\Bigl(\rho H_p+\frac{64\tau}{\rho^2}\Bigr), \] and the compensating atom contributes $|\int g|\,\bar k_{\xi,x_\xi}(z)\le C\|\sigma\|_{\mathrm b,\xi}$. This proves (ii). \proofstep{3}{Variation away from the core.} Let $f(x)=\bar k_{\xi,x}\one_{\mathcal O_\xi}\in L^1_q(\mathcal O_\xi)$. For $|x|\ge2\rho/9$, we have \[ \|f(x)\|+\|\partial_xf(x)\|\le\int\widetilde\nu_\xi(\dd d)\int\bigl(k_\tau(x;y)+|\partial_xk_\tau(x;y)|\bigr)\one_{\{d-y\in\mathcal O_\xi\}}\bigl(1+|d-y|^q\bigr)\dd y. \] If $d\in\mathcal S_\xi$ and $d-y\in\mathcal O_\xi$, then $(1+\theta)|y|\ge r_p/2-r_p/4$, which gives $|y|\ge r_p/5$; further, $|d|\le2$ implies that $1+|d-y|^q\le3(1+|y|^q)$. By \Cref{lem:jets-kernel-bounds}(b), the part of the integral with $d\in\mathcal S_\xi$ is at most $Ce^{-(2\rho/9)r_p/(10\tau)}\le Ce^{-cp^{7/4}}$. For the part with $d\notin\mathcal S_\xi$, we bound $1+|d-y|^q\le(1+|d|^q)(1+|y|^q)$ and use \Cref{lem:jets-kernel-bounds}(a) and \eqref{eq:jets-holder}, which give at most $Ce^{-a_1\sqrt p}$. Arguing as in Step 1, $f$ is Lipschitz on $S_{\mathrm c}$ and bounded and Lipschitz on $S_{\mathrm i}$ with constants at most $Ce^{-a_1\sqrt p}$. By \Cref{lem:jets-components}(iv) with $V=L^1_q(\mathcal O_\xi)$, the core and intermediate components contribute at most $Cpe^{-a_1\sqrt p}(\|\sigma\|_{\mathrm c,\xi}+\|\sigma\|_{\mathrm i,\xi})$ to the left side of (iii). The boundary component contributes at most $\|\widetilde{\mathcal L}_\xi\Pi^\xi_{\mathrm b}\sigma\|_{q,\TV}\le Cp\|\sigma\|_{\mathrm b,\xi}$ by Step 1. This proves (iii). \proofstep{4}{Comparison with the norm of $X_q$.} For real $\tau$ we have $\|\mathcal K_\tau\sigma\|_\TV\le\|\sigma\|_\TV$, and therefore $\|\widetilde{\mathcal L}_\xi\sigma\|_\TV\le p\|\sigma\|_\TV$. Since $\sup_xk_\tau(x;y)=(e|y|)^{-1}$, we have $|\bar k_{\xi,x}(z)|\le H_p+8/(e\rho)$ for all $x\neq0$ and $|z|\le R_0$. Then the density of $\widetilde{\mathcal L}_\xi\sigma$ on $[-R_0,R_0]$ is at most $Cp\|\sigma\|_\TV$. Part (iv) now follows from \Cref{lem:jets-components}(iii). \end{proof} \begin{lemma}\label{lem:jets-invertible} Let $\gamma=\min\{c_{\mathrm b},c_{\mathrm i}\}/4$. There exists $p_2(\Xi)\ge p_1(\Xi)$ such that for all $p\ge p_2(\Xi)$, $\xi\in\Xi'$, and $\sigma\in X_q^0$, \begin{equation}\label{eq:jets-contraction} N_{\xi,\gamma}(\widetilde{\mathcal L}_\xi\sigma)\le\tfrac12N_{\xi,\gamma}(\sigma). \end{equation} Consequently, there exists $C_p<\infty$ such that $\|\widetilde{\mathcal L}_\xi^n\|_{X_q^0\to X_q^0}\le C_p2^{-n}$ for all $n\ge0$, and $I-\widetilde{\mathcal L}_\xi$ has a bounded inverse on $X_q^0$, given by the norm-convergent series $\sum_{n\ge0}\widetilde{\mathcal L}_\xi^n$. \end{lemma} \begin{proof} We set $v_\xi(\sigma)=(c,b,i)$ and $v_\xi(\widetilde{\mathcal L}_\xi\sigma)=(c',b',i')$. By \Cref{lem:jets-conjugate}, \begin{align*} N_{\xi,\gamma}(\widetilde{\mathcal L}_\xi\sigma)&=c'+e^{\gamma p}b'+e^{\gamma\sqrt p}i'\\ &\le(\varepsilon_pc+P_pb+P_pi)+e^{\gamma p}(\delta_{\mathrm b,p}c+\varepsilon_pb+\delta_{\mathrm b,p}i)\\ &\qquad+e^{\gamma\sqrt p}\bigl(\delta_{\mathrm i,p}c+P_pb+(\delta_{\mathrm i,p}+\delta_{\mathrm b,p})i\bigr)\\ &=k_{\mathrm c}\,c+k_{\mathrm b}\,e^{\gamma p}b+k_{\mathrm i}\,e^{\gamma\sqrt p}i, \end{align*} where \begin{align*} k_{\mathrm c}&=\varepsilon_p+e^{\gamma p}\delta_{\mathrm b,p}+e^{\gamma\sqrt p}\delta_{\mathrm i,p},\qquad k_{\mathrm b}=P_pe^{-\gamma p}+\varepsilon_p+P_pe^{-\gamma(p-\sqrt p)},\\ k_{\mathrm i}&=P_pe^{-\gamma\sqrt p}+\delta_{\mathrm b,p}e^{\gamma(p-\sqrt p)}+\delta_{\mathrm i,p}+\delta_{\mathrm b,p}. \end{align*} Since $\gamma\le c_{\mathrm b}/4$ and $\gamma\le c_{\mathrm i}/4$, every term is at most $C/p$ or $p^{A_{\mathrm{col}}}e^{-\gamma\sqrt p}$, and $\max\{k_{\mathrm c},k_{\mathrm b},k_{\mathrm i}\}\le1/2$ for $p$ larger than a threshold depending only on $\Xi$. This proves \eqref{eq:jets-contraction}. By \Cref{lem:jets-L-bounds}(i),(ii) and $N_{\xi,\gamma}\ge\|\cdot\|_{\mathrm{cbi}}$, we have $\|\widetilde{\mathcal L}_\xi\mu\|_{X_q}\le CpN_{\xi,\gamma}(\mu)$ for $\mu\in X_q^0$. For $n\ge2$, this, \eqref{eq:jets-contraction}, and \Cref{lem:jets-L-bounds}(iv) give \[ \|\widetilde{\mathcal L}_\xi^n\sigma\|_{X_q}\le CpN_{\xi,\gamma}(\widetilde{\mathcal L}_\xi^{n-1}\sigma)\le Cp\,2^{-(n-2)}N_{\xi,\gamma}(\widetilde{\mathcal L}_\xi\sigma)\le C^2p^2e^{\gamma p}2^{-(n-2)}\|\sigma\|_{X_q}. \] The cases $n=0,1$ follow from the boundedness of $\widetilde{\mathcal L}_\xi$ on $X_q$, which holds by \Cref{lem:jets-holomorphic}(i). The Neumann series therefore converges in operator norm on the Banach space $X_q^0$, and its sum inverts $I-\widetilde{\mathcal L}_\xi$. \end{proof} \begin{remark}\label{rem:jets-two-norms} The mixed seminorm $N_{\xi,\gamma}$ is weaker than $\|\cdot\|_{X_q}$: it measures the core and intermediate parts of a measure only through Lipschitz test functions. The operator $\widetilde{\mathcal L}_\xi$ is a contraction for the weak seminorm and maps it back into the strong norm, because its kernel is smooth in the input variable away from zero. The proof above uses only these two properties. \end{remark} \section{Analyticity of the principal law}\label{sec:jets-analytic-sec} We first show that the principal law depends continuously on $\xi$ in the norm of $X_q$. This is used to identify the analytic branch produced by the implicit function theorem with the principal law. \begin{lemma}\label{lem:jets-continuity} Let $p\ge p_1(\Xi)$. Then $\xi\mapsto\widetilde\nu_\xi$ is continuous from $\Xi'$ to $X_q$. \end{lemma} \begin{proof} Let $\xi_n\to\xi$ in $\Xi'$, and write $\nu_n=\widetilde\nu_{\xi_n}$, $\nu=\widetilde\nu_\xi$, $\tau_n=\tau(\xi_n)$, and $\tau_0=\tau(\xi)$. By \Cref{lem:jets-holomorphic}(iv), $\nu_n=\delta_1*\mathsf R\,\mathsf P_p(\mathcal K_{\tau_n}\nu_n)$ and $\nu=\delta_1*\mathsf R\,\mathsf P_p(\mathcal K_{\tau_0}\nu)$. Since $\mathsf P_p\colon Z_q\to W_q$ is continuous and $\mu\mapsto\delta_1*\mathsf R\mu$ is bounded from $W_q$ to $X_q$, it suffices to show that $\mathcal K_{\tau_n}\nu_n\to\mathcal K_{\tau_0}\nu$ in $Z_q$. We write \[ \mathcal K_{\tau_n}\nu_n-\mathcal K_{\tau_0}\nu=(\mathcal K_{\tau_n}-\mathcal K_{\tau_0})\nu_n+\mathcal K_{\tau_0}(\nu_n-\nu). \] By \Cref{lem:jets-law}(i),(ii), $\sup_n\|\nu_n\|_{X_q}<\infty$, and the first term tends to zero by the continuity of $\tau\mapsto\mathcal K_\tau$ in operator norm (\Cref{lem:jets-kernel}(ii)). For the second term, we note that \Cref{lem:jets-kernel-bounds}(a),(c) with $m\in\{0,1\}$ and $r=0$ give \begin{equation}\label{eq:jets-Z-kernel} \begin{gathered} \|k_{\tau}(x;\cdot)\|_{Z_q}\le C(1+|x|^{-q}),\\ \|k_\tau(x;\cdot)-k_\tau(s;\cdot)\|_{Z_q}\le C\,\frac{|x-s|}{\min\{|x|,|s|\}}\Bigl(1+\frac{1}{\min\{|x|,|s|\}^{q}}\Bigr) \end{gathered} \end{equation} for $x,s$ of the same sign, where $\tau=\tau_0$ and we identify $k_\tau(x;\cdot)$ with the measure it is the density of. The second bound follows from the bounds for $\partial_xk_\tau$ and the fundamental theorem of calculus in $x$. Let $\eps>0$. Let $0<\deltaM)\le CH_p\delta^{1-q}+\frac{C}{M}. \] We fix $\delta$ and $M$ so that the right side is at most $\eps$. The $Z_q$-valued function $F(x)=\psi(x)k_{\tau_0}(x;\cdot)$ vanishes for $|x|\le\delta/2$ and $|x|\ge2M$ and is uniformly continuous by \eqref{eq:jets-Z-kernel}. Let $r>0$ be such that $\|F(x)-F(s)\|_{Z_q}\le\eps$ for $|x-s|0$ and $\tau(\xi)\in U$. \end{enumerate} \end{theorem} \begin{proof} Let $p_{\mathrm{an}}(\Xi)=p_2(\Xi)$, which does not depend on $q$. Part (i) is \Cref{lem:jets-holomorphic}(iv). Fix $\xi_0\in\Xi$, set $\tau_0=\tau(\xi_0)$ and $\nu_0=\widetilde\nu_{\xi_0}$, and define \[ F\colon\Omega\times X_q^0\to X_q^0,\qquad F(\tau,\sigma)=\sigma-\widetilde\Phi_\tau(\nu_0+\sigma)+\nu_0. \] Since $\widetilde\Phi_\tau(\nu_0+\sigma)$ has mass one by \Cref{lem:jets-holomorphic}(ii), $F$ takes values in $X_q^0$. By \Cref{lem:jets-holomorphic}(i), $F$ is holomorphic. Its partial derivatives are \[ D_\sigma F(\tau,\sigma)=I-D_\nu\widetilde\Phi_\tau(\nu_0+\sigma),\qquad D_\tau F(\tau,\sigma)h=-h\,p\,\widetilde\Phi_\tau(\nu_0+\sigma)*\mathsf R(\partial_\tau\mathcal K_\tau)(\nu_0+\sigma), \] where the second formula follows from \eqref{eq:jets-CP-derivative} with $m=1$ and the chain rule. These are continuous functions of $(\tau,\sigma)$ in operator norm: by \Cref{lem:jets-CP}, the map $\alpha\mapsto D\mathsf P_p(\alpha)$ has a derivative that is bounded on bounded sets, which implies that it is locally Lipschitz in operator norm, and $\tau\mapsto\mathcal K_\tau$ and $\tau\mapsto\partial_\tau\mathcal K_\tau$ are continuous in operator norm by \Cref{lem:jets-kernel}(ii). We conclude that $F$ is continuously Fr\'echet differentiable when $\Omega\subset\C$ is regarded as an open subset of $\R^2$. Further, $F(\tau_0,0)=0$ by \Cref{lem:jets-holomorphic}(iv), and $D_\sigma F(\tau_0,0)=I-\widetilde{\mathcal L}_{\xi_0}$ is invertible on $X_q^0$ by \Cref{lem:jets-invertible}. By the implicit function theorem in Banach spaces \cite[Section~15]{Deimling1985nonlinear}, there exist an open disc $U\subset\Omega$ centered at $\tau_0$, an open neighborhood $V$ of $0$ in $X_q^0$, and a continuously differentiable map $\sigma\colon U\to V$ such that, for $(\tau,\sigma')\in U\times V$, we have $F(\tau,\sigma')=0$ if and only if $\sigma'=\sigma(\tau)$. The uniqueness in this pointwise form is part of the standard proof, and it can be checked directly. Let $A=D_\sigma F(\tau_0,0)$. By the continuity of $D_\sigma F$, after replacing $V$ by a small ball around $0$ and then shrinking $U$ so that $\sigma(U)\subset V$, we have $\|I-A^{-1}D_\sigma F(\tau,\sigma')\|\le1/2$ on $U\times V$. By the mean value inequality, $\sigma'\mapsto\sigma'-A^{-1}F(\tau,\sigma')$ then has Lipschitz constant at most $1/2$ on $V$ for each $\tau\in U$, so it has at most one fixed point in $V$, and the zeros of $F(\tau,\cdot)$ in $V$ are exactly its fixed points. Shrinking $U$, we may assume that $D_\sigma F(\tau,\sigma(\tau))$ is invertible for $\tau\in U$, since the invertible operators form an open set. Differentiating the identity $F(\tau,\sigma(\tau))=0$, we obtain \[ D\sigma(\tau)h=-\bigl(D_\sigma F(\tau,\sigma(\tau))\bigr)^{-1}D_\tau F(\tau,\sigma(\tau))h,\qquad h\in\C. \] The right side is complex-linear in $h$, so $\sigma$ is complex differentiable on $U$, that is, holomorphic. We set $\nu(\tau)=\nu_0+\sigma(\tau)$, which lies in $\mathcal A_q$ and satisfies $\widetilde\Phi_\tau(\nu(\tau))=\nu(\tau)$. Since $\Xi\subset\operatorname{int}\Xi'$, \Cref{lem:jets-continuity} and the continuity of $\xi\mapsto\tau(\xi)$ give $\eta>0$ such that, for all real $\xi$ with $|\xi-\xi_0|<\eta$, we have $\xi\in\Xi'$, $\widetilde\nu_\xi-\nu_0\in V$, and $\tau(\xi)\in U$. For such $\xi$, we have $F(\tau(\xi),\widetilde\nu_\xi-\nu_0)=0$ by \Cref{lem:jets-holomorphic}(iv), and the uniqueness in the implicit function theorem gives $\widetilde\nu_\xi=\nu(\tau(\xi))$. This proves (iii), after shrinking $U$ so that its real points are of the form $\tau(\xi)$ with $|\xi-\xi_0|<\eta$. By the power series expansion recalled at the beginning of \Cref{sec:jets-spaces}, $\tau\mapsto\nu(\tau)$ is real-analytic on $U\cap\R$. Since $\xi\mapsto\tau(\xi)=(p\xi)^{-2}$ extends holomorphically to a complex neighborhood of $(0,\infty)$, the map $\xi\mapsto\nu(\tau(\xi))=\widetilde\nu_\xi$ extends holomorphically to a complex neighborhood of $(\xi_0-\eta,\xi_0+\eta)$, and it is real-analytic there. The union of these intervals over $\xi_0\in\Xi$ is an open set containing the compact interval $\Xi$, and it contains an open interval $I\supset\Xi$. This proves (ii); the reformulations in $E=\xi\sqrt p$ and in $\tau$ hold because $\xi\mapsto\xi\sqrt p$ and $\xi\mapsto\tau(\xi)$ are real-analytic with real-analytic inverses on $(0,\infty)$. \end{proof} \begin{remark}\label{rem:jets-analytic-scope} The constants in the proof of \Cref{thm:jets-analytic} depend on $p$, for instance through the factor $e^{\gamma p}$ in \Cref{lem:jets-invertible}, and the theorem gives no bound on the derivatives that is uniform in $p$. Such bounds are proved in \Cref{sec:jets-quant}. By \Cref{thm:jets-analytic}(ii), the derivatives $\partial_\xi^j\widetilde\nu_{p,\xi\sqrt p}$ exist in $X_q$ for all $j\ge1$; they are real measures of total mass zero, since $\widetilde\nu_{p,\xi\sqrt p}$ is a probability measure for every real $\xi$, and they have no atoms, since the atom $e^{-p}\delta_1$ does not depend on $\xi$. \end{remark} \section{Analyticity of the free energy at \texorpdfstring{$s=1$}{s=1}}\label{sec:jets-critical} \subsection{The transfer operator in the scaling \texorpdfstring{$g=1/E$}{g=1/E}}\label{sec:jets-angular} We use the transfer matrices $M(b,\beta)$, the vectors $v_x$, the projective maps $f_{b,\beta}$, and the transfer operators $\mathcal Q_s$ on $C(\RP)$ of \Cref{def:press-operators}. Let $p\ge p_1(\Xi)$, $\xi\in\Xi'$, and $E=\xi\sqrt p$. By \Cref{cor:stab-halfline}, the law $\pi$ of $D=B/g(E)$, where $B\sim\nu_{p,E}$, is a real solution at $E$. With the scaling $g=g(E)$, the law of $gD=B$ is $\nu_{p,E}$ and the rate is $K_g=p/\theta$; this is the $B$-normalization, and we write $\mathcal Q_{E,1}$ for the corresponding critical transfer operator. With the scaling $g'=1/E$, the law of $g'D=\widetilde B$ is $\widetilde\nu_\xi$ and the rate is $K_{g'}=pE^2=1/\tau(\xi)$. We denote the corresponding critical transfer operator by $\widetilde{\mathcal Q}_\xi$: \begin{equation}\label{eq:jets-Qtilde} (\widetilde{\mathcal Q}_\xi F)(x)=p\int\widetilde\nu_\xi(\dd d)\int_0^\infty\frac{1}{\tau}e^{-\beta/\tau}\,\|M(d,\beta)v_x\|^{-1}F(f_{d,\beta}(x))\dd\beta,\qquad\tau=\tau(\xi). \end{equation} All dependence on $\xi$ is in the law $\widetilde\nu_\xi$ and in the exponential density; the function $F$ is evaluated at real projective points. Since $g'/g=(Eg)^{-1}=(1+\theta)^{-1}$, we obtain from \Cref{lem:press-gauge}(ii) that \begin{equation}\label{eq:jets-gauge-similarity} \begin{gathered} \widetilde{\mathcal Q}_\xi=\Theta_\xi^{-1}\mathcal Q_{E,1}\Theta_\xi,\qquad(\Theta_\xi F)(y)=\|S_\xi v_y\|^{-1}F(S_\xi\cdot y),\\ S_\xi=\operatorname{diag}\bigl((1+\theta)^{1/2},(1+\theta)^{-1/2}\bigr), \end{gathered} \end{equation} where $\Theta_\xi$ is a positive bounded operator on $C(\RP)$ with positive bounded inverse. At the level of matrices, this is the constant diagonal conjugacy $M(B,\theta Y/p)=S_\xi^{-1}M(\widetilde B,\tau Y)S_\xi$, which follows from \eqref{eq:press-gauge-conj} with $\gamma=1+\theta$ and \eqref{eq:jets-tau-theta}. In particular, the two operators have the same spectrum and the same Perron--Frobenius eigenvalue, and by \Cref{def:press-pressure} this eigenvalue is the exponential of the free energy at $E=\xi\sqrt p$ and $s=1$, which we denote by $\lambda_p(\xi)$. The two scalings differ in one respect that matters here. In the $B$-normalization the law $\nu_{p,E}$ has its atom at $1+\theta(\xi)$, which moves with $\xi$, and this atom produces an atom in the row of $\mathcal Q_{E,1}$ at $x=0$: since $\|M(b,\beta)v_0\|^{-1}=\beta^{1/2}(1+b^2)^{-1/2}$ and $f_{b,\beta}(0)=1/b$, the measure $F\mapsto(\mathcal Q_{E,1}F)(0)$ has an atom at $1/(1+\theta(\xi))$ of mass $pe^{-p}\E[\beta^{1/2}](1+(1+\theta)^2)^{-1/2}$, and it has no atom at $1/(1+\theta(\xi'))$ for $\xi'\neq\xi$. Since $\xi\mapsto\theta(\xi)$ is strictly monotone, $\|\mathcal Q_{\xi\sqrt p,1}-\mathcal Q_{\xi'\sqrt p,1}\|$ is bounded below by the mass of this atom, and $\xi\mapsto\mathcal Q_{\xi\sqrt p,1}$ is not norm-continuous; by \Cref{prop:press-continuity}, its third power is norm-continuous (see also \Cref{rem:press-third-power}). In the scaling $g=1/E$, the atom of $\widetilde\nu_\xi$ is at $1$ for every $\xi$, and we show that $\xi\mapsto\widetilde{\mathcal Q}_\xi$ itself is real-analytic in operator norm. \subsection{Analyticity of the critical growth rate}\label{sec:jets-critical-proof} \begin{proposition}\label{prop:jets-critical-analytic} Let $\Xi\subset(0,\infty)$ be compact, and let $\widehat\Xi$ denote its convex hull. There exists $p_{\mathrm{cr}}(\Xi)$ such that for every $p\ge p_{\mathrm{cr}}(\Xi)$ the following holds. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $\xi\in\widehat\Xi$, $\lambda_p(\xi)$ is the Perron--Frobenius eigenvalue of the critical transfer operator $\widetilde{\mathcal Q}_\xi$ in the scaling $g=1/E$, and of the critical transfer operator $\mathcal Q_{\xi\sqrt p,1}$ in the $B$-normalization. \item The maps $\xi\mapsto\widetilde{\mathcal Q}_\xi$ and $\xi\mapsto\mathcal Q_{\xi\sqrt p,1}^3$ are continuous on $\widehat\Xi$ in the operator norm of $C(\RP)$. Moreover, there exists an open interval $I\supset\widehat\Xi$ such that $\xi\mapsto\widetilde{\mathcal Q}_\xi$ is real-analytic from $I$ to the bounded operators on $C(\RP)$ with the operator norm. \item The function $\xi\mapsto\lambda_p(\xi)$ is real-analytic on $I$. In particular, it is continuous on $\Xi$ and real-analytic on the interior of $\Xi$. \end{enumerate} \end{proposition} \begin{proof} We may assume that $\Xi$ is an interval, and we set $p_{\mathrm{cr}}(\Xi)=p_{\mathrm{an}}(\Xi)$. We fix $q=3/4$ and apply \Cref{thm:jets-analytic} with this $q$. Let $p\ge p_{\mathrm{cr}}(\Xi)$. \proofstep{1}{Identification of $\lambda_p$ and continuity.} Part (i) for $\xi\in\Xi'$ is the discussion after \eqref{eq:jets-Qtilde}; the hypotheses \eqref{eq:press-hyp} hold for $\nu_{p,E}$ and $\widetilde\nu_\xi$ by \Cref{lem:press-gauge}(i), since both are rescalings of the real solution $\pi$. Next we apply \Cref{prop:press-continuity} on the interval of energies $\sqrt p\,\Xi'$, with $\nu_E=\nu_{p,E}$ and $K(E)=p/\theta$. The map $E\mapsto K(E)$ is continuous by \Cref{lem:jets-parameters}, $E\mapsto\nu_{p,E}$ is weakly continuous by \ref{item:jets-S1}, and by the tail bound in \eqref{eq:jets-class}, we have $\sup_{\xi\in\Xi'}\int|b|^{3/4}\nu_{p,\xi\sqrt p}(\dd b)<\infty$, as in the proof of \Cref{lem:jets-law}(ii). Since $q_0=3/4>1/2$, \Cref{prop:press-continuity} shows that $\xi\mapsto\mathcal Q_{\xi\sqrt p,1}^3$ is norm-continuous and $\xi\mapsto\lambda_p(\xi)$ is continuous on $\Xi'$. \proofstep{2}{Complex extension of the operator.} Fix $\xi_0\in\Xi$, let $\tau_0=\tau(\xi_0)$, and let $U$ and $\tau\mapsto\nu(\tau)$ be as in \Cref{thm:jets-analytic}(iii). Shrinking $U$, we may assume that its radius is at most $\tau_0/2$. Then for $\tau\in U$, the number $a(\tau)=\Re(1/\tau)=\Re\tau/|\tau|^2$ lies in $[2/(9\tau_0),2/\tau_0]$, and $|\tau^{-1}e^{-\beta/\tau}|=(|\tau|a(\tau))^{-1}a(\tau)e^{-a(\tau)\beta}$. For a complex measure $\mu\in\mathcal M_q$, $\tau\in U$, and a bounded Borel function $F$ on $\RP$, define \[ (\mathcal Q(\mu,\tau)F)(x)=p\int\mu(\dd d)\int_0^\infty\frac1\tau e^{-\beta/\tau}\|M(d,\beta)v_x\|^{-1}F(f_{d,\beta}(x))\dd\beta. \] By \Cref{lem:press-row-moment} with $s=1$ and $K=a(\tau)$, whose constant $C_{1,K}$ is bounded for $K$ in a compact subset of $(0,\infty)$, and by $|d|^{1/2}\le1+|d|^q$, this is a bounded operator on the space $\mathcal B(\RP)$ of bounded Borel functions with the supremum norm, and $\|\mathcal Q(\mu,\tau)\|\le C\|\mu\|_{q,\TV}$ for $\tau\in U$, where $C$ depends on $p$ and $\tau_0$. We claim that $\tau\mapsto\mathcal Q(\mu,\tau)$ is holomorphic on $U$ in the operator norm of $\mathcal B(\RP)$, with derivative given by the same formula with $\partial_\tau(\tau^{-1}e^{-\beta/\tau})$ in place of $\tau^{-1}e^{-\beta/\tau}$, and that the first-order Taylor remainder at $\tau$ in the direction $h$ has norm at most $C|h|^2\|\mu\|_{q,\TV}$. To see this, write $\kappa=1/\tau$ and $e_\kappa(\beta)=\kappa e^{-\kappa\beta}=\tau^{-1}e^{-\beta/\tau}$. The image of $U$ under $\tau\mapsto1/\tau$ is a disc, on which $\Re\kappa\ge2/(9\tau_0)$ and $|\kappa|\le2/\tau_0$. Then $|\partial_\kappa^2e_\kappa(\beta)|=|(-2\beta+\kappa\beta^2)e^{-\kappa\beta}|\le C\,a_*e^{-a_*\beta}$ with $a_*=1/(9\tau_0)$. Applying \Cref{lem:press-row-moment} with $K=a_*$ to the remainder $e_{\kappa+h}-e_\kappa-h\partial_\kappa e_\kappa=h^2\int_0^1(1-t)\,\partial_\kappa^2e_{\kappa+th}\dd t$ proves the claim, as in the proof of \Cref{lem:jets-kernel}. Since $\mathcal Q(\mu,\tau)$ is linear in $\mu$ with $\|\mathcal Q(\mu,\tau)\|\le C\|\mu\|_{q,\TV}$, the map $(\mu,\tau)\mapsto\mathcal Q(\mu,\tau)$ is holomorphic on $\mathcal M_q\times U$: the first-order remainder at $(\mu,\tau)$ in the direction $(\mu',h)$ is $\mathcal Q(\mu',\tau+h)-\mathcal Q(\mu',\tau)$ plus the Taylor remainder in $\tau$ at $\mu$, whose norms are $O(|h|\,\|\mu'\|_{q,\TV})$ and $O(|h|^2)$. By \Cref{thm:jets-analytic}(iii) and the inequality $\|\cdot\|_{q,\TV}\le\|\cdot\|_{X_q}$, the map $\tau\mapsto\mathcal Q(\tau)=\mathcal Q(\nu(\tau),\tau)$ is holomorphic on $U$. Next, $\mathcal Q(\tau)$ maps $C(\RP)$ into itself. Let $F\in C(\RP)$ and $x_n\to x$ in $\RP$. Consider the finite positive measure $m_\tau(\dd d,\dd\beta)=|\nu(\tau)|(\dd d)\,|\tau^{-1}e^{-\beta/\tau}|\dd\beta$ and the functions $g_n(d,\beta)=\|M(d,\beta)v_{x_n}\|^{-1}F(f_{d,\beta}(x_n))$. For fixed $(d,\beta)$ with $\beta>0$, these converge to the same expression $g(d,\beta)$ at $x$, by the continuity noted after \eqref{eq:press-norm}. By \Cref{lem:press-row-moment} with $s=5/4$, we have $\int|g_n|^{5/4}\dd m_\tau\le C\|F\|_\infty^{5/4}\int(1+|d|^{5/8})\,|\nu(\tau)|(\dd d)$, which is finite and independent of $n$ because $5/80$ is an eigenvalue of $T_0$ whose Riesz projection $\Pi_0$ has rank one, it has an eigenfunction $\psi_1$ with $\min\psi_1>0$, and there exists $r_0<\lambda_0$ with $|z|\le r_0$ for every $z\in\sigma(T_0)\setminus\{\lambda_0\}$. Since $\psi_1$ is an eigenfunction, $\Pi_0\psi_1=\psi_1$. Let $\ell$ be the functional $F\mapsto\int F\dd\ell_1$, where $\ell_1$ is the measure of \Cref{thm:press-perron}(iii), so that $\ell(\Pi_0\psi_1)=1$. Fix $\delta\in(0,(\lambda_0-r_0)/3)$, and let $\eps>0$ be given by \Cref{lem:press-perturbation} for $T_0$, $\mu_0=\lambda_0$, $r_0$, $\delta$, $\psi=\psi_1$, and $\ell$. By Step 2, there exists an open disc $U_1\subset U$ centered at $\tau_0$ with $\|\mathcal Q(\tau)-T_0\|<\eps$ for $\tau\in U_1$. By \Cref{lem:press-perturbation}(iii), the function $\tau\mapsto\mu(\mathcal Q(\tau))$ is holomorphic on $U_1$. Let $\xi>0$ with $\tau(\xi)\in U_1$. By Step 1 and \Cref{thm:press-perron}(ii), $\lambda_p(\xi)$ is the spectral radius of $\mathcal Q(\tau(\xi))=\widetilde{\mathcal Q}_\xi$ and belongs to its spectrum. By \Cref{lem:press-perturbation}(i), $\mu(\mathcal Q(\tau(\xi)))$ is the unique point of maximal modulus in this spectrum, and we conclude that $\lambda_p(\xi)=\mu(\mathcal Q(\tau(\xi)))$. Then $\lambda_p$ coincides near $\xi_0$ with the composition of the holomorphic function $\tau\mapsto\mu(\mathcal Q(\tau))$ and the real-analytic function $\xi\mapsto\tau(\xi)$, and it is real-analytic near $\xi_0$. The union over $\xi_0\in\Xi$ of these neighborhoods is an open set containing the compact interval $\Xi$, and it contains an open interval $I'\supset\Xi$. Replacing $I$ by $I\cap I'$, which does not affect (ii), we obtain (iii); continuity on $\Xi$ also follows from Step 1. \end{proof} \begin{remark}\label{rem:jets-crossings} By \Cref{prop:jets-critical-analytic}(iii), if $\lambda_p-1$ does not vanish identically on the interval $I$, then the equation $\lambda_p(\xi)=1$ has only finitely many solutions in $\Xi$. \Cref{sec:uniq} proves the stronger statement that it has exactly one solution in a suitable window when $p$ is large. \end{remark} \section{Quantitative derivative bounds}\label{sec:jets-quant} \subsection{Statements}\label{sec:jets-quant-statements} In this section, we bound the derivatives $\sigma_j=\partial_\xi^j\widetilde\nu_\xi$ for $1\le j\le4$, which exist in $X_q$ by \Cref{thm:jets-analytic}. The constants in these bounds are explicit in $p$. We proceed by induction on $j$, as follows. Differentiating the fixed point equation $\widetilde\nu_\xi=\widetilde\Phi_{\tau(\xi)}(\widetilde\nu_\xi)$ $j$ times, we obtain $(I-\widetilde{\mathcal L}_\xi)\sigma_j=R_j$ (\Cref{lem:jets-structure}). The source $R_j$ is a sum of convolutions of the principal law with at most $j$ jump measures. Each jump measure is a derivative of the jump kernel, integrated against the principal law or against a lower-order derivative. The derivatives fall only on the exponential jump kernels, never on the principal law, and they produce the polynomial factors of \Cref{lem:jets-score}. We bound $R_j$ in three strong norms: the weighted total variation, the density near zero, and the weighted variation away from the core (\Cref{lem:jets-tensor}). The last two bounds are exponentially small. Indeed, near zero the principal law has an exponentially small density, and an output that leaves the core from the strict core requires a large jump. Inverting the component matrix of \ref{item:jets-S4}, we obtain bounds on the components of $\sigma_j$ (\Cref{lem:jets-matrix}), and the identity $\sigma_j=R_j+\widetilde{\mathcal L}_\xi\sigma_j$ and \Cref{lem:jets-L-bounds} then give the strong norms of $\sigma_j$. Finally, we show that the core parts of $\sigma_1$ and $\sigma_2$ are of order $p^{-1}$ when tested against Lipschitz and $C^2$ functions, respectively. The reason is that each derivative of the kernel in $\xi$ or in the input variable costs a factor $|\partial_\xi\tau|\le Cp^{-2}$ or $\tau\le Cp^{-2}$, while the prefactor is $p$. We fix a function $\eta\in C^2(\R)$ with $0\le\eta\le1$, $\eta=1$ on $[-1/2,1/2]$, and $\supp\eta\subset[-1,1]$, and set \begin{equation}\label{eq:jets-smooth-cutoff} \hat\chi_\xi(x)=\eta\Bigl(\frac{(1+\theta)x-1}{r_p}\Bigr). \end{equation} Then $\hat\chi_\xi=1$ on $\R\setminus\mathcal O_\xi$, the support of $\hat\chi_\xi$ is contained in $\{|(1+\theta)x-1|\le r_p\}$, which lies in the interior of $J_0$, and $|\hat\chi_\xi^{(k)}|\le Cr_p^{-k}$ for $k=1,2$. \begin{definition}\label{def:jets-smooth-norm} For a finite real measure $\sigma$ on $\R$, the \emph{smooth core norm} is \[ n_\xi(\sigma)=\sup\Bigl\{\Bigl|\int\hat\chi_\xi\varphi\dd\sigma\Bigr|:\varphi\in C^2(J_0),\ \varphi(x_\xi)=0,\ \|\varphi'\|_{L^\infty(J_0)}+\|\varphi''\|_{L^\infty(J_0)}\le1\Bigr\}, \] where $\hat\chi_\xi\varphi$ is extended by zero outside $J_0$. \end{definition} \begin{proposition}\label{prop:jets-quantitative} Let $\Xi\subset(0,\infty)$ be a compact interval and $q\in(0,1)$. There exist constants $A,a,C>0$ and $p_{\mathrm{der}}(\Xi,q)\ge p_{\mathrm{an}}(\Xi)$, depending only on $\Xi$ and $q$, such that for all $p\ge p_{\mathrm{der}}(\Xi,q)$ and $\xi\in\Xi$ the following holds. For $j\in\{1,2,3,4\}$, the derivative $\sigma_j=\partial_\xi^j\widetilde\nu_{p,\xi\sqrt p}\in X_q^0$ satisfies: \begin{enumerate}[label=\textup{(\roman*)}] \item $\|\sigma_j\|_{q,\TV}\le p^A$; \item $\|\sigma_j\|_{\mathrm b,\xi}\le p^Ae^{-ap}$, and the density of $\sigma_j$ on $\mathcal Z$ is bounded by $p^Ae^{-ap}$; \item $\|\sigma_j\|_{\mathrm i,\xi}\le p^Ae^{-a\sqrt p}$; \item $\int_{\mathcal O_\xi}(1+|x|^q)\,|\sigma_j|(\dd x)\le p^Ae^{-a\sqrt p}$. \end{enumerate} Further, \begin{enumerate}[label=\textup{(\roman*)},resume] \item $\|\sigma_1\|_{\mathrm c,\xi}\le C/p$; \item $n_\xi(\sigma_2)\le C/p$; \item $\int_{\mathcal O_\xi}(1+|x|^q)\,\widetilde\nu_\xi(\dd x)\le Ce^{-a\sqrt p}$. \end{enumerate} \end{proposition} We choose the constants in the following order. We first fix $\Xi$ and $q$. The constants of \ref{item:jets-S1}--\ref{item:jets-S4} for the window $\Xi'$ are then given, and they depend only on $\Xi$. Next, we set $a=\min\{a_1,c_{\mathrm b},c_{\mathrm i}\}$, with $a_1$ from \eqref{eq:jets-a1}, and $A=A_4$, where \begin{equation}\label{eq:jets-exponents} A_0=0,\qquad A_j=4A_{j-1}+A_{\mathrm{st}}+3A_{\mathrm{col}}+13\quad(1\le j\le4); \end{equation} Finally, we choose $p_{\mathrm{der}}$ large depending on all of the above. In particular, $A$ depends only on $A_{\mathrm{st}}$ and $A_{\mathrm{col}}$, and $a$ depends only on $q$, $a_{\mathrm{st}}$, $c_{\mathrm b}$, and $c_{\mathrm i}$. \begin{corollary}\label{cor:jets-tests} Let $\Xi\subset(0,\infty)$ be a compact interval and let $J\subset\R$ be compact. There exist $A,a,C>0$ and $p_{\mathrm{test}}(\Xi,J)$ such that for all $p\ge p_{\mathrm{test}}(\Xi,J)$, $\xi\in\Xi$, $j\in\{1,2\}$, and $\varphi\in C^2(\R)$ supported in $J$, \begin{equation}\label{eq:jets-tests} \bigl|\partial_\xi^j\E\varphi(B)\bigr|\le\Bigl(\frac Cp+p^Ae^{-a\sqrt p}\Bigr)\|\varphi\|_{C^2},\qquad\|\varphi\|_{C^2}=\|\varphi\|_\infty+\|\varphi'\|_\infty+\|\varphi''\|_\infty, \end{equation} where $B\sim\nu_{p,\xi\sqrt p}$, so that $B=(1+\theta)\widetilde B$ with $\widetilde B\sim\widetilde\nu_{p,\xi\sqrt p}$. More generally, let $I_0\supset\Xi$ be an open interval and let $\varphi\colon I_0\times\R\to\R$ be such that $\xi\mapsto\varphi(\xi,\cdot)$ is twice continuously differentiable from $I_0$ to the space of $C^2$ functions supported in $J$ with the norm $\|\cdot\|_{C^2}$. Then, for $j\in\{1,2\}$, \begin{equation}\label{eq:jets-tests-param} \Bigl|\partial_\xi^j\E\varphi(\xi,B)-\E\bigl[(\partial_\xi^j\varphi)(\xi,B)\bigr]\Bigr|\le\Bigl(\frac Cp+p^Ae^{-a\sqrt p}\Bigr)\sum_{i=0}^{j-1}\binom ji\bigl\|\partial_\xi^i\varphi(\xi,\cdot)\bigr\|_{C^2}. \end{equation} \end{corollary} We note that the support of the test function in \Cref{cor:jets-tests} may be much larger than the core region $\{|b-1|\le2r_p\}$. For instance, it may be a neighborhood of $1$ of radius $1/\log p$, or any fixed compact set. We control the part of the test function outside the core with the weighted variation bound in \Cref{prop:jets-quantitative}(iv). \subsection{The differentiated fixed point equation}\label{sec:jets-structure} For $r\ge0$ we write $\mathcal K^{(r)}_\xi$ for the $r$-th derivative of $\xi\mapsto\mathcal K_{\tau(\xi)}$ as a map into the bounded operators from $X_q$ to $Z_q$; by \Cref{lem:jets-kernel}(ii) and the chain rule, it is the integral operator with kernel $\partial_\xi^rk_{\tau(\xi)}(x;y)$, and $\mathcal K^{(0)}_\xi=\mathcal K_{\tau(\xi)}$. \begin{lemma}\label{lem:jets-structure} Let $p\ge p_{\mathrm{an}}(\Xi)$, $q\in(0,1)$, and $\xi\in\Xi$, and write $\nu=\widetilde\nu_\xi$ and $\sigma_j=\partial_\xi^j\widetilde\nu_\xi$. For every $j\ge1$, \begin{equation}\label{eq:jets-differentiated} (I-\widetilde{\mathcal L}_\xi)\sigma_j=R_j, \end{equation} where $R_j$ is a finite sum, over an index set depending only on $j$, of terms of the form \begin{equation}\label{eq:jets-selected-tensor} c\,p^n\,\nu*\mathsf R\mathcal K^{(r_1)}_\xi\mu_1*\dots*\mathsf R\mathcal K^{(r_n)}_\xi\mu_n, \end{equation} in which $c$ is a positive integer depending only on $j$ and on the term, $1\le n\le j$, $r_1+\dots+r_n\le j$, and each $\mu_i$ belongs to $\{\nu,\sigma_1,\dots,\sigma_{j-1}\}$. In particular, \begin{align*} R_1&=p\,\nu*\mathsf R\mathcal K^{(1)}_\xi\nu,\\ R_2&=p\,\nu*\mathsf R\mathcal K^{(2)}_\xi\nu+p^2\,\nu*\mathsf R\mathcal K^{(1)}_\xi\nu*\mathsf R\mathcal K^{(1)}_\xi\nu+2p\,\nu*\mathsf R\mathcal K^{(1)}_\xi\sigma_1\\ &\quad+2p^2\,\nu*\mathsf R\mathcal K^{(1)}_\xi\nu*\mathsf R\mathcal K^{(0)}_\xi\sigma_1+p^2\,\nu*\mathsf R\mathcal K^{(0)}_\xi\sigma_1*\mathsf R\mathcal K^{(0)}_\xi\sigma_1. \end{align*} \end{lemma} \begin{proof} Let $I$ be the interval of \Cref{thm:jets-analytic}(ii) and set $\mathcal F(\xi,\mu)=\widetilde\Phi_{\tau(\xi)}(\mu)$ for $\xi\in I$ and $\mu\in X_q$. The map $\mathcal F$ is infinitely differentiable, as the composition of the infinitely differentiable maps $\xi\mapsto\mathcal K_{\tau(\xi)}$ (\Cref{lem:jets-kernel}(ii)), $(\mathcal K,\mu)\mapsto\mathcal K\mu$ (bounded and bilinear), $\mathsf P_p$ (\Cref{lem:jets-CP}), and the bounded linear map $\mu\mapsto\delta_1*\mathsf R\mu$. The map $\nu(\xi)=\widetilde\nu_\xi$ is infinitely differentiable from $I$ to $X_q$ by \Cref{thm:jets-analytic}(ii), and $\nu(\xi)=\mathcal F(\xi,\nu(\xi))$ by \Cref{lem:jets-holomorphic}(iv). We claim that for $j\ge1$, \begin{equation}\label{eq:jets-faa} \partial_\xi^j\bigl[\mathcal F(\xi,\nu(\xi))\bigr]=\sum c_{r,m,j_1,\dots,j_m}\,\partial_\xi^rD_\mu^m\mathcal F(\xi,\nu(\xi))\bigl[\nu^{(j_1)},\dots,\nu^{(j_m)}\bigr], \end{equation} where the sum is over $r,m\ge0$ and $j_1,\dots,j_m\ge1$ with $r+j_1+\dots+j_m=j$, the coefficients are nonnegative integers, and the only term containing $\nu^{(j)}$ is the term $(r,m,j_1)=(0,1,j)$, with coefficient one. For $j=1$, this is the chain rule. If it holds for $j$, we differentiate each term once more. By the chain rule, the derivative of the term indexed by $(r,m,j_1,\dots,j_m)$ is the sum of the terms indexed by $(r+1,m,j_1,\dots,j_m)$, by $(r,m+1,j_1,\dots,j_m,1)$, and by $(r,m,j_1,\dots,j_\ell+1,\dots,j_m)$ for $\ell\le m$. All of them satisfy $r+\sum j_\ell=j+1$. A term containing $\nu^{(j+1)}$ arises only from a term containing $\nu^{(j)}$ by raising that index, which proves the claim. Since $\partial_\xi^j[\mathcal F(\xi,\nu(\xi))]=\sigma_j$ and $D_\mu\mathcal F(\xi,\nu)=\widetilde{\mathcal L}_\xi$, \eqref{eq:jets-faa} implies \eqref{eq:jets-differentiated} with $R_j$ equal to the sum of all other terms. It remains to compute $\partial_\xi^rD^m_\mu\mathcal F(\xi,\mu)[\mu_1,\dots,\mu_m]$ for fixed $\mu,\mu_1,\dots,\mu_m$. By \eqref{eq:jets-Dm-lawmap}, \[ D^m_\mu\mathcal F(\xi,\mu)[\mu_1,\dots,\mu_m]=p^m\,\mathcal F(\xi,\mu)*\mathsf R\mathcal K^{(0)}_\xi\mu_1*\dots*\mathsf R\mathcal K^{(0)}_\xi\mu_m. \] By \eqref{eq:jets-CP-derivative} with $m=1$ and the chain rule, $\partial_\xi\mathcal F(\xi,\mu)=p\,\mathcal F(\xi,\mu)*\mathsf R\mathcal K^{(1)}_\xi\mu$, and $\partial_\xi\mathcal K^{(r)}_\xi=\mathcal K^{(r+1)}_\xi$. Since convolution is a bounded multilinear operation on $\mathcal M_q$, the Leibniz rule applies, and by induction on $r$ we find that $\partial_\xi^rD^m_\mu\mathcal F(\xi,\mu)[\mu_1,\dots,\mu_m]$ is a sum, with positive integer coefficients depending only on $r$ and $m$, of the terms \[ p^{m+n'}\,\mathcal F(\xi,\mu)*\mathsf R\mathcal K^{(r_1)}_\xi\mu_1*\dots*\mathsf R\mathcal K^{(r_m)}_\xi\mu_m*\mathsf R\mathcal K^{(s_1)}_\xi\mu*\dots*\mathsf R\mathcal K^{(s_{n'})}_\xi\mu, \] with $n'\ge0$, $s_1,\dots,s_{n'}\ge1$, and $r_1+\dots+r_m+s_1+\dots+s_{n'}=r$. At $\mu=\nu$ we have $\mathcal F(\xi,\nu)=\nu$. In each term of $R_j$, the number of jump factors is $n=m+n'\le m+r\le j$, and $n\ge1$ because $r+\sum j_\ell=j\ge1$ forces $m\ge1$ or $r\ge1$. The inputs are $\nu$ or $\nu^{(j_\ell)}=\sigma_{j_\ell}$ with $j_\ell\le j-1$, since the only term with $j_\ell=j$ has been removed. The formulas for $R_1$ and $R_2$ follow from the same computation. \end{proof} \subsection{Estimates for selected jumps}\label{sec:jets-tensor} For a finite real measure $\mu$ on $\R$ whose restriction to $\mathcal Z$ is absolutely continuous with a bounded density, we set \[ h(\mu)=\operatorname*{ess\,sup}_{x\in\mathcal Z}\Bigl|\frac{\dd\mu}{\dd x}(x)\Bigr|,\qquad\|\mu\|_{q,\natural}=\|\mu\|_{q,\TV}+h(\mu). \] By \Cref{lem:jets-law}(i),(ii), we have $h(\widetilde\nu_\xi)\le H_p$ and $\|\widetilde\nu_\xi\|_{q,\natural}\le C$. In this subsection, the notation $\mathcal K^{(r)}_\xi$ also denotes its integral-kernel extension to finite real measures $\mu$ with $\|\mu\|_{q,\natural}<\infty$. \begin{lemma}\label{lem:jets-tensor} There exists $C>0$, depending only on $q$ and $\Xi$, such that the following holds for $p\ge p_1(\Xi)$ and $\xi\in\Xi'$. Let $1\le n\le4$, let $r_1,\dots,r_n\ge0$ with $r_1+\dots+r_n\le4$, and let $\mu_1,\dots,\mu_n$ be finite real measures with $\|\mu_i\|_{q,\natural}<\infty$. Let \[ T=\widetilde\nu_\xi*\mathsf R\mathcal K^{(r_1)}_\xi\mu_1*\dots*\mathsf R\mathcal K^{(r_n)}_\xi\mu_n,\qquad \Pi=\prod_{i=1}^n\|\mu_i\|_{q,\natural},\qquad \Sigma_h=\sum_{i=1}^nh(\mu_i)\prod_{k\neq i}\|\mu_k\|_{q,\natural}. \] Then $T$ is absolutely continuous, and \begin{align} \|T\|_{q,\TV}&\le C\,\Pi,\label{eq:jets-tensor-TV}\\ \operatorname*{ess\,sup}_{|z|\le R_0}\Bigl|\frac{\dd T}{\dd z}(z)\Bigr|&\le C\bigl(H_p+e^{-p^2/C}\bigr)\Pi+C\,\Sigma_h,\label{eq:jets-tensor-density}\\ \int_{\mathcal O_\xi}(1+|z|^q)\,|T|(\dd z)&\le C\bigl(e^{-a_1\sqrt p}+e^{-p^{7/4}/C}\bigr)\Pi+C\,\Sigma_h.\label{eq:jets-tensor-offcore} \end{align} \end{lemma} \begin{proof} We abbreviate $\nu=\widetilde\nu_\xi$ and $\tau=\tau(\xi)\le4(p\xi_-)^{-2}$, and we let $\ell_i(y)=\int\partial_\xi^{r_i}k_\tau(x;y)\,\mu_i(\dd x)$ be the density of $\mathcal K^{(r_i)}_\xi\mu_i$. We first record three bounds on $\ell_i$, which follow from \Cref{lem:jets-kernel-bounds} with $m=0$ after splitting the integral over $x$ into $x\in\mathcal Z$, where $\mu_i$ has density at most $h(\mu_i)$, and $|x|\ge\rho/5$: \begin{align} \int(1+|y|^q)|\ell_i(y)|\dd y&\le C\int\bigl(1+(\tau/|x|)^q\bigr)|\mu_i|(\dd x)\le C\|\mu_i\|_{q,\natural},\label{eq:jets-ell-1}\\ \operatorname*{ess\,sup}_{|y|\ge\rho/32}|\ell_i(y)|&\le Ce^{-p^2/C}\|\mu_i\|_\TV+Ch(\mu_i),\label{eq:jets-ell-2}\\ \int_{|y|\ge r_p/20}(1+|y|^q)|\ell_i(y)|\dd y&\le Ce^{-p^{7/4}/C}\|\mu_i\|_\TV+Ch(\mu_i).\label{eq:jets-ell-3} \end{align} For \eqref{eq:jets-ell-1} we used \Cref{lem:jets-kernel-bounds}(a) and $\int_{\mathcal Z}|x|^{-q}\dd x<\infty$. For \eqref{eq:jets-ell-2}, the part with $|x|\ge\rho/5$ is bounded by \Cref{lem:jets-kernel-bounds}(c), which gives $C(32/\rho)e^{-(\rho/5)(\rho/32)/(2\tau)}$, and the part with $x\in\mathcal Z$ by \Cref{lem:jets-kernel-bounds}(d), which gives $Ch(\mu_i)\tau(32/\rho)^2$. For \eqref{eq:jets-ell-3}, the part with $|x|\ge\rho/5$ is bounded by \Cref{lem:jets-kernel-bounds}(b), with exponent $(\rho/5)(r_p/20)/(2\tau)\ge p^{7/4}/C$, and the part with $x\in\mathcal Z$ by \Cref{lem:jets-kernel-bounds}(a). The measure $T$ is absolutely continuous since $n\ge1$, with density $z\mapsto\int\nu(\dd d)\,G(d-z)$, where $G=\ell_1*\dots*\ell_n$. The bound \eqref{eq:jets-tensor-TV} follows from \Cref{lem:jets-complete} and \eqref{eq:jets-ell-1}, since $\|\nu\|_{q,\TV}\le C$. For \eqref{eq:jets-tensor-density}, consider $|z|\le R_0$. The part of the integral with $d\in(-3\rho/4,3\rho/4)$ is at most $H_p\|G\|_{L^1}\le H_p\prod_i\|\ell_i\|_{L^1}\le CH_p\Pi$. For the remaining $d$, we have $|d-z|\ge\rho/8$. The argument in the proof of \Cref{lem:jets-onejump} shows that for almost every $w$ with $|w|\ge\rho/8$, \[ |G(w)|\le\sum_{i=1}^n\operatorname*{ess\,sup}_{|y|\ge|w|/n}|\ell_i(y)|\prod_{k\ne i}\|\ell_k\|_{L^1}, \] and $|w|/n\ge\rho/32$. By Fubini's theorem, this bound holds at $w=d-z$ for $\nu$-almost every $d$ and Lebesgue-almost every $z$. By \eqref{eq:jets-ell-1} and \eqref{eq:jets-ell-2}, the remaining part is therefore at most $Ce^{-p^2/C}\Pi+C\Sigma_h$ for almost every $z$. For \eqref{eq:jets-tensor-offcore}, we have \begin{multline*} \int_{\mathcal O_\xi}(1+|z|^q)\,|T|(\dd z)\\ \le\int\nu(\dd d)\int\prod_{i=1}^n|\ell_i(y_i)|\,\one\Bigl\{d-\sum_iy_i\in\mathcal O_\xi\Bigr\}\Bigl(1+\Bigl|d-\sum_iy_i\Bigr|^q\Bigr)\dd y_1\cdots\dd y_n. \end{multline*} We use $1+|d-\sum_iy_i|^q\le(1+|d|^q)\prod_i(1+|y_i|^q)$. The part with $d\notin\mathcal S_\xi$ is at most $C\int_{\mathcal S_\xi^c}(1+|d|^q)\,\nu(\dd d)\,\Pi\le Ce^{-a_1\sqrt p}\Pi$ by \eqref{eq:jets-ell-1} and \eqref{eq:jets-holder}. If $d\in\mathcal S_\xi$ and $d-\sum_iy_i\in\mathcal O_\xi$, then $(1+\theta)|\sum_iy_i|>r_p/4$, which gives $|\sum_iy_i|\ge r_p/5$, and some $|y_i|\ge r_p/(5n)\ge r_p/20$; further, $|d|\le2$. The part with $d\in\mathcal S_\xi$ is therefore at most \[ 3\sum_{i=1}^n\int_{|y|\ge r_p/20}(1+|y|^q)|\ell_i(y)|\dd y\prod_{k\ne i}\int(1+|y|^q)|\ell_k(y)|\dd y, \] which is at most $Ce^{-p^{7/4}/C}\Pi+C\Sigma_h$ by \eqref{eq:jets-ell-1} and \eqref{eq:jets-ell-3}. \end{proof} \subsection{Inversion of the component matrix}\label{sec:jets-matrix} \begin{lemma}\label{lem:jets-matrix} Let $P\ge1$, $\varepsilon\in[0,1/4]$, and $\delta_{\mathrm b},\delta_{\mathrm i}\ge0$ with $P^3(\delta_{\mathrm b}+\delta_{\mathrm i})\le1/16$. Let $v=(v_{\mathrm c},v_{\mathrm b},v_{\mathrm i})$ and $s=(s_{\mathrm c},s_{\mathrm b},s_{\mathrm i})$ belong to $[0,\infty)^3$ and satisfy, entrywise, \[ v\le Mv+s,\qquad M=\begin{pmatrix}\varepsilon&P&P\\\delta_{\mathrm b}&\varepsilon&\delta_{\mathrm b}\\\delta_{\mathrm i}&P&\delta_{\mathrm i}+\delta_{\mathrm b}\end{pmatrix}. \] Then \begin{align*} v_{\mathrm c}&\le3s_{\mathrm c}+11P^2(s_{\mathrm b}+s_{\mathrm i}),\\ v_{\mathrm b}&\le6\delta_{\mathrm b}s_{\mathrm c}+4s_{\mathrm b}+25P^2\delta_{\mathrm b}s_{\mathrm i},\\ v_{\mathrm i}&\le6(\delta_{\mathrm i}+P\delta_{\mathrm b})s_{\mathrm c}+4Ps_{\mathrm b}+4s_{\mathrm i}. \end{align*} \end{lemma} \begin{proof} The hypotheses give $\delta_{\mathrm b},\delta_{\mathrm i},P\delta_{\mathrm b},P\delta_{\mathrm i},P^2\delta_{\mathrm b}\le1/16$. Since $\varepsilon\le1/4$, the second inequality gives \begin{equation}\label{eq:jets-mat1} v_{\mathrm b}\le\tfrac43\bigl(\delta_{\mathrm b}v_{\mathrm c}+\delta_{\mathrm b}v_{\mathrm i}+s_{\mathrm b}\bigr). \end{equation} Inserting this into the third inequality, we find that the coefficient of $v_{\mathrm i}$ on the right side is $4P\delta_{\mathrm b}/3+\delta_{\mathrm i}+\delta_{\mathrm b}\le1/4$, and therefore \begin{equation}\label{eq:jets-mat2} v_{\mathrm i}\le\tfrac43\Bigl(\bigl(\delta_{\mathrm i}+\tfrac43P\delta_{\mathrm b}\bigr)v_{\mathrm c}+\tfrac43Ps_{\mathrm b}+s_{\mathrm i}\Bigr)\le2(\delta_{\mathrm i}+P\delta_{\mathrm b})v_{\mathrm c}+2Ps_{\mathrm b}+2s_{\mathrm i}. \end{equation} Inserting \eqref{eq:jets-mat2} into \eqref{eq:jets-mat1} and using $2\delta_{\mathrm i}+2P\delta_{\mathrm b}\le1/4$, we obtain \begin{equation}\label{eq:jets-mat3} v_{\mathrm b}\le2\delta_{\mathrm b}v_{\mathrm c}+2s_{\mathrm b}+3\delta_{\mathrm b}s_{\mathrm i}. \end{equation} Inserting \eqref{eq:jets-mat2} and \eqref{eq:jets-mat3} into the first inequality, the coefficient of $v_{\mathrm c}$ on the right side is $\varepsilon+2P\delta_{\mathrm b}+2P\delta_{\mathrm i}+2P^2\delta_{\mathrm b}\le5/8$, and we find \[ \tfrac38v_{\mathrm c}\le s_{\mathrm c}+(2P+2P^2)s_{\mathrm b}+(3P\delta_{\mathrm b}+2P)s_{\mathrm i}\le s_{\mathrm c}+4P^2s_{\mathrm b}+3Ps_{\mathrm i}, \] which gives the bound for $v_{\mathrm c}$. Inserting it into \eqref{eq:jets-mat3} and \eqref{eq:jets-mat2}, and using $22P^2\delta_{\mathrm b}\le2$ and $22P^2(\delta_{\mathrm i}+P\delta_{\mathrm b})\le22P^3(\delta_{\mathrm i}+\delta_{\mathrm b})\le2$, we obtain the bounds for $v_{\mathrm b}$ and $v_{\mathrm i}$. \end{proof} \subsection{Proof of the quantitative bounds}\label{sec:jets-quant-proof} We use the following representation of the pairing of a test function with a selected tensor. Let $\psi$ be bounded and Borel, let $T$ be as in \Cref{lem:jets-tensor}, and let $\widetilde D\sim\widetilde\nu_\xi$ and $T_1,\dots,T_n\sim\Exp(1)$ be independent. By Fubini's theorem, \begin{equation}\label{eq:jets-dual-tensor} \begin{gathered} \int\psi\dd T=\int\!\!\cdots\!\!\int G_\psi(x_1,\dots,x_n)\,\mu_1(\dd x_1)\cdots\mu_n(\dd x_n),\\ G_\psi(x_1,\dots,x_n)=\partial_{\eta_1}^{r_1}\cdots\partial_{\eta_n}^{r_n}\,\E\,\psi\Bigl(\widetilde D-\sum_{i=1}^n\frac{\tau(\eta_i)T_i}{x_i}\Bigr)\Big|_{\eta_1=\dots=\eta_n=\xi}, \end{gathered} \end{equation} where the law of $\widetilde D$ is held fixed when differentiating in $\eta_i$. To justify \eqref{eq:jets-dual-tensor}, write the expectation as $\int\widetilde\nu_\xi(\dd d)\int\psi(d-\sum_iy_i)\prod_ik_{\tau(\eta_i)}(x_i;y_i)\dd y$ and differentiate under the integral sign; this is allowed because, by \Cref{lem:jets-score}, for $\eta_i$ in a compact neighborhood of $\xi$ in $(0,\infty)$ the functions $|\partial_{\eta_i}^{r}k_{\tau(\eta_i)}(x_i;\cdot)|$ are dominated by a fixed integrable function of $y_i$. In particular, $|G_\psi|\le\|\psi\|_\infty\prod_i\int|\partial_\xi^{r_i}k_\tau(x_i;y)|\dd y\le C\|\psi\|_\infty$ for all $x_1,\dots,x_n\neq0$, since the substitution $u=|x||y|/\tau$ and \Cref{lem:jets-score} give $\int|\partial_\xi^rk_\tau(x;y)|\dd y=\xi^{-r}\int_0^\infty|P_{0,r}(u)|e^{-u}\dd u$, which does not depend on $x$. If $\psi\in C^2$ with bounded derivatives, we may also differentiate under the expectation in \eqref{eq:jets-dual-tensor} when the total differentiation order is at most two. \begin{proof}[Proof of \Cref{prop:jets-quantitative}] Fix $\Xi$ and $q$, and let $a$ and $A_0,\dots,A_4$ be as in \eqref{eq:jets-exponents}. Throughout the proof, $p\ge p_{\mathrm{der}}$, where $p_{\mathrm{der}}\ge p_{\mathrm{an}}(\Xi)$ is large enough for the finitely many conditions stated below, each of which depends only on $\Xi$, $q$, and the constants of \ref{item:jets-S1}--\ref{item:jets-S4}. Fix $\xi\in\Xi$ and write $\nu=\widetilde\nu_\xi$, $\tau=\tau(\xi)$, $\mathcal O=\mathcal O_\xi$, and $\|\cdot\|_{\mathrm c}$, $\|\cdot\|_{\mathrm b}$, $\|\cdot\|_{\mathrm i}$ for the norms of \Cref{def:jets-components}. Recall that $|\partial_\xi^k\tau|\le Cp^{-2}$ for $k\le4$ by \Cref{lem:jets-parameters}. \proofstep{1}{Small quantities.} Since $a\le a_10$ we write $\lambda_p(\xi)=\exp\varphi_{\xi\sqrt p}(1)$, where $\varphi_E$ is the free energy of $\nu_{p,E}$ (\Cref{def:press-pressure}). The goal of this section is to prove the following two results. The first identifies the leading behavior of $\lambda_p$ on compact scaled windows. The second shows that the free energy at $s=1$ is negative at all sufficiently large energies. \begin{theorem}\label{thm:asym-leading} Fix a compact set $\Xi\subset(0,\infty)$. Then \begin{equation*} \lim_{p\to\infty}\,\sup_{\xi\in\Xi}\,\Bigl|\lambda_p(\xi)-\frac{\mustar}{\xi}\Bigr|=0 . \end{equation*} \end{theorem} Here $\lambda_p(\xi)$ is defined for $p\ge p_0(\min\Xi)$, the threshold of \Cref{cor:stab-halfline} (on the window, $\nu_{p,\xi\sqrt p}$ is the law of \Cref{thm:stab-window}), and $\mustar=\sqrt\pi/2$. \begin{proposition}\label{prop:asym-entry} Let $M_0$ be the numerical constant of \Cref{cor:stab-halfline}. There exist a numerical constant $M_{\mathrm{ext}}\ge M_0$ and a numerical threshold $p_{\mathrm{ext}}>1$ such that, for every $p\ge p_{\mathrm{ext}}$ and every $E\ge M_{\mathrm{ext}}\sqrt p$, the principal solution $\nu_{p,E}$ satisfies \begin{equation}\label{eq:asym-entry-statement} \varphi_E(1)\le\varphi_E\bigl(1-1/\log p\bigr)\le-\log 2 . \end{equation} \end{proposition} \Cref{thm:asym-leading} is used in \Cref{sec:uniq} to locate the critical energy, and \Cref{prop:asym-entry} is used in \Cref{sec:uniq,sec:main} to show that the free energy at $s=1$ has no zero at energies above the scaled windows. We now explain how the proofs go. Both rest on one expansion of the free energy. For $s<1$, \Cref{prop:press-identification} expresses $\varphi_E(s)$ as the exponential growth rate of the path moments $\Phi_L(s)$. In \Cref{sec:asym-normalized} we write $\Phi_L(s)$ through the iterates of a normalized operator $\mathsf T^*_{p,s,\xi}$, which differs from the operator $\mathcal A_s$ of \Cref{def:press-operators} by an explicit scalar $\alpha_{p,s}(\xi)$. We find that $\varphi_E(s)=\log\alpha_{p,s}(\xi)+\log r_{p,s}(\xi)$, where $r_{p,s}(\xi)$ is the exponential growth rate of the normalized iterates. As $s\uparrow1$, the scalar $\alpha_{p,s}(\xi)$ converges to $\mustar/a_p(\xi)$, and $a_p(\xi)\to\xi$ as $p\to\infty$. It therefore suffices to show that $r_{p,s}(\xi)=1+o(1)$ for $s$ close to one, uniformly in $\xi\in\Xi$. The lower bound $r_{p,s}(\xi)\ge1-o(1)$ is elementary. Since the principal law concentrates near one, a test function supported near one is a subsolution for $\mathsf T^*_{p,s,\xi}$ (\Cref{sec:asym-lower}). The upper bound is the main work. There are two parts of the state space on which the normalized operator has growth close to one. The first is the core, near $x=1$. The second is the tail, where $x$ is of order $K=p^2a_p(\xi)^2$. There a state $x$ produces a state of the same order when the side variable $B$ nearly cancels the term $T_sx/K$, and at $s=1$ the expected weight of such near-cancellations is also close to one. We cover the real line by six regions (\Cref{def:asym-charts}), estimate the norm of every block of the operator between two regions (\Cref{prop:asym-blocks}), and show that every closed chain of blocks other than the two diagonal blocks of the core and the tail has a small product. A diagonal balancing of the resulting six-by-six matrix (\Cref{lem:asym-balancing}) then produces a supersolution with eigenvalue $1+o(1)$. The same construction applies, with uniform constants, at energies $E\ge M\sqrt p$ for large $M$, if the exponent is $s_p=1-1/\log p$. This choice keeps $p^{1-s_p}=e$ bounded, and it replaces the losses of order $\log K$ in the capped integrals by losses of order $\log p$; this matters because $K$ is unbounded at such energies. Since the scalar $\alpha_{p,s_p}$ is small there, the free energy at $s_p$ is negative, and \Cref{prop:asym-entry} follows from the monotonicity of the free energy on $(0,1]$. \section{The normalized transfer operator}\label{sec:asym-normalized} Throughout this section, $p>1$ and $E>2$. We use the normalizations of \Cref{def:rde-normalizations}: \begin{equation}\label{eq:asym-normalizations} g(E)=\frac{E-\sqrt{E^2-4}}{2}, \qquad \theta=g(E)^2, \qquad K=\frac p\theta . \end{equation} When $E=\xi\sqrt p$, we also write \begin{equation}\label{eq:asym-ap} a_p(\xi)=\frac{1}{\sqrt p\,g(\xi\sqrt p)}=\frac{\xi+\sqrt{\xi^2-4/p}}{2}, \qquad\text{so that}\qquad K=p^2a_p(\xi)^2 . \end{equation} Let $\pi$ be a real solution at energy $E$ (\Cref{def:rde-real}), let $D$ have law $\pi$, and let $\nu$ be the law of $B=g(E)D$. Since $g(E)E=1+\theta$ and $g(E)^2=\theta$, the law $\nu$ solves \begin{equation}\label{eq:asym-B-equation} B\eqdist 1+\theta-\frac\theta p\sum_{j=1}^N\frac{Y_j}{B_j}, \qquad N\sim\Pois(p),\quad Y_j\sim\Exp(1), \end{equation} with all variables on the right side independent. Conversely, if a probability measure $\nu$ on $\R\setminus\{0\}$ solves \eqref{eq:asym-B-equation}, then the law of $B/g(E)$ is a real solution at $E$. We denote by $\nu^x$ the law of $X=B^{-1}$. In the notation of \Cref{lem:press-gauge} and \Cref{lem:press-chain}(ii), $\nu$ is the law $\nu_g$ with $g=g(E)$, the rate $K_g=p/g^2$ equals $K$, and $\nu^x$ is the law $\pi_g^x$; the variable $\beta=g^2Y/p$ with $Y\sim\Exp(1)$ equals $Y/K$. By \Cref{lem:press-gauge}(i), $\nu$ satisfies \eqref{eq:press-hyp}. \begin{definition}\label{def:asym-normalized} For $s\in(0,2)$, set $\mu_s=\Gamma(1+s/2)$ and let $T_s$ be a random variable with density \begin{equation}\label{eq:asym-omega} \omega_s(t)=\frac{t^{s/2}e^{-t}}{\mu_s}\,\one_{\{t>0\}} . \end{equation} Let $B\sim\nu$ and $T_s$ be independent. For $x\in\R$, set \begin{equation}\label{eq:asym-Dx} D_x=B-\frac{T_s x}{K}, \qquad Y_x=D_x^{-1}. \end{equation} For a measurable function $F\colon\R\to[0,\infty]$, define \begin{equation}\label{eq:asym-Tstar} (\mathsf T^*F)(x)=|x|^s\,\E\bigl[F(Y_x)\bigr], \qquad x\in\R, \end{equation} and, for $E=\xi\sqrt p$, set \begin{equation}\label{eq:asym-alpha} \alpha_{p,s}(\xi)=p^{1-s}\mu_s\,a_p(\xi)^{-s}. \end{equation} When $\nu=\nu_{p,E}$ is the principal solution, we write $\mathsf T^*_{p,s,\xi}$ for $\mathsf T^*$. \end{definition} The variable $Y_x$ is defined almost surely. Indeed, for $x\ne0$ the variable $D_x$ has a density because $T_s$ has one and is independent of $B$, and for $x=0$ we have $D_0=B$, which has no atom at zero because $\pi$ is a law on $\R\setminus\{0\}$. We also write $e_s(x)=|x|^s$. \begin{lemma}\label{lem:asym-palm-normalization} Let $\pi$ be a real solution at $E=\xi\sqrt p>2$, and let $\nu$, $\nu^x$ be as above. Let $\mathcal A_s$ be the transfer operator in the $x$-coordinate of \Cref{def:press-operators}(vi), defined with the law $\nu$ and the rate $K$. For every $s\in(0,2)$ and every Borel $F\colon\R\to[0,\infty]$, we have \begin{equation}\label{eq:asym-A-is-alpha-T} \mathcal A_sF=\alpha_{p,s}(\xi)\,\mathsf T^*F . \end{equation} Moreover, for $s\in(0,1)$ and $L\ge1$, the path moment of \Cref{def:press-path-moment} satisfies \begin{equation}\label{eq:asym-Phi-normalized} \Phi_L(s;E)=g(E)^s\,\alpha_{p,s}(\xi)^L\int(\mathsf T^*)^Le_s\dd\nu^x , \end{equation} the limit \begin{equation}\label{eq:asym-r-definition} r_{\pi,s}=\lim_{L\to\infty}\Bigl(\int(\mathsf T^*)^Le_s\dd\nu^x\Bigr)^{1/L} \end{equation} exists in $(0,\infty)$, and \begin{equation}\label{eq:asym-pressure-split} \varphi_\pi(s)=\log\alpha_{p,s}(\xi)+\log r_{\pi,s}. \end{equation} \end{lemma} \begin{proof} Let $Y\sim\Exp(1)$. For every measurable $G\ge0$, the definition of $\omega_s$ gives \begin{equation}\label{eq:asym-tilt} \E\bigl[Y^{s/2}G(Y)\bigr]=\int_0^\infty y^{s/2}e^{-y}G(y)\dd y =\mu_s\int_0^\infty\omega_s(y)G(y)\dd y=\mu_s\,\E\bigl[G(T_s)\bigr]. \end{equation} By \Cref{def:press-operators}(iii) and (vi) with $\beta=Y/K$, we have \begin{equation*} (\mathcal A_sF)(x)=p\,\E\Bigl[\Bigl(\frac YK\Bigr)^{s/2}|x|^sF\Bigl(\Bigl(B-\frac{Yx}{K}\Bigr)^{-1}\Bigr)\Bigr] =pK^{-s/2}\mu_s\,|x|^s\,\E\bigl[F(Y_x)\bigr], \end{equation*} where in the second equality we conditioned on $B$ and applied \eqref{eq:asym-tilt}. By \eqref{eq:asym-ap}, we have $pK^{-s/2}\mu_s=p\,(p\,a_p(\xi))^{-s}\mu_s=\alpha_{p,s}(\xi)$, which proves \eqref{eq:asym-A-is-alpha-T}. Iterating \eqref{eq:asym-A-is-alpha-T} and inserting it into the definition $\Phi_L(s;E)=g^s\int\mathcal A_s^{(g)L}e_s\dd\pi_g^x$ of \Cref{def:press-path-moment} with $g=g(E)$, for which $\mathcal A_s^{(g)}=\mathcal A_s$ and $\pi_g^x=\nu^x$, we obtain \eqref{eq:asym-Phi-normalized}. Let $s\in(0,1)$. By \Cref{prop:press-identification}(iii), the law $\nu$ and the rate $K$ satisfy the hypotheses of \Cref{prop:press-identification}, namely \eqref{eq:press-hyp} and the fractional cancellation estimate \eqref{eq:press-fc} for some $q\in(s,1)$. Part (i) of that proposition gives $C\ge1$ with \begin{equation*} C^{-1}\lambda(s)^L\le\int\mathcal A_s^Le_s\dd\nu^x\le C\lambda(s)^L,\qquad L\ge1, \end{equation*} where $\lambda(s)$ is the Perron--Frobenius eigenvalue of \Cref{thm:press-perron} for $\nu$ and $K$. By \Cref{def:press-pressure}(iii), $\lambda(s)=e^{\varphi_\pi(s)}$. Dividing by $\alpha_{p,s}(\xi)^L$ and using \eqref{eq:asym-A-is-alpha-T}, we find \begin{equation*} C^{-1}\Bigl(\frac{e^{\varphi_\pi(s)}}{\alpha_{p,s}(\xi)}\Bigr)^L\le\int(\mathsf T^*)^Le_s\dd\nu^x\le C\Bigl(\frac{e^{\varphi_\pi(s)}}{\alpha_{p,s}(\xi)}\Bigr)^L . \end{equation*} Taking $L$-th roots, we conclude that the limit in \eqref{eq:asym-r-definition} exists, equals $e^{\varphi_\pi(s)}/\alpha_{p,s}(\xi)\in(0,\infty)$, and satisfies \eqref{eq:asym-pressure-split}. \end{proof} When $\pi$ is the principal solution at $E=\xi\sqrt p$, that is, the law of $B/g(E)$ under $\nu_{p,E}$, we write $r_{p,s}(\xi)=r_{\pi,s}$. Then \eqref{eq:asym-pressure-split} reads $\varphi_E(s)=\log\alpha_{p,s}(\xi)+\log r_{p,s}(\xi)$. \begin{remark}\label{rem:asym-palm} By \Cref{rem:press-path-moment} and \Cref{prop:loc-columns}, $\Phi_L(s;E)$ is the expected generation-$L$ sum of the $s$-th powers of the formal inverse column. In this interpretation each generation contributes the factor $p$ from the Mecke formula, the factor $p^{-s/2}Y^{s/2}$ from $|h_e|^s$, and the factor $g(E)^s$ from the normalization $B=g(E)D$. The tilt $Y^{s/2}$ turns the exponential mark into $T_s$, and the remaining scalar is $\alpha_{p,s}(\xi)=p^{1-s/2}\mu_sg(E)^s$, which agrees with \eqref{eq:asym-alpha} because $g(E)=(\sqrt p\,a_p(\xi))^{-1}$. We do not use this interpretation below. \end{remark} \begin{lemma}\label{lem:asym-critical} Let $\pi$ be a real solution at $E=\xi\sqrt p>2$. Then $\mu_1=\Gamma(3/2)=\mustar$, the limit $\lim_{s\uparrow1}r_{\pi,s}$ exists, and \begin{equation}\label{eq:asym-critical-formula} \exp\varphi_\pi(1)=\lim_{s\uparrow1}\alpha_{p,s}(\xi)\,r_{\pi,s} =\frac{\mustar}{a_p(\xi)}\,\lim_{s\uparrow1}r_{\pi,s}. \end{equation} \end{lemma} \begin{proof} For a standard complex Gaussian $w$, the variable $|w|^2$ is exponential with mean one. Then \begin{equation*} \mustar=\E|w|=\int_0^\infty y^{1/2}e^{-y}\dd y=\Gamma(3/2)=\mu_1, \end{equation*} and $\Gamma(3/2)=\Gamma(1/2)/2=\sqrt\pi/2$. By \Cref{thm:press-perron}(ii) and (iv), $s\mapsto \lambda(s)$ is positive and real-analytic on $(0,2)$. In particular, $\varphi_\pi(s)=\log\lambda(s)$ is continuous at $s=1$ (see also \Cref{prop:press-properties}(i)). Using \eqref{eq:asym-pressure-split}, we have $\alpha_{p,s}(\xi)r_{\pi,s}=\exp\varphi_\pi(s)\to\exp \varphi_\pi(1)$ as $s\uparrow1$. Since $\Gamma$ is continuous, $\alpha_{p,s}(\xi)\to\mu_1a_p(\xi)^{-1} =\mustar/a_p(\xi)>0$. Dividing, we conclude that $r_{\pi,s}$ converges as $s\uparrow1$ and that \eqref{eq:asym-critical-formula} holds. \end{proof} By \Cref{lem:asym-critical}, the proof of \Cref{thm:asym-leading} reduces to showing that $\lim_{s\uparrow1}r_{\pi,s}=1+o(1)$ for the principal solution, uniformly on the window. The rest of this section is devoted to two-sided bounds on $r_{\pi,s}$. \section{Inputs from the principal solution}\label{sec:asym-inputs} We use the following properties of the principal solution from \Cref{sec:stab}. Let $\mathbf q_{\mathrm{can}}=(1,0,1,1/16)$ be the canonical tuple of \Cref{def:stab-class}, and for $\theta>0$ let $\mathfrak S_{p,\theta}=\mathfrak S_{p,\theta}(\mathbf q_{\mathrm{can}})$ be the corresponding class. Every $\nu\in\mathfrak S_{p,\theta}$ has the form $e^{-p}\delta_{1+\theta}+f(b)\dd b$ and satisfies the core, boundary and tail bounds of \Cref{def:stab-class} with $(a,A,K',\rho)=(1,0,1,1/16)$: \begin{align} \nu\bigl(\{b:|b-1|>p^{-1/4}\}\bigr)&\le e^{-\sqrt p},\label{eq:asym-core-barrier}\\ f(b)&\le e^{-p}\quad\text{for almost every }b\in(-1/16,1/16),\label{eq:asym-boundary-density}\\ \nu\bigl(\{b:|b|>L\}\bigr)&\le\frac{e^{-p}}{L}+e^{-pL},\qquad L\ge3.\label{eq:asym-tail-barrier} \end{align} By \Cref{cor:stab-halfline}, with the numerical constants $M_0=3$ and $c_{\mathrm{ent}}=1$, we have $g(E)^2\le c_{\mathrm{ent}}/p$ for $E\ge M_0\sqrt p$, and for every $\xi_0>0$ there exists $p_0(\xi_0)$ such that for every $p\ge p_0(\xi_0)$ and $E\ge\xi_0\sqrt p$ the following hold: the principal solution $\nu_{p,E}$ is the unique fixed point of the law map in $\mathfrak S_{p,\theta(E)}$, where $\theta(E)=g(E)^2$; it is the law of \Cref{thm:stab-window} if $E\le(M_0+1)\sqrt p$, and the fixed point of \Cref{prop:stab-entry} with $\theta=\theta(E)$ if $E\ge M_0\sqrt p$; and the law of $B/g(E)$ under $\nu_{p,E}$ is a real solution at $E$. In particular, for $p\ge p_0(\xi_0)$, the bounds \eqref{eq:asym-core-barrier}--\eqref{eq:asym-tail-barrier} hold for $\nu_{p,E}$ with numerical constants, uniformly in $E\ge\xi_0\sqrt p$. On compact scaled windows, part (iii) of \Cref{thm:stab-window} gives the same bounds with halved right sides, and part (v) gives the real-solution property. We do not use the other properties of \Cref{thm:stab-window,prop:stab-entry,cor:stab-halfline}. To treat both regimes at once, we isolate the inequalities used below. \begin{definition}\label{def:asym-admissible} Let $\mathfrak c=(c_0,A_0,K_0,\rho_0,c_*,C_*)$ be constants with $A_0\ge0$, $\rho_0\in(0,1/2]$, and $c_0,K_0,c_*,C_*>0$, and let $p\ge3$. For $K\ge1$ and $s\in(0,1)$, set \begin{equation}\label{eq:asym-ell} \ell_{K,s}=1+\min\Bigl\{\log K,\ \frac1{1-s}\Bigr\}. \end{equation} A triple $(\nu,K,s)$, consisting of a Borel probability measure $\nu$ on $\R$, a number $K\ge1$ and an exponent $s$, is \emph{$\mathfrak c$-admissible at $p$} if the following hold. \begin{enumerate}[label=\textup{(A\arabic*)}] \item\label{it:asym-A1} $\nu(\{b:|b-1|>p^{-1/4}\})\le e^{-c_0\sqrt p}$. \item\label{it:asym-A2} The restriction of $\nu$ to $[-\rho_0,\rho_0]$ has a density bounded by $p^{A_0}e^{-c_0p}$. \item\label{it:asym-A3} $\nu(\{b:|b|>L\})\le K_0p^{A_0}e^{-c_0p}L^{-1}+e^{-c_0pL}$ for every $L\ge3$. \item\label{it:asym-A4} $K\ge c_*p^2$. \item\label{it:asym-A5} $3/4\le s<1$ and $\ell_{K,s}\le C_*\log p$. \end{enumerate} \end{definition} Conditions \ref{it:asym-A1}--\ref{it:asym-A3} are the core bound, the boundary density bound and the tail bound. In the scaled windows, \ref{it:asym-A5} holds for all $s$ because $\log K\le 3\log p$. At large energies $K$ is unbounded, and \ref{it:asym-A5} is obtained from the term $(1-s)^{-1}$ at $s=1-1/\log p$. \begin{lemma}\label{lem:asym-admissible} The following statements hold. \begin{enumerate}[label=\textup{(\roman*)}] \item Fix a compact set $\Xi\subset(0,\infty)$ and write $\xi_-=\min\Xi$, $\xi_+=\max\Xi$. There exist constants $\mathfrak c_\Xi$ and a threshold $p_\Xi\ge p_0(\xi_-)$ such that for every $p\ge p_\Xi$, $\xi\in\Xi$ and $s\in[3/4,1)$, the triple $(\nu_{p,\xi\sqrt p}, p^2a_p(\xi)^2,s)$ is $\mathfrak c_\Xi$-admissible at $p$. \item Let $M\ge M_0$ and $s_p=1-1/\log p$. There exist numerical constants $\mathfrak c_{\mathrm{ext}}$ such that for every $p\ge\max\{p_0(M),e^4\}$ and every $E\ge M\sqrt p$, the triple $(\nu_{p,E},p/g(E)^2,s_p)$ is $\mathfrak c_{\mathrm{ext}}$-admissible at $p$. \end{enumerate} \end{lemma} \begin{proof} For (i), we take \begin{equation*} p_\Xi=\max\bigl\{3,\,p_0(\xi_-),\,16\xi_-^{-2},\,e\,\xi_+^2\bigr\}, \qquad \mathfrak c_\Xi=\bigl(1,0,1,1/32,\xi_-^2/4,3\bigr). \end{equation*} Conditions \ref{it:asym-A1}--\ref{it:asym-A3} follow from \eqref{eq:asym-core-barrier}--\eqref{eq:asym-tail-barrier}, since $\xi\sqrt p\ge\xi_-\sqrt p$ and $p\ge p_0(\xi_-)$; for \ref{it:asym-A2} we used that the density bound on $(-1/16,1/16)$ implies the same bound on $[-1/32,1/32]$, and that $\nu_{p,E}$ has no atom there. Since $p\ge16\xi_-^{-2}$, we have $\xi^2-4/p\ge0$, and \eqref{eq:asym-ap} gives $\xi/2\le a_p(\xi)\le\xi$. Then $K=p^2a_p(\xi)^2$ satisfies $\xi_-^2p^2/4\le K\le\xi_+^2p^2$, which gives \ref{it:asym-A4} and $K\ge1$, and \begin{equation*} \ell_{K,s}\le1+\log K\le1+2\log p+2\log\xi_+\le3\log p, \end{equation*} where the last inequality holds because $p\ge e\,\xi_+^2$. This proves \ref{it:asym-A5}. For (ii), we take $\mathfrak c_{\mathrm{ext}}=(1,0,1,1/32,1,2)$. Conditions \ref{it:asym-A1}--\ref{it:asym-A3} hold as in (i), since $E\ge M\sqrt p$ and $p\ge p_0(M)$. Since $M\ge3$, we have $\theta=g(E)^2\le4/E^2\le4/(9p)\le1/p$, and $K=p/\theta\ge p^2$. Since $p\ge e^4$, we have $s_p\ge3/4$, and $\ell_{K,s_p}\le1+(1-s_p)^{-1}=1+\log p\le2\log p$. \end{proof} \paragraph{Conventions.} In \Cref{sec:asym-elementary,sec:asym-lower,sec:asym-charts,sec:asym-blocks,sec:asym-upper}, the constants $\mathfrak c$ are fixed, and $(\nu,K,s)$ is $\mathfrak c$-admissible at $p$. Unless stated otherwise, $C$ and $c$ denote positive constants depending only on $\mathfrak c$, and $p_1$ denotes a threshold depending only on $\mathfrak c$; we increase $p_1$ finitely many times, and each time we say which condition it must satisfy. We write $\ell=\ell_{K,s}$. We set \begin{equation}\label{eq:asym-side-interval} I_*=\Bigl[\frac{99}{100},\frac{101}{100}\Bigr], \qquad \chi=\one_{I_*}, \end{equation} and we always take $p_1$ so large that $p_1^{-1/4}\le1/100$. Then $\{b:\chi(b)=0\}\subset\{b:|b-1|> p^{-1/4}\}$ for $p\ge p_1$, and \ref{it:asym-A1} gives \begin{equation}\label{eq:asym-irregular-mass} \nu\bigl(\{b:\chi(b)=0\}\bigr)\le e^{-c_0\sqrt p}. \end{equation} For $x\in\R$ we write $m=|x|/K$ and $\sigma=\sgn x$ (with $\sigma=1$ if $x=0$), so that $D_x=B-\sigma mT_s$. \section{Elementary estimates}\label{sec:asym-elementary} We begin with properties of the tilted variable $T_s$. \begin{lemma}\label{lem:asym-gamma} Let $s\in[3/4,1]$. Then the following statements hold. \begin{enumerate}[label=\textup{(\roman*)}] \item $0.88\le\mu_s\le1$, and for every $t>0$, \begin{equation}\label{eq:asym-omega-bounds} \omega_s(t)\le2t^{s/2}e^{-t}\le2t^{s/2}, \qquad \omega_s(t)\le2e^{-t/2}. \end{equation} \item For every $\tau>0$, we have $\P(T_s\le\tau)\le2\tau^{1+s/2}$ and $\P(T_s\ge\tau)\le4e^{-\tau/2}$. \item $\E[T_s^{-s}]=\Gamma(1-s/2)/\Gamma(1+s/2)\le3$. \item For $b>0$, $u>0$ and $d0$, we have \begin{equation}\label{eq:asym-omega-mass} \int_0^\infty u^{1-s}\omega_s(ub)\dd u=\frac{\Gamma(2-s/2)}{\Gamma(1+s/2)}\,b^{s-2}, \end{equation} and there exists a numerical constant $C_\Gamma$ such that $\Gamma(2-s/2)/\Gamma(1+s/2)\le1+C_\Gamma(1-s)$. \end{enumerate} \end{lemma} \begin{proof} On $(0,\infty)$ the function $\Gamma$ is positive with minimum larger than $0.88$, and it is log-convex with $\Gamma(1)=\Gamma(2)=1$; log-convexity implies that $\Gamma\le1$ on $[1,2]$. Since $1+s/2\in[1,2]$, this implies $0.88\le\mu_s\le1$. The first bound in \eqref{eq:asym-omega-bounds} follows from $1/\mu_s\le 2$. For the second, if $t\le1$ then $t^{s/2}e^{-t}\le e^{-t/2}$, and if $t\ge1$ then $t^{s/2}e^{-t}\le t^{1/2}e^{-t/2}e^{-t/2}\le e^{-1/2}e^{-t/2}$, because $t\mapsto t^{1/2}e^{-t/2}$ is decreasing on $[1,\infty)$. In both cases $\omega_s(t)\le2e^{-t/2}$. For (ii), we integrate the two bounds in \eqref{eq:asym-omega-bounds} and find $\P(T_s\le\tau)\le\int_0^\tau2t^{s/2}\dd t\le2\tau^{1+s/2}$ and $\P(T_s\ge\tau)\le\int_\tau^\infty2e^{-t/2}\dd t=4e^{-\tau/2}$. For (iii), we have $\E[T_s^{-s}]=\mu_s^{-1}\int_0^\infty t^{-s/2}e^{-t}\dd t=\Gamma(1-s/2)/\mu_s$. Since $1-s/2\in[1/2,5/8]$ and $\Gamma$ is decreasing on $(0,1]$, we have $\Gamma(1-s/2)\le\Gamma(1/2)=\sqrt\pi$, and $\sqrt\pi/0.88\le3$. Identity \eqref{eq:asym-omega-shift} follows from the definition \eqref{eq:asym-omega}, since $(u(b-d))^{s/2}=(ub)^{s/2}(1-d/b)^{s/2}$ and $e^{-u(b-d)}=e^{-ub}e^{ud}$. For (v), the substitution $w=ub$ gives \begin{equation*} \int_0^\infty u^{1-s}\omega_s(ub)\dd u =\frac{b^{s-2}}{\mu_s}\int_0^\infty w^{1-s/2}e^{-w}\dd w =\frac{\Gamma(2-s/2)}{\Gamma(1+s/2)}\,b^{s-2}. \end{equation*} The function $s\mapsto\Gamma(2-s/2)/\Gamma(1+s/2)$ is smooth on $[3/4,1]$ and equals one at $s=1$. The mean value theorem then implies the last claim, with $C_\Gamma$ the maximum of the absolute value of its derivative on $[3/4,1]$. \end{proof} We next introduce the capped weight. For $K\ge1$ and $s\in(0,1)$, set \begin{equation}\label{eq:asym-capped} v(y)=\frac{K^s|y|^s}{K^s+|y|^s}, \qquad M_K(d)=\frac{K^s}{1+K^s|d|^s}, \qquad y,d\in\R . \end{equation} Then $v(d^{-1})=M_K(d)$ for $d\ne0$; we extend $v$ by $v(\pm\infty)=K^s=M_K(0)$. The weight $v$ agrees with $|y|^s$ up to a factor two when $|y|\le K$, and it is capped at $K^s$ for larger $|y|$. The following estimate controls the integral of $M_K$ near its peak; the term $\min\{\log K,(1-s)^{-1}\}$ in $\ell_{K,s}$ comes from it. \begin{lemma}\label{lem:asym-capped} Let $K\ge1$ and $s\in(0,1)$. Then \begin{equation}\label{eq:asym-v-comparison} \tfrac12\min\{|y|^s,K^s\}\le v(y)\le\min\{|y|^s,K^s\}, \qquad M_K(d)\le\min\{K^s,|d|^{-s}\}. \end{equation} Moreover, for every interval $I\subset\R$ of length at most two, \begin{equation}\label{eq:asym-master} \int_IM_K(d)\dd d\le2\ell_{K,s}. \end{equation} \end{lemma} \begin{proof} The bounds \eqref{eq:asym-v-comparison} follow from $\max\{K^s,|y|^s\}\le K^s+|y|^s\le 2\max\{K^s,|y|^s\}$ and from $1+K^s|d|^s\ge\max\{1,K^s|d|^s\}$. Since $M_K$ is even and nonincreasing in $|d|$, its integral over an interval of length at most two is at most its integral over $[-1,1]$. Substituting $w=K|d|$, we have \begin{equation*} \int_{-1}^1M_K(d)\dd d=2K^{s-1}\int_0^K\frac{\dd w}{1+w^s} \le2K^{s-1}\Bigl(1+\int_1^Kw^{-s}\dd w\Bigr) =2K^{s-1}+2\,\frac{1-K^{s-1}}{1-s}. \end{equation*} We have $K^{s-1}\le1$. Writing $K^{s-1}=e^{-y}$ with $y=(1-s)\log K\ge0$ and using $1-e^{-y}\le \min\{1,y\}$, we find $(1-K^{s-1})/(1-s)\le\min\{(1-s)^{-1},\log K\}$. This proves \eqref{eq:asym-master}. \end{proof} The next lemma bounds the contribution of side variables outside $I_*$. It is the only place where the boundary density bound and the tail bound are used. Since such contributions occur in every block below, we state it for all $x\in\R$. \begin{lemma}\label{lem:asym-irregular} There exist $C_N$ and $p_1$, depending only on $\mathfrak c$, such that for $p\ge p_1$ and every $x\in\R$, \begin{equation}\label{eq:asym-irregular} \Bigl(1+\Bigl|\frac xK\Bigr|^s\Bigr)\,\E\bigl[(1-\chi(B))\,M_K(D_x)\bigr] \le C_N\,\ell\,e^{-c_0\sqrt p}. \end{equation} This uses \ref{it:asym-A1}, \ref{it:asym-A2} and \ref{it:asym-A3}. \end{lemma} \begin{proof} Recall that $D_x=B-\sigma mT_s$ with $m=|x|/K$. We consider the cases $m<1$ and $m\ge1$ separately. In both cases we use \eqref{eq:asym-irregular-mass}. \proofstep{1}{The case $m<1$.} We claim that \begin{equation}\label{eq:asym-irr-small-m} \E\bigl[(1-\chi(B))M_K(D_x)\bigr]\le2\ell p^{A_0}e^{-c_0p}+8\ell\rho_0^{-1}e^{-c_0\sqrt p}. \end{equation} Since $1+m^s\le2$, this implies \eqref{eq:asym-irregular} for $m<1$ after enlarging $p_1$ so that $p^{A_0}e^{-c_0p}\le e^{-c_0\sqrt p}$. We first bound the contribution of $|B|\le\rho_0$, dropping the factor $1-\chi(B)$. Fix $t\ge0$. By \ref{it:asym-A2} and \eqref{eq:asym-master} applied to the interval $[-\rho_0,\rho_0]-\sigma mt$ of length $2\rho_0\le1$, \begin{equation*} \int_{[-\rho_0,\rho_0]}M_K(b-\sigma mt)\,\nu(\dd b) \le p^{A_0}e^{-c_0p}\int_{-\rho_0}^{\rho_0}M_K(b-\sigma mt)\dd b \le2\ell p^{A_0}e^{-c_0p}. \end{equation*} Since $T_s$ is independent of $B$, we may integrate this bound over the law of $T_s$ (Tonelli's theorem), and we find $\E[M_K(D_x);|B|\le\rho_0]\le2\ell p^{A_0}e^{-c_0p}$. Next fix $b$ with $|b|>\rho_0$. We show that $\E[M_K(b-\sigma mT_s)]\le8\ell/\rho_0$. If $m=0$ or $\sigma b<0$, then $|b-\sigma mT_s|\ge|b|>\rho_0$, so $M_K(b-\sigma mT_s)\le\rho_0^{-s}\le \rho_0^{-1}$ by \eqref{eq:asym-v-comparison}. Suppose $m>0$ and $\sigma b>0$, so that $|b-\sigma mT_s|=\bigl||b|-mT_s\bigr|$. On the event $\bigl||b|-mT_s\bigr|>\rho_0/2$, we have $M_K\le(\rho_0/2)^{-s}\le2\rho_0^{-1}$. On its complement $T_s\ge(|b|-\rho_0/2)/m\ge\rho_0/(2m)$. Changing variables from $t$ to $d=|b|-mt$, and using \eqref{eq:asym-omega-bounds} and \eqref{eq:asym-master}, we obtain \begin{align*} \E\bigl[M_K(b-\sigma mT_s);\,\bigl||b|-mT_s\bigr|\le\rho_0/2\bigr] &=\frac1m\int_{|d|\le\rho_0/2}\omega_s\Bigl(\frac{|b|-d}{m}\Bigr)M_K(d)\dd d\\ &\le\frac1m\,2e^{-\rho_0/(4m)}\cdot2\ell\le\frac{16\ell}{e\rho_0}. \end{align*} In the last inequality we used that $m^{-1}e^{-\rho_0/(4m)}\le4/(e\rho_0)$ for all $m>0$. Adding the two parts, we have $\E[M_K(b-\sigma mT_s)]\le(2+16/e)\ell\rho_0^{-1}\le8\ell\rho_0^{-1}$, since $\ell\ge1$. Integrating this bound against $(1-\chi(b))\one_{\{|b|>\rho_0\}}\nu(\dd b)$, whose total mass is at most $e^{-c_0\sqrt p}$ by \eqref{eq:asym-irregular-mass}, and adding the contribution of $|B|\le\rho_0$, we obtain \eqref{eq:asym-irr-small-m}. \proofstep{2}{A conditional bound for $m\ge1$.} Let $m\ge1$, fix $b\in\R$ and write $\tau=|b|$. We claim that \begin{equation}\label{eq:asym-irr-cond-large-m} (1+m^s)\,\E\bigl[M_K(b-\sigma mT_s)\bigr] \le8\ell(1+\tau)^{1/2}+12+12\,m^s\min\bigl\{1,(\tau/m)^{1+s/2}\bigr\}. \end{equation} We write $d=b-\sigma mT_s$ and split according to the events $\{|d|\le1\}$, $\{|d|>1,\,T_s\ge2\tau/m\}$ and $\{|d|>1,\,T_s<2\tau/m\}$. On $\{|d|\le1\}$, we change variables from $t$ to $d=b-\sigma mt$, which has Jacobian $1/m$; the corresponding value of $t$ is $\sigma(b-d)/m\le(\tau+1)/m$. Using the first bound in \eqref{eq:asym-omega-bounds} and \eqref{eq:asym-master}, we find \begin{align*} \E\bigl[M_K(d);|d|\le1\bigr]&=\frac1m\int_{|d|\le1}\omega_s\Bigl(\frac{\sigma(b-d)}m\Bigr)M_K(d)\dd d\\ &\le\frac2m\Bigl(\frac{\tau+1}{m}\Bigr)^{s/2}\cdot2\ell \le4\ell\,m^{-1-s/2}(1+\tau)^{1/2}. \end{align*} On $\{T_s\ge2\tau/m\}$, we have $|d|\ge mT_s-\tau\ge mT_s/2$, and \eqref{eq:asym-v-comparison} implies that $M_K(d)\le(mT_s/2)^{-s}$. By \Cref{lem:asym-gamma}(iii), the contribution of this event is at most $2m^{-s}\E[T_s^{-s}]\le6m^{-s}$. On $\{|d|>1,\,T_s<2\tau/m\}$, we have $M_K(d)\le1$, and by \Cref{lem:asym-gamma}(ii) the contribution is at most $\min\{1,2(2\tau/m)^{1+s/2}\}\le\min\{1,6(\tau/m)^{1+s/2}\}$. Multiplying the sum of the three contributions by $1+m^s\le2m^s$ and using $m^{s/2-1}\le1$, we obtain \eqref{eq:asym-irr-cond-large-m}. \proofstep{3}{Two tail integrals.} We record two consequences of \ref{it:asym-A3}. For $q\in(0,2]$ and $M\ge3$, we have by the layer cake formula that \begin{equation}\label{eq:asym-layer-cake} \int_{\{3<|b|\le M\}}|b|^q\,\nu(\dd b)\le3^q\nu(\{|b|>3\})+\int_3^MqL^{q-1}\nu(\{|b|>L\})\dd L . \end{equation} We enlarge $p_1$ so that $c_0p\ge1$, which implies $e^{-c_0pL}\le e^{-c_0p}e^{1-L}$ for $L\ge1$. Inserting \ref{it:asym-A3} and this bound into \eqref{eq:asym-layer-cake}, and using $e^{-c_0p}\le p^{A_0}e^{-c_0p}$, we obtain, for $q=1/2$ and $M=\infty$, and for $q=1+s/2$ and $M=m\ge3$, \begin{equation}\label{eq:asym-tail-moments} \int_{\{|b|>3\}}|b|^{1/2}\nu(\dd b)\le Cp^{A_0}e^{-c_0p}, \qquad \int_{\{3<|b|\le m\}}|b|^{1+s/2}\nu(\dd b)\le Cp^{A_0}e^{-c_0p}(1+m^{s/2}). \end{equation} For the second bound we used $\int_3^mL^{s/2-1}\dd L\le(2/s)m^{s/2}$. \proofstep{4}{The case $m\ge1$.} We integrate \eqref{eq:asym-irr-cond-large-m} against $\nu_{\mathrm{irr}}(\dd b)=(1-\chi(b))\nu(\dd b)$. By \eqref{eq:asym-irregular-mass} and the first bound in \eqref{eq:asym-tail-moments}, \begin{align*} \int\bigl(8\ell(1+|b|)^{1/2}+12\bigr)\nu_{\mathrm{irr}}(\dd b) &\le(16\ell+12)e^{-c_0\sqrt p}+16\ell\int_{\{|b|>3\}}|b|^{1/2}\nu(\dd b)\\ &\le C\ell\bigl(e^{-c_0\sqrt p}+p^{A_0}e^{-c_0p}\bigr). \end{align*} For the last term in \eqref{eq:asym-irr-cond-large-m}, we split the range of $b$ into $\{|b|\le 3\}$, $\{3<|b|\le m\}$ and $\{|b|>\max\{3,m\}\}$. On the first set, $\min\{1,(|b|/m)^{1+s/2}\}\le (3/m)^{1+s/2}$, and \eqref{eq:asym-irregular-mass} bounds its contribution by $12\cdot3^{3/2} m^{s/2-1}e^{-c_0\sqrt p}\le63e^{-c_0\sqrt p}$. On the second set, which is empty unless $m>3$, the second bound in \eqref{eq:asym-tail-moments} bounds the contribution by \begin{equation*} 12\,m^{s}m^{-1-s/2}\,Cp^{A_0}e^{-c_0p}(1+m^{s/2})\le24C\,p^{A_0}e^{-c_0p}, \end{equation*} since $m^{s-1}\le1$. On the third set, \ref{it:asym-A3} bounds the contribution by \begin{equation*} 12\,m^s\Bigl(\frac{K_0p^{A_0}e^{-c_0p}}{\max\{3,m\}}+e^{-c_0p\max\{3,m\}}\Bigr) \le12K_0p^{A_0}e^{-c_0p}+36e^{-3c_0p}\le Cp^{A_0}e^{-c_0p}. \end{equation*} Here we used $m^s\le\max\{3,m\}$, and that $y\mapsto ye^{-c_0py}$ is decreasing on $[3,\infty)$ because $c_0p\ge1$. Adding these bounds, and enlarging $p_1$ so that $p^{A_0}e^{-c_0p}\le e^{-c_0\sqrt p}$, we conclude that \eqref{eq:asym-irregular} holds for $m\ge1$. This completes the proof. \end{proof} We now introduce the profile that defines the similarity on the tail region. For $u>0$, set \begin{equation}\label{eq:asym-profile} \varrho_s(u)=\E\bigl[\chi(B)\,\omega_s(uB)\bigr], \qquad \phi_s(u)=u^{1-s}\varrho_s(u). \end{equation} In the tail region, we will see that the function $x\mapsto K^s\phi_s(K/x)$ is an approximate eigenfunction of $\mathsf T^*$ with eigenvalue $K^{s-1}\int_0^\infty\phi_s$. \begin{lemma}\label{lem:asym-profile} There exists $p_1$, depending only on $\mathfrak c$, such that for $p\ge p_1$ the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item For every $u>0$, \begin{equation}\label{eq:asym-profile-bounds} \tfrac13\min\{1,u\}^{3/4}e^{-3u/2}\le\phi_s(u)\le(1+u^s)\phi_s(u)\le5 . \end{equation} \item We have \begin{equation}\label{eq:asym-profile-mass} \int_0^\infty\phi_s(u)\dd u\le\min\Bigl\{\bigl(1+C_\Gamma(1-s)\bigr)\bigl(1+4p^{-1/4}\bigr),\,2\Bigr\} . \end{equation} \end{enumerate} This uses \ref{it:asym-A1}. \end{lemma} \begin{proof} For the lower bound in (i), we restrict the expectation defining $\varrho_s$ to the event $\{|B-1|\le p^{-1/4}\}$, which is contained in $\{\chi(B)=1\}$ and has probability at least $1/2$ by \ref{it:asym-A1}, after enlarging $p_1$ so that $e^{-c_0\sqrt p}\le1/2$. For $b\in[1/2,3/2]$ we have $\omega_s(ub)\ge(ub)^{s/2}e^{-ub}\ge(u/2)^{s/2}e^{-3u/2}$, because $\mu_s\le1$. If $u\le2$ then $(u/2)^{s/2}\ge(u/2)^{1/2}$, and if $u\ge2$ then $(u/2)^{s/2}\ge1$. This gives $\varrho_s(u)\ge2^{-3/2}\min\{1,u\}^{1/2}e^{-3u/2}$. Since $u^{1-s}\ge\min\{1,u\}^{1/4}$ for $s\ge3/4$, the lower bound in \eqref{eq:asym-profile-bounds} follows. For the upper bound, \eqref{eq:asym-omega-bounds} and $\chi=\one_{I_*}$ give $\varrho_s(u)\le\sup_{b\in I_*}2(ub)^{s/2}e^{-ub}\le2(1.01u)^{s/2}e^{-0.99u}$. If $u\le1$, then $(1+u^s)u^{1-s}\varrho_s(u)\le2\cdot2\cdot1.01\le5$. If $u\ge1$, then $(1+u^s)u^{1-s}\varrho_s(u)\le4.04\,u^{3/2}e^{-0.99u}\le5$, since $u^{3/2}e^{-0.99u}\le1$ for all $u\ge0$. For (ii), Tonelli's theorem and \eqref{eq:asym-omega-mass} give \begin{equation*} \int_0^\infty\phi_s(u)\dd u=\frac{\Gamma(2-s/2)}{\Gamma(1+s/2)}\,\E\bigl[\chi(B)B^{s-2}\bigr]. \end{equation*} On $\{|B-1|\le p^{-1/4}\}$, we have $B^{s-2}\le(1-p^{-1/4})^{-2}\le1+3p^{-1/4}$, because $p^{-1/4}\le1/100$. On $\{\chi(B)=1,|B-1|>p^{-1/4}\}$, we have $B^{s-2}\le0.99^{-2}\le1.03$, and this event has probability at most $e^{-c_0\sqrt p}$ by \ref{it:asym-A1}. Then $\E[\chi(B)B^{s-2}]\le1+3p^{-1/4}+1.03e^{-c_0\sqrt p}\le1+4p^{-1/4}$ after enlarging $p_1$. Together with \Cref{lem:asym-gamma}(v), this proves the first bound in \eqref{eq:asym-profile-mass}. For the second, we have $\Gamma(2-s/2)\le1$ and $\Gamma(1+s/2)\ge0.88$ by \Cref{lem:asym-gamma}(i), and $\E[\chi(B)B^{s-2}]\le0.99^{s-2}\le1.03$ because $B\ge0.99$ on $\{\chi(B)=1\}$. Then the left side of \eqref{eq:asym-profile-mass} is at most $1.03/0.88\le2$. \end{proof} \section{The lower bound}\label{sec:asym-lower} The lower bound uses only the concentration of $B$ near one: a function supported near one is a subsolution for $\mathsf T^*$. \begin{lemma}\label{lem:asym-lower} There exists $p_1$, depending only on $\mathfrak c$, such that the following holds for $p\ge p_1$. Let $\pi$ be a real solution at $E=\xi\sqrt p>2$, let $\nu$ be the law of $g(E)D$, and suppose that $(\nu,K,s)$ is $\mathfrak c$-admissible at $p$, where $K=p/g(E)^2$. Then \begin{equation}\label{eq:asym-lower} r_{\pi,s}\ge1-2p^{-1/4}. \end{equation} This uses \ref{it:asym-A1} and \ref{it:asym-A4}. \end{lemma} \begin{proof} Let $S=[1/2,2]$ and $f(x)=|x|^s\one_S(x)$. Fix $x\in S$, and consider the event \begin{equation*} G=\bigl\{|B-1|\le p^{-1/4}\bigr\}\cap\bigl\{T_s\le K^{1/2}\bigr\}. \end{equation*} On $G$, we have $0\le T_sx/K\le2K^{-1/2}$ and $D_x\in[1-p^{-1/4}-2K^{-1/2},1+p^{-1/4}]$. By \ref{it:asym-A4}, we have $2K^{-1/2}\le2c_*^{-1/2}p^{-1}\le p^{-1/4}$ after enlarging $p_1$. On $G$, this gives $D_x\in[1-2p^{-1/4},1+p^{-1/4}]\subset[1/2,2]$. In particular, $Y_x\in S$ and $|Y_x|^s\ge (1+p^{-1/4})^{-s}\ge1-p^{-1/4}$ on $G$. By \ref{it:asym-A1} and \Cref{lem:asym-gamma}(ii), \begin{equation*} \P(G^c)\le e^{-c_0\sqrt p}+4e^{-\sqrt K/2}\le p^{-1/4}, \end{equation*} after enlarging $p_1$ and using \ref{it:asym-A4}. We conclude that for $x\in S$, \begin{equation*} (\mathsf T^*f)(x)=|x|^s\,\E\bigl[|Y_x|^s\one_S(Y_x)\bigr]\ge|x|^s(1-p^{-1/4})\P(G) \ge(1-2p^{-1/4})f(x). \end{equation*} For $x\notin S$, we have $f(x)=0\le(\mathsf T^*f)(x)$. Therefore $\mathsf T^*f\ge(1-2p^{-1/4})f$ on $\R$. Since $\mathsf T^*$ is positive and $e_s\ge f$, we have by induction that $(\mathsf T^*)^Le_s\ge(\mathsf T^*)^Lf\ge(1-2p^{-1/4})^Lf$ for every $L\ge1$. Integrating against $\nu^x$, we obtain \begin{equation*} \int(\mathsf T^*)^Le_s\dd\nu^x\ge(1-2p^{-1/4})^L\int f\dd\nu^x, \qquad \int f\dd\nu^x=\E\bigl[|B|^{-s};B\in S\bigr]\ge\tfrac12\,\nu(S)\ge\tfrac14, \end{equation*} where we used that $X=B^{-1}\in S$ if and only if $B\in S$, and that $\nu(S)\ge1-e^{-c_0\sqrt p}\ge 1/2$ by \ref{it:asym-A1}. Taking $L$-th roots, letting $L\to\infty$ and using \eqref{eq:asym-r-definition}, we conclude that \eqref{eq:asym-lower} holds. \end{proof} \section{Six regions and the block matrix}\label{sec:asym-charts} For the upper bound, we conjugate $\mathsf T^*$ by the capped weight $v$ of \eqref{eq:asym-capped}. For a measurable function $F\colon\R\to[0,\infty]$, or a bounded measurable $F$, define \begin{equation}\label{eq:asym-L} (\mathsf LF)(x)=\Bigl(1+\Bigl|\frac xK\Bigr|^s\Bigr)\,\E\bigl[M_K(D_x)\,F(Y_x)\bigr], \qquad x\in\R . \end{equation} Since $v(Y_x)=M_K(D_x)$ and $v(x)(1+|x/K|^s)=|x|^s$, we have \begin{equation}\label{eq:asym-L-conjugation} \mathsf T^*(vF)=v\,\mathsf LF \end{equation} for every measurable $F\ge0$; at $x=0$ both sides vanish. The point of the cap is that $M_K\le K^s$: a single step of $\mathsf L$ is finite even when $D_x$ is close to zero. The price is the logarithmic factor $\ell$ in \eqref{eq:asym-master}. We now cover the real line by six regions. For a state $x>0$ we use the reciprocal coordinate $u=K/x$, and for $x<0$ we write $x=-K/u$. The two scales of the construction are \begin{equation}\label{eq:asym-scales} R=\log p, \qquad u_*=p^{-1}. \end{equation} We always take $p_1$ so large that $R\ge2$ and $R^2\le K$ for $p\ge p_1$; this is possible by \ref{it:asym-A4}. \begin{definition}\label{def:asym-charts} Let $\mathscr C=\{\mathrm c,\mathrm t,0,+,+\infty,-\}$. The \emph{regions} are the sets \begin{equation*} \begin{gathered} S_{\mathrm c}=\bigl[\tfrac12,2\bigr],\qquad S_0=[-R,R]\setminus S_{\mathrm c},\qquad S_-=(-\infty,-R),\\ S_+=\bigl(R,K/R\bigr),\qquad S_{\mathrm t}=\bigl[K/R,\,K/u_*\bigr],\qquad S_{+\infty}=\bigl(K/u_*,\infty\bigr). \end{gathered} \end{equation*} We call $S_{\mathrm c}$ the core region, $S_{\mathrm t}$ the tail region, $S_0$ the bounded region, $S_+$ the intermediate region, $S_{+\infty}$ the supertail region, and $S_-$ the negative region. The labels $\mathrm c$ and $\mathrm t$ are \emph{recurrent}, and the other four labels are \emph{transient}. The similarity functions are $r_i\equiv1$ on $S_i$ for $i\ne\mathrm t$, and \begin{equation}\label{eq:asym-rt} r_{\mathrm t}(x)=\bigl(1+u^s\bigr)\phi_s(u),\qquad u=K/x,\quad x\in S_{\mathrm t}. \end{equation} \end{definition} Since $2\le R$ and $R0$ with $w>0$, \begin{equation}\label{eq:asym-tail-weight} v(y)\,(1+w^s)\phi_s(w)=\frac{K^s}{1+w^s}\,(1+w^s)\phi_s(w)=K^s\phi_s(w). \end{equation} In the tail region the weight $v\,r_{\mathrm t}$ is therefore the profile $x\mapsto K^s\phi_s(K/x)$. \begin{definition}\label{def:asym-blocks} For $i,j\in\mathscr C$ and $x\in S_i$, set \begin{equation}\label{eq:asym-block-function} a_{ij}(x)=r_i(x)^{-1}\bigl(\mathsf L(r_j\one_{S_j})\bigr)(x), \qquad a_{ij}=\sup_{x\in S_i}a_{ij}(x)\in[0,\infty]. \end{equation} We call $a_{ij}$ the \emph{block norm} from $i$ to $j$, and $\mathsf A=(a_{ij})_{i,j\in\mathscr C}$ the block matrix; the current region indexes the row and the future region indexes the column. Since $\mathsf L$ is positive, $a_{ij}$ is the norm, on bounded Borel functions with the supremum norm, of the operator $q\mapsto r_i^{-1}\mathsf L(r_j\one_{S_j}q)|_{S_i}$. We split $a_{ij}(x)=a^{\mathrm{reg}}_{ij}(x)+a^{\mathrm{irr}}_{ij}(x)$, where the regular part $a^{\mathrm{reg}}_{ij}(x)$ is defined as $a_{ij}(x)$ with the factor $\chi(B)$ inserted in the expectation in \eqref{eq:asym-L}, and the irregular part $a^{\mathrm{irr}}_{ij}(x)$ with the factor $1-\chi(B)$ inserted. \end{definition} A vector $d$ with positive entries and $\mathsf Ad\le\lambda d$ entrywise gives a supersolution for $\mathsf T^*$, and a supersolution bounds the growth rate. \begin{lemma}\label{lem:asym-supersolution} Suppose that $a_{ij}<\infty$ for all $i,j\in\mathscr C$, and that there exist $\lambda>0$ and $d_i>0$, $i\in\mathscr C$, with \begin{equation}\label{eq:asym-row-condition} \sum_{j\in\mathscr C}a_{ij}d_j\le\lambda d_i, \qquad i\in\mathscr C . \end{equation} Then the function $h=v\sum_{i\in\mathscr C}d_ir_i\one_{S_i}$ satisfies $\mathsf T^*h\le\lambda h$ on $\R$. If moreover $\nu$ is the law of $g(E)D$ for a real solution $\pi$ at $E=\xi\sqrt p>2$ and $K=p/g(E)^2$, then $r_{\pi,s}\le\lambda$. \end{lemma} \begin{proof} Let $w=\sum_jd_jr_j\one_{S_j}$, so that $h=vw$. For $x\in S_i$, by the definition of $a_{ij}$ we have \begin{equation*} (\mathsf Lw)(x)=\sum_jd_j\bigl(\mathsf L(r_j\one_{S_j})\bigr)(x)\le r_i(x)\sum_ja_{ij}d_j \le\lambda d_ir_i(x)=\lambda w(x). \end{equation*} By \eqref{eq:asym-L-conjugation}, $\mathsf T^*h=v\,\mathsf Lw\le\lambda vw=\lambda h$. For the second claim, we use the fractional cancellation estimate \eqref{eq:press-fc} with $q=s$, which holds for the law $\nu$ of $g(E)D$ by \Cref{prop:rde-properties}; the variable $T_q$ in \eqref{eq:press-fc} with $q=s$ is our $T_s$. Then there exists a constant $C_\pi<\infty$ with $\E|B-\sigma mT_s|^{-s}\le C_\pi\min\{1,m^{-s}\}$ for all $m\ge0$ and $\sigma\in\{-1,1\}$. With $m=|x|/K$ and \eqref{eq:asym-v-comparison}, this gives \begin{equation*} (\mathsf T^*e_s)(x)=|x|^s\,\E|D_x|^{-s}\le C_\pi\min\{|x|^s,K^s\}\le2C_\pi v(x) \le\frac{2C_\pi}{w_{\min}}\,h(x), \end{equation*} where $w_{\min}=\min_id_i\inf_{S_i}r_i>0$ by \eqref{eq:asym-rt-bounds}. By positivity and induction, $(\mathsf T^*)^Le_s\le2C_\pi w_{\min}^{-1}\lambda^{L-1}h$ for $L\ge1$. Further, $h\le w_{\max}v\le w_{\max}e_s$ with $w_{\max}=\max_id_i\sup_{S_i}r_i$, and $\int e_s\dd\nu^x= \E|B|^{-s}<\infty$ by \Cref{lem:press-gauge}(i). Then \begin{equation*} \int(\mathsf T^*)^Le_s\dd\nu^x\le\frac{2C_\pi w_{\max}}{w_{\min}}\,\E|B|^{-s}\,\lambda^{L-1}, \end{equation*} and taking $L$-th roots in \eqref{eq:asym-r-definition}, we conclude that $r_{\pi,s}\le\lambda$. \end{proof} \section{Block estimates}\label{sec:asym-blocks} In this section we bound every block norm. We first collect the result; the proofs, one row of the block matrix at a time, follow. Recall that $\ell=\ell_{K,s}$ and $R=\log p$, $u_*=p^{-1}$. Set \begin{equation}\label{eq:asym-small-quantities} \varepsilon_K=40Ke^{-\sqrt K/2}, \qquad E_{\mathrm{irr}}=15\,C_N\,\ell\,p^{9/4}e^{-c_0\sqrt p}, \end{equation} where $C_N$ is the constant of \Cref{lem:asym-irregular}. \begin{proposition}\label{prop:asym-blocks} There exist a numerical constant $C_+$ and a threshold $p_1$ depending only on $\mathfrak c$ such that, for $p\ge p_1$, the block norms satisfy $a_{ij}\le\bar a_{ij}$ for all $i,j\in\mathscr C$, where the matrix $\bar{\mathsf A}=(\bar a_{ij})$ is given in \Cref{tab:asym-blocks}. \end{proposition} \begin{table}[ht] \centering \renewcommand{\arraystretch}{1.25} \begin{tabular}{@{}lll@{}} \toprule Current region $i$ & Future region $j$ & Bound $\bar a_{ij}$ \\ \midrule $\mathrm c$ & $\mathrm c$ & $1+4p^{-1/4}$ \\ $\mathrm c$ or $0$ & $j\ne\mathrm c$ & $\varepsilon_K+E_{\mathrm{irr}}$ \\ $0$ or $+$ & $\mathrm c$ & $4$ \\ $+$ & $j\ne\mathrm c$ & $C_+\,\ell\,p^{-1/5}+E_{\mathrm{irr}}$ \\ $+\infty$ & $\mathrm c$ & $8p^{-1/2}+E_{\mathrm{irr}}$ \\ $+\infty$ & $0$ & $14+E_{\mathrm{irr}}$ \\ $+\infty$ & $\mathrm t,+,+\infty,-$ & $80\,\ell\,p^{-1/2}+E_{\mathrm{irr}}$ \\ $-$ & $\mathrm c$ or $0$ & $6+E_{\mathrm{irr}}$ \\ $-$ & $\mathrm t,+,+\infty,-$ & $E_{\mathrm{irr}}$ \\ $\mathrm t$ & $\mathrm t$ & $K^{s-1}e^{R^2/K}\bigl(1+C_\Gamma(1-s)\bigr)\bigl(1+4p^{-1/4}\bigr)+E_{\mathrm{irr}}$ \\ $\mathrm t$ & $\mathrm c$ & $3p^{1.01}+E_{\mathrm{irr}}$ \\ $\mathrm t$ & $0$ & $2p^{1/2}\log p+4p+E_{\mathrm{irr}}$ \\ $\mathrm t$ & $+,+\infty,-$ & $12\,\ell+E_{\mathrm{irr}}$ \\ \bottomrule \end{tabular} \caption{Bounds on the block norms $a_{ij}$ for $\mathfrak c$-admissible data and $p\ge p_1$. The quantities $\varepsilon_K$ and $E_{\mathrm{irr}}$ are defined in \eqref{eq:asym-small-quantities}. The two recurrent diagonal entries are close to one (the tail entry for $s$ close to one); the only transient blocks of order one are $+\infty\to0$ and $-\to0$, and both end in the bounded region, whose exits other than to the core are bounded by $\varepsilon_K+E_{\mathrm{irr}}$.} \label{tab:asym-blocks} \end{table} Before the proof, we explain the structure of \Cref{tab:asym-blocks}. The quantities $\varepsilon_K$ and $E_{\mathrm{irr}}$ decay faster than every power of $p$ by \ref{it:asym-A4} and \ref{it:asym-A5}, while the factors $\ell\le C_*\log p$ and the large entries $3p^{1.01}$ and $4p$ are at most polynomial. From a bounded state, the output is in the core unless $T_s$ is larger than $\sqrt K$, which explains the entries $\varepsilon_K$. From an intermediate state, an output outside the core requires $T_s$ of order $u\ge R$, which costs $e^{-cR}=p^{-c}$. From a supertail state $x=K/u$ with $u\sqrt K\}$. Using $M_K\le K^s\le K$, $r_j\le5$ and \Cref{lem:asym-gamma}(ii), for $j\ne\mathrm c$ we obtain \begin{equation*} a^{\mathrm{reg}}_{ij}(x)\le(1+m^s)\cdot5K\cdot\P\bigl(T_s>\sqrt K\bigr)\le2\cdot5K\cdot4e^{-\sqrt K/2} =\varepsilon_K .\qedhere \end{equation*} \end{proof} In the remaining rows we use the following change of variables. Let $u>0$, $b>0$, and let $F\ge0$ be measurable. Substituting $d=b-t/u$, so that $t=u(b-d)$ and $\dd t=u\dd d$, we have \begin{equation}\label{eq:asym-change-variables} \E\Bigl[F\Bigl(b-\frac{T_s}{u}\Bigr)\Bigr]=\int_0^\infty\omega_s(t)F\Bigl(b-\frac tu\Bigr)\dd t =u\int_{-\infty}^b\omega_s\bigl(u(b-d)\bigr)F(d)\dd d . \end{equation} For $x=K/u>0$ and $B=b$, the variable $b-T_s/u$ is $D_x$. \begin{lemma}\label{lem:asym-row-intermediate} There exist a numerical constant $C_+$ and $p_1$ such that for $p\ge p_1$ the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item $a_{+\mathrm c}\le4$. \item For $j\ne\mathrm c$ and $x\in S_+$, we have $a^{\mathrm{reg}}_{+j}(x)\le C_+\ell e^{-R/5}$. \end{enumerate} \end{lemma} \begin{proof} Let $x\in S_+$ and $u=K/x\in(R,K/R)$, so that $m=u^{-1}\le1$, $1+m^s\le2$ and $r_+(x)=1$. Part (i) follows as in the proof of \Cref{lem:asym-row-bounded}(ii). For (ii), fix $b\in I_*$. Since $D_x=b-T_s/u\le b\le101/100<2$, the event $Y_x\notin S_{\mathrm c}$ forces $D_x<1/2$, hence $T_s>u(b-1/2)\ge0.49u$. Using $r_j\le5$, we have \begin{equation*} a^{\mathrm{reg}}_{+j}(x)\le10\sup_{b\in I_*}\E\bigl[M_K(b-T_s/u);\,T_s>0.49u\bigr]. \end{equation*} On the event $\{|b-T_s/u|\le1\}$, we use \eqref{eq:asym-change-variables}, the bound $\omega_s(t)\le2e^{-t/2}$ for $t>0.49u$, and \eqref{eq:asym-master}: \begin{equation*} \E\bigl[M_K(b-T_s/u);\,T_s>0.49u,\,|b-T_s/u|\le1\bigr]\le u\cdot2e^{-0.245u}\int_{-1}^1M_K(d)\dd d \le4\ell\,ue^{-0.245u}. \end{equation*} On the complement, $M_K\le1$, and \Cref{lem:asym-gamma}(ii) implies that this part is at most $\P(T_s>0.49u)\le4e^{-0.245u}$. Adding, we find $a^{\mathrm{reg}}_{+j}(x)\le40\ell(1+u)e^{-0.245u}\le C_+\ell e^{-u/5}\le C_+\ell e^{-R/5}$, with $C_+=40\sup_{u\ge0}(1+u)e^{-0.045u}$. \end{proof} \begin{lemma}\label{lem:asym-row-supertail} For $p\ge p_1$ and $x\in S_{+\infty}$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item $a^{\mathrm{reg}}_{+\infty,\mathrm c}(x)\le8u_*^{1/2}$. \item $a^{\mathrm{reg}}_{+\infty,0}(x)\le14$. \item For $j\in\{\mathrm t,+,+\infty,-\}$, we have $a^{\mathrm{reg}}_{+\infty,j}(x)\le80\ell u_*^{1/2}$. \end{enumerate} \end{lemma} \begin{proof} Let $x\in S_{+\infty}$ and $u=K/x\in(0,u_*)$, so that $m=u^{-1}$, $1+m^s\le2u^{-s}$ and $r_{+\infty}(x)=1$. Fix $b\in I_*$; we bound the conditional expectation given $B=b$, uniformly in $b$. For (i), $Y_x\in S_{\mathrm c}$ requires $D_x\ge1/2$, hence $T_s\le u(b-1/2)\le u$. Since $M_K(D_x)\le2$ on $\{Y_x\in S_{\mathrm c}\}$, \Cref{lem:asym-gamma}(ii) gives $a^{\mathrm{reg}}_{+\infty,\mathrm c}(x)\le2u^{-s}\cdot2\cdot2u^{1+s/2}=8u^{1-s/2}\le8u_*^{1/2}$, since $u<1$ and $1-s/2\ge1/2$. For (ii), $Y_x\in S_0$ implies $|Y_x|\le R$, that is $|D_x|\ge1/R$, equivalently $|ub-T_s|\ge u/R$. By \eqref{eq:asym-v-comparison}, \begin{equation*} (1+m^s)M_K(D_x)\le(1+u^{-s})|b-T_s/u|^{-s}=(1+u^s)|ub-T_s|^{-s}\le2|ub-T_s|^{-s}. \end{equation*} On $\{T_s\ge2ub\}$, we have $|ub-T_s|\ge T_s/2$, and \Cref{lem:asym-gamma}(iii) bounds the contribution by $2\cdot2\cdot3=12$. On $\{T_s<2ub\}$, we have $T_s<3u$ and $\omega_s(T_s)\le 2(3u)^{s/2}$, and the variable $z=|ub-T_s|$ ranges over $[u/R,ub]$, each value being attained at most twice. Then \begin{align*} \E\bigl[2|ub-T_s|^{-s};\,T_s<2ub,\,|ub-T_s|\ge u/R\bigr] &\le2\cdot2(3u)^{s/2}\cdot2\int_{u/R}^{ub}z^{-s}\dd z\\ &\le24\,u^{1-s/2}\log(2R), \end{align*} where we used $\int_{u/R}^{ub}z^{-s}\dd z\le(ub)^{1-s}\log(bR)$ and $b\le2$. Since $uR$, that is $|D_x|<1/R$. Using $r_j\le5$, \eqref{eq:asym-change-variables}, the bound $\omega_s(u(b-d))\le 2(2u)^{s/2}$ for $|d|<1/R$ (since $u(b-d)\le2u$), and \eqref{eq:asym-master}, we obtain \begin{align*} a^{\mathrm{reg}}_{+\infty,j}(x)&\le2u^{-s}\cdot5\cdot u\int_{|d|<1/R}2(2u)^{s/2}M_K(d)\dd d\\ &\le2u^{-s}\cdot5\cdot u\cdot4u^{s/2}\cdot2\ell=80\ell u^{1-s/2}\le80\ell u_*^{1/2}.\qedhere \end{align*} \end{proof} \begin{lemma}\label{lem:asym-row-negative} For $p\ge p_1$ and $x\in S_-$, we have $a^{\mathrm{reg}}_{-,\mathrm c}(x)+a^{\mathrm{reg}}_{-,0}(x) \le6$, and $a^{\mathrm{reg}}_{-,j}(x)=0$ for $j\in\{\mathrm t,+,+\infty,-\}$. \end{lemma} \begin{proof} Let $x=-K/u\in S_-$, with $u\in(0,K/R)$, so that $m=u^{-1}$ and $r_-(x)=1$. If $\chi(B)=1$, then $D_x=B+T_s/u\ge0.99$ and $Y_x\in(0,1/0.99]\subset S_0\cup S_{\mathrm c}$. This proves the second claim. For the first, \eqref{eq:asym-v-comparison} gives \begin{equation*} (1+m^s)M_K(D_x)\le(1+u^{-s})(B+T_s/u)^{-s}=(1+u^s)(uB+T_s)^{-s}. \end{equation*} If $u\le1$, this is at most $2T_s^{-s}$, whose expectation is at most $6$ by \Cref{lem:asym-gamma}(iii). If $u>1$, it is at most $2u^s(0.99u)^{-s}\le3$. \end{proof} For the tail row, we write $x=K/u$ with $u\in[u_*,R]$. Since $(1+u^{-s})/r_{\mathrm t}(x)= (1+u^{-s})/((1+u^s)u^{1-s}\varrho_s(u))=1/(u\varrho_s(u))$, the change of variables \eqref{eq:asym-change-variables} gives, for every measurable $F\ge0$, \begin{equation}\label{eq:asym-tail-row-formula} \frac{1+m^s}{r_{\mathrm t}(x)}\,\E\bigl[\chi(B)M_K(D_x)F(Y_x)\bigr] =\frac1{\varrho_s(u)}\,\E\Bigl[\chi(B)\int_{-\infty}^B\omega_s\bigl(u(B-d)\bigr)M_K(d)F(d^{-1})\dd d\Bigr]. \end{equation} By \eqref{eq:asym-omega-shift}, for $b\in I_*$ and $dR$, that is $|d|<1/R$, and $r_j=1$ for these regions. For $|d|<1/R$ we have $W_u(d)\le(1+2/R)^{s/2}e^{u/R}\le2e\le6$, since $u\le R$ and $R\ge2$. By \eqref{eq:asym-master}, the sum in (iv) is at most $6\int_{|d|<1/R}M_K(d)\dd d\le 12\ell$. \end{proof} Combining \Cref{lem:asym-row-tail}(i) with \Cref{lem:asym-profile}(ii), we find \begin{equation}\label{eq:asym-tail-diagonal} a^{\mathrm{reg}}_{\mathrm{tt}}\le K^{s-1}e^{R^2/K}\bigl(1+C_\Gamma(1-s)\bigr)\bigl(1+4p^{-1/4}\bigr). \end{equation} This is the diagonal entry of the tail region in \Cref{tab:asym-blocks}. The factor $K^{s-1}$ is at most one, and it tends to one as $s\uparrow1$ for fixed $K$; the profile mass tends to a number close to one. This is the second source of growth of $\mathsf T^*$ at $s=1$. \section{Balancing a nonnegative matrix}\label{sec:asym-balancing} The following lemma bounds the growth of a nonnegative matrix whose large entries are confined to a set of diagonal positions and whose other closed chains have small products. It is the standard feasibility criterion for a system of difference constraints: such a system is solvable if and only if no cycle has negative total weight, and a solution is given by shortest-path potentials. \begin{lemma}\label{lem:asym-balancing} Let $n\ge1$, let $\bar{\mathsf A}=(\bar a_{ij})_{i,j=1}^n$ have nonnegative entries, and let $\Delta\subset\{1,\dots,n\}$. Call $(i,j)$ an \emph{allowed edge} if $\bar a_{ij}>0$ and $(i,j)\ne (k,k)$ for every $k\in\Delta$. A \emph{simple cycle} of length $k\ge1$ is a sequence $(i_1,\dots,i_k)$ of distinct indices such that $(i_1,i_2),\dots,(i_{k-1},i_k),(i_k,i_1)$ are allowed edges, and its \emph{weight} is the product of $\bar a$ over these $k$ edges. Let $q$ be the maximum of $(\text{weight})^{1/k}$ over all simple cycles, with $q=0$ if there are none. Then for every $\eps>q$ there exist $d_1,\dots,d_n\in(0,1]$ such that $\bar a_{ij}d_j\le\eps d_i$ whenever $(i,j)\ne(k,k)$ for all $k\in\Delta$. In particular, \begin{equation}\label{eq:asym-balanced-rows} \sum_{j=1}^n\bar a_{ij}d_j\le\Bigl(\max_{k\in\Delta}\bar a_{kk}+n\eps\Bigr)d_i, \qquad i=1,\dots,n, \end{equation} where the maximum over the empty set is zero. \end{lemma} \begin{proof} Since $\eps>q\ge0$, the number $\log\eps$ is finite. Give each allowed edge $(i,j)$ the weight $w_{ij}=\log\eps-\log\bar a_{ij}$. A walk of length $k\ge0$ is a sequence $i_0,i_1,\dots,i_k$ such that each $(i_{l-1},i_l)$ is an allowed edge, and its weight is $\sum_{l=1}^kw_{i_{l-1}i_l}$; a walk of length zero has weight zero. A walk is closed if $k\ge1$ and $i_k=i_0$. \proofstep{1}{Closed walks have positive weight.} We argue by induction on the length $k$. If $i_0,\dots,i_{k-1}$ are distinct, the closed walk is a simple cycle of some weight $\Pi>0$, and its weight as a walk is $k\log\eps-\log\Pi=k(\log\eps- \log\Pi^{1/k})>0$, since $\Pi^{1/k}\le q<\eps$. Otherwise there exist $0\le a0$, and we have taken $L_0=L_1=1$. The choice of the two scales enters only through the factors $p^{-L_0/5}$ and $p^{-L_1/2}$ in $\theta_2$ and $\theta_3$, which must beat the logarithmic exit bound $13C_*\log p$ from the tail, and through polynomial factors $p^{O(L_0+L_1)}$ in $\bar E_{\mathrm{irr}}$ and in the large entries, which are absorbed by $\theta_1$. No order of choice between $L_0$ and $L_1$ is needed, because the exits from the tail into the transient regions other than the bounded region are bounded by $12\ell$ through the capped estimate \eqref{eq:asym-master}. \end{remark} \begin{proposition}\label{prop:asym-upper} There exist $p_1$, depending only on $\mathfrak c$, and $\delta_p=6(Q_p^{1/6}+p^{-1})$ such that the following holds for $p\ge p_1$. Let $\pi$ be a real solution at $E=\xi\sqrt p>2$, let $\nu$ be the law of $g(E)D$, let $K=p/g(E)^2$, and suppose that $(\nu,K,s)$ is $\mathfrak c$-admissible at $p$. Then \begin{equation}\label{eq:asym-upper} r_{\pi,s}\le\max\Bigl\{1+4p^{-1/4},\ K^{s-1}e^{R^2/K}\bigl(1+C_\Gamma(1-s)\bigr)\bigl(1+4p^{-1/4} \bigr)+\bar E_{\mathrm{irr}}\Bigr\}+\delta_p . \end{equation} \end{proposition} \begin{proof} By \Cref{prop:asym-blocks}, $a_{ij}\le\bar a_{ij}<\infty$ for all $i,j$, and the two diagonal entries $\bar a_{\mathrm{cc}}$ and $\bar a_{\mathrm{tt}}$ are at most the two terms in the maximum in \eqref{eq:asym-upper}, since $E_{\mathrm{irr}}\le\bar E_{\mathrm{irr}}$. Let $\eps=Q_p^{1/6}+p^{-1}$, which is larger than $q$ by \Cref{prop:asym-cycles}. \Cref{lem:asym-balancing} with $n=6$ and $\Delta=\{\mathrm c,\mathrm t\}$ gives $d_i\in(0,1]$ with \begin{equation*} \sum_ja_{ij}d_j\le\sum_j\bar a_{ij}d_j\le\bigl(\max\{\bar a_{\mathrm{cc}},\bar a_{\mathrm{tt}}\}+6\eps\bigr)d_i, \qquad i\in\mathscr C . \end{equation*} \Cref{lem:asym-supersolution} with $\lambda=\max\{\bar a_{\mathrm{cc}},\bar a_{\mathrm{tt}}\}+6\eps$ now implies that $r_{\pi,s}\le\lambda$, which is \eqref{eq:asym-upper}. \end{proof} \section{Proof of the leading asymptotics}\label{sec:asym-proof-leading} \begin{proof}[Proof of \Cref{thm:asym-leading}] Fix a compact set $\Xi\subset(0,\infty)$, and let $\xi_-=\min\Xi$. Let $\mathfrak c_\Xi$ and $p_\Xi$ be given by \Cref{lem:asym-admissible}(i), and let $p_1=p_1(\mathfrak c_\Xi)$ be so large that \Cref{lem:asym-lower,prop:asym-upper} hold. Let $p\ge\max\{p_\Xi,p_1\}$ and $\xi\in\Xi$, and set $E=\xi\sqrt p$. Let $\pi$ be the law of $B/g(E)$ under $\nu_{p,E}$; it is a real solution at $E$ because $p\ge p_0(\xi_-)$ (\Cref{sec:asym-inputs}), and $\lambda_p(\xi)=\exp\varphi_\pi(1)$. For every $s\in[3/4,1)$, the triple $(\nu_{p,E},p^2a_p(\xi)^2,s)$ is $\mathfrak c_\Xi$-admissible at $p$. Using $K^{s-1}\le1$ and $R^2/K\le(\log p)^2/(c_*p^2)$ with $c_*=\xi_-^2/4$, \Cref{lem:asym-lower,prop:asym-upper} give \begin{align*} 1-2p^{-1/4}\le r_{\pi,s}\le\max\Bigl\{&1+4p^{-1/4},\\ &e^{(\log p)^2/(c_*p^2)}\bigl(1+C_\Gamma(1-s)\bigr)\bigl(1+4p^{-1/4}\bigr)+\bar E_{\mathrm{irr}}\Bigr\} +\delta_p . \end{align*} Letting $s\uparrow1$ and using \Cref{lem:asym-critical}, we obtain \begin{equation}\label{eq:asym-two-sided} \frac{\mustar}{a_p(\xi)}\bigl(1-2p^{-1/4}\bigr)\le\lambda_p(\xi)\le\frac{\mustar}{a_p(\xi)} \bigl(1+\delta'_p\bigr), \end{equation} where \begin{equation*} \delta'_p=\max\Bigl\{4p^{-1/4},\ e^{(\log p)^2/(c_*p^2)}\bigl(1+4p^{-1/4}\bigr)-1+\bar E_{\mathrm{irr}}\Bigr\}+\delta_p \end{equation*} depends only on $\Xi$ and $p$ and tends to zero as $p\to\infty$. It remains to compare $a_p(\xi)$ with $\xi$. By \eqref{eq:asym-ap}, \begin{equation*} 0\le\xi-a_p(\xi)=\frac{\xi-\sqrt{\xi^2-4/p}}{2}=\frac{2}{p\bigl(\xi+\sqrt{\xi^2-4/p}\bigr)} \le\frac{2}{p\xi}, \end{equation*} and $a_p(\xi)\ge\xi/2$. Together these give $|a_p(\xi)^{-1}-\xi^{-1}|\le4/(p\xi_-^3)$ and $a_p(\xi)^{-1}\le2/\xi_-$. Combining these bounds with \eqref{eq:asym-two-sided}, we find \begin{equation*} \sup_{\xi\in\Xi}\Bigl|\lambda_p(\xi)-\frac{\mustar}{\xi}\Bigr| \le\frac{4\mustar}{p\xi_-^3}+\frac{2\mustar}{\xi_-}\max\bigl\{2p^{-1/4},\delta'_p\bigr\}, \end{equation*} which tends to zero as $p\to\infty$. \end{proof} \section{The free energy at large energies}\label{sec:asym-entry} At energies $E\ge M\sqrt p$ with $M$ large, the parameter $\theta=g(E)^2$ is at most $4/(M^2p)$ and $K=p/\theta$ is unbounded as $E\to\infty$. We work at the exponent \begin{equation}\label{eq:asym-sp} s_p=1-\frac{\kappa}{\log p}, \qquad \kappa=1 . \end{equation} At this exponent $p^{1-s_p}=e^\kappa$, and the capped integrals of \Cref{lem:asym-capped} are bounded by $2\ell_{K,s_p}\le2(1+\kappa^{-1}\log p)$ uniformly in $K$. We fix the constants \begin{equation}\label{eq:asym-entry-constants} c_1=(4e)^{-8/3}, \qquad M_{\mathrm{ext}}=\max\bigl\{M_0,\,2c_1^{-1/2}\bigr\}=2(4e)^{4/3}, \end{equation} where $M_0=3$ is the constant of \Cref{cor:stab-halfline}. These constants satisfy $c_1\le c_{\mathrm{ent}}=1$, $e^\kappa c_1^{3/8}=1/4$, and $4/M_{\mathrm{ext}}^2=c_1$. By \Cref{rem:stab-M0}, the principal law at energies $E\ge M_{\mathrm{ext}}\sqrt p$ is not affected by the choice of a constant larger than $M_0$. \begin{remark}\label{rem:asym-entry-uniform} In the large-energy regime, the estimates of \Cref{sec:asym-elementary,sec:asym-lower,sec:asym-charts,sec:asym-blocks,sec:asym-upper} hold uniformly in $E\ge M_{\mathrm{ext}}\sqrt p$, that is, in $\theta\in(0,c_1/p]$, because they depend on $K$ only through the following quantities. The term $\varepsilon_K=40Ke^{-\sqrt K/2}$ is decreasing for $K\ge16$, and the terms $R^2/K$, $K^{-1/2}$ and $K^{s-1}$ are nonincreasing in $K$; we bound each by its value at $K=c_*p^2$. The capped integrals are bounded by $2\ell_{K,s}$, and at $s=s_p$ we have $\ell_{K,s_p}\le1+\kappa^{-1}\log p$ for every $K\ge1$. At a fixed $s<1$ the second point would also hold, but then $p^{1-s}$ would be unbounded; at $s=1$ the capped integrals grow like $\log K$. The exponent $s_p$ is chosen between these two cases. In addition, since $K\ge p^2$, the tail diagonal entry satisfies \begin{equation*} \bar a_{\mathrm{tt}}\le K^{s_p-1}e^{R^2/K}\bigl(1+C_\Gamma(1-s_p)\bigr)\bigl(1+4p^{-1/4}\bigr) +\bar E_{\mathrm{irr}}\le e^{-2\kappa+o(1)}, \end{equation*} because $K^{s_p-1}=\exp(-\kappa\log K/\log p)\le e^{-2\kappa}$. In this regime the tail region is strictly subcritical, and the growth of $\mathsf T^*$ comes from the core alone. \end{remark} \begin{proof}[Proof of \Cref{prop:asym-entry}] Let $\mathfrak c_{\mathrm{ext}}$ be the constants of \Cref{lem:asym-admissible}(ii), and let $p_1=p_1(\mathfrak c_{\mathrm{ext}})$ be so large that \Cref{prop:asym-upper} holds. Let $p_0(M_{\mathrm {ext}})$ be the threshold of \Cref{cor:stab-halfline} for $\xi_0=M_{\mathrm{ext}}$. Let $p_{\mathrm{ext}}\ge\max\{e^4,p_0(M_{\mathrm{ext}}),p_1\}$ be so large that for $p\ge p_{\mathrm{ext}}$, \begin{equation}\label{eq:asym-entry-pext} \max\Bigl\{1+4p^{-1/4},\ e^{-2}e^{(\log p)^2/p^2}\Bigl(1+\frac{C_\Gamma}{\log p}\Bigr)\bigl(1+4p^{-1/4} \bigr)+\bar E_{\mathrm{irr}}\Bigr\}+\delta_p\le2 . \end{equation} All these thresholds are numerical. Let $p\ge p_{\mathrm{ext}}$ and $E\ge M_{\mathrm{ext}}\sqrt p$, and set $\theta=g(E)^2$ and $K=p/\theta$. Since $g(E)=2/(E+\sqrt{E^2-4})\le2/E$, we have \begin{equation*} \theta\le\frac{4}{E^2}\le\frac{4}{M_{\mathrm{ext}}^2\,p}=\frac{c_1}{p}. \end{equation*} Since $M_{\mathrm{ext}}\ge M_0$ and $p\ge p_0(M_{\mathrm{ext}})$, \Cref{cor:stab-halfline} shows that $\nu_{p,E}$ is the fixed point of \Cref{prop:stab-entry} with parameter $\theta$, and that the law $\pi$ of $B/g(E)$ under $\nu_{p,E}$ is a real solution at $E$. By \Cref{lem:asym-admissible}(ii) with $M=M_{\mathrm{ext}}$, the triple $(\nu_{p,E},K,s_p)$ is $\mathfrak c_{\mathrm{ext}}$-admissible at $p$, with $c_*=1$. Since $K\ge p^2$, we have $K^{s_p-1}\le e^{-2}$ and $R^2/K\le(\log p)^2/p^2$. Then \Cref{prop:asym-upper} and \eqref{eq:asym-entry-pext} imply that $r_{\pi,s_p}\le2$. Next we bound the scalar. With $\xi=E/\sqrt p$, we have $a_p(\xi)^{-1}=\sqrt p\,g(E)=(p\theta)^{1/2}$. Then \begin{equation*} \alpha_{p,s_p}(\xi)=p^{1-s_p}\mu_{s_p}(p\theta)^{s_p/2}\le e\cdot1\cdot c_1^{s_p/2}\le e\,c_1^{3/8} \le\frac14, \end{equation*} where we used $p^{1-s_p}=e$, $\mu_{s_p}\le1$ (\Cref{lem:asym-gamma}(i)), $p\theta\le c_1\le1$ and $s_p\ge3/4$. By \Cref{lem:asym-palm-normalization}, we conclude that \begin{equation*} \varphi_E(s_p)=\log\alpha_{p,s_p}(\xi)+\log r_{\pi,s_p}\le\log\frac14+\log2=-\log2 . \end{equation*} Finally, since $\varphi_E$ is nonincreasing on $(0,1]$ by \Cref{prop:press-properties}(ii), we have $\varphi_E(1)\le\varphi_E(s_p)$. This proves \eqref{eq:asym-entry-statement}. \end{proof} \chapter{Uniqueness of the critical energy}\label{sec:uniq} In this section we prove that, for every fixed $\xi_0\in(0,\mustar)$ and all large $p$, the free energy at $s=1$ of the principal solution vanishes at exactly one energy in $[\xi_0\sqrt p,\infty)$, and that this zero is transverse. Recall from \Cref{sec:stab} that for every compact window $\Xi\subset(0,\infty)$ and $p\ge p_{\mathrm{st}}(\Xi)$, the principal solution $\nu_{p,E}$ is defined at every energy $E=\xi\sqrt p$ with $\xi\in\Xi$ (\Cref{thm:stab-window}), and that we write \begin{equation*} \lambda_p(\xi)=\exp\varphi_{\xi\sqrt p}(1) \end{equation*} for the exponential of its free energy at $s=1$. By \Cref{thm:asym-leading}, $\lambda_p(\xi)$ converges to $\mustar/\xi$ uniformly on compact windows. This places every solution of $\lambda_p(\xi)=1$ near $\mustar$, but it does not exclude several crossings of the level one in a neighborhood of $\mustar$ whose size tends to zero as $p\to\infty$. The analyticity of $\lambda_p$ (\Cref{prop:jets-critical-analytic}) gives no bound that is uniform in $p$ either. The main task of this section is to show that $\lambda_p$ is strictly decreasing near $\mustar$. \section{Main results and strategy}\label{sec:uniq-results} \begin{theorem}\label{thm:uniq-window} Let $\Xi_0\subset(0,\infty)$ be a compact interval with $\mustar$ in its interior. There exists $p(\Xi_0)$ such that the following holds for every $p\ge p(\Xi_0)$. The equation $\lambda_p(\xi)=1$ has exactly one solution $\xi_c(p)\in\Xi_0$. For $\xi\in\Xi_0$, we have $\lambda_p(\xi)>1$ if $\xi<\xi_c(p)$ and $\lambda_p(\xi)<1$ if $\xi>\xi_c(p)$. The function $\lambda_p$ is continuously differentiable in a neighborhood of $\xi_c(p)$, and $\partial_\xi\lambda_p(\xi_c(p))<0$. Finally, $\xi_c(p)\to\mustar$ as $p\to\infty$. \end{theorem} \begin{corollary}\label{cor:uniq-halfline} Fix $\xi_0\in(0,\mustar)$. There exists $p(\xi_0)$ such that for every $p\ge p(\xi_0)$ the following holds, where $\varphi_E(1)$ denotes the free energy at $s=1$ of the principal solution $\nu_{p,E}$ of \Cref{cor:stab-halfline}. There exists a unique $E_c(p)\in[\xi_0\sqrt p,\infty)$ with $\varphi_{E_c(p)}(1)=0$. Further, $\varphi_E(1)>0$ for $E\in[\xi_0\sqrt p,E_c(p))$ and $\varphi_E(1)<0$ for $E\in(E_c(p),\infty)$. The function $E\mapsto \varphi_E(1)$ is continuously differentiable in a neighborhood of $E_c(p)$, with negative derivative at $E_c(p)$. Finally, $E_c(p)/\sqrt p\to\mustar$ as $p\to\infty$. \end{corollary} \subsection{Strategy}\label{sec:uniq-strategy} We work on a fixed compact window $\Xi$ around $\mustar$ and in the scaling $g=1/E$, that is, with $\widetilde B=D/E$, in which the squared modulus $Y$ enters through $\beta=\tau Y$ with $\tau=1/(pE^2)$. The proof has four steps. First, we construct an $L^2$ version of the critical transfer operator, on a Hilbert space $\mathscr L$ that does not depend on $\xi$. Its states are pairs $(x,\beta)$, where $x$ is the reciprocal of the projective coordinate and $\beta$ is the rescaled squared modulus that the next step will use. The operator $\mathsf A_\xi$ sends a function $F$ to an average of $F(y-\beta/x,\beta')$ over the side variable $y$ and a fresh rescaled squared modulus $\beta'$. The maps $(x,\beta)\mapsto y-\beta/x$ do not depend on $\xi$; the parameter enters only through the law $\nu_\xi$ of $\widetilde B$ and through a density factor. Since $\nu_\xi$ has a fixed atom, the derivative bounds of \Cref{sec:jets} show that $\xi\mapsto \mathsf A_\xi$ is four times differentiable in operator norm, with derivatives bounded by a power of $p$. There exists an involution $J$ of $\mathscr L$, which does not depend on $\xi$, with $J\mathsf A_\xi J=\mathsf A_\xi^*$. We prove in \Cref{lem:uniq-hilbert} that the spectral radius of $\mathsf A_\xi$ is $\lambda_p(\xi)$, and that it is an eigenvalue with a positive eigenvector. Second, we decompose the square as \begin{equation*} \mathsf A_\xi^2=G_\xi+h_{\mathrm c}\otimes\ell_{\mathrm c}+h_{\mathrm t}\otimes\ell_{\mathrm t}, \end{equation*} where all three terms are positive operators. The first rank-one term (the core component) is supported on an ordinary set of states with $x$ close to one and small $\beta$; there the kernel of $\mathsf A_\xi^2$ is close to a product kernel, and we subtract a rank-one minorant of it. The second rank-one term (the tail component) is the image of the first under $J$. The remainder $G_\xi$ has spectral radius $o(1)$. To see this, we split the state space into four blocks and observe that the transitions of $G_\xi$ with norm larger than a negative power of $p$ form an acyclic graph, apart from two diagonal blocks of norm $O(p^{-1}\log^{D}p)$. In particular, $(\lambda-G_\xi)^{-1}$ is a positive operator with norm bounded by a power of $p$, for $\lambda\in[1/2,2]$. Third, for $\lambda>r(G_\xi)$, a positive eigenvector of $\mathsf A_\xi^2$ with eigenvalue $\lambda$ corresponds to a positive eigenvector of the $2\times2$ renewal matrix $T(\xi,\lambda)$ with entries $\lambda\langle\ell_a,(\lambda-G_\xi)^{-1}h_b\rangle$. The square of $\lambda_p(\xi)$ is therefore the solution of $\lambda=r(T(\xi,\lambda))$, where $r(T)$ is the Perron--Frobenius eigenvalue. The symmetry under $J$ makes the two diagonal entries of $T$ equal, so that $r(T)=T_{\mathrm{cc}}+(T_{\mathrm{ct}}T_{\mathrm{tc}})^{1/2}$. The diagonal entry is close to $(\mustar/\xi)^2$ and its derivative is $-2\mustar^2/\xi^3+o(1)$. The entry $T_{\mathrm{ct}}$, which measures transitions from the core component to the tail component, is exponentially small, while $T_{\mathrm{tc}}$ is bounded by a power of $p$. Fourth, we control $\partial_\xi(T_{\mathrm{ct}}T_{\mathrm{tc}})^{1/2}$ without dividing by any small quantity: neither by $T_{\mathrm{ct}}$, nor by the gap between the two eigenvalues of $T$, nor by a spectral gap of $\mathsf A_\xi$. For a nonnegative function $f$, Glaeser's inequality bounds $|f'|$ by $(2f\|f''\|_\infty)^{1/2}$ plus a boundary term, and the second derivatives of $T_{\mathrm{ct}}$ and $T_{\mathrm{tc}}$ are controlled by interpolating between the bounds on their values and polynomial bounds on their fourth derivatives. Since the entries of $T$ are nonincreasing in $\lambda$, implicit differentiation of $\lambda=r(T(\xi,\lambda))$ has a denominator at least one. This gives $\partial_\xi\lambda_p<0$ near $\mustar$. The abstract form of this last step is \Cref{prop:uniq-renewal-criterion}. The section is organized as follows. \Cref{sec:uniq-elementary} collects elementary facts on positive operators and two inequalities for real functions. \Cref{sec:uniq-renewal} states and proves the abstract renewal criterion. \Cref{sec:uniq-inputs} fixes the parameters and lists the inputs from earlier sections. \Cref{sec:uniq-hilbert} constructs the $L^2$ version of the transfer operator. \Cref{sec:uniq-minorant} constructs the decomposition of the square, \Cref{sec:uniq-cutoffs} proves the bounds on the remainder, and \Cref{sec:uniq-matrix} analyzes the renewal matrix. \Cref{sec:uniq-proofs} contains the proofs of \Cref{thm:uniq-window} and \Cref{cor:uniq-halfline}. \section{Elementary lemmas}\label{sec:uniq-elementary} \subsection{Positive operators} Throughout this section, $\mathscr H$ denotes a real Hilbert space of the form $L^2(\Omega,\mathfrak m)$ for a $\sigma$-finite measure space $(\Omega,\mathfrak m)$, with inner product $\langle\cdot,\cdot\rangle$. All operators are bounded. Spectra and spectral radii $r(\cdot)$ refer to the complexifications of the operators; by the spectral radius formula \cite[Theorem VI.6]{ReedSimon1980methods}, $r(X)=\lim_{n\to\infty}\|X^n\|^{1/n}$. For $f,g\in\mathscr H$, we write $f\otimes g$ for the rank-one operator $F\mapsto\langle g,F\rangle f$. \begin{definition}\label{def:uniq-positive} An operator $X$ on $\mathscr H$ is \emph{positive}, written $X\succeq0$, if $XF\ge0$ almost everywhere whenever $F\ge0$ almost everywhere. We write $X\preceq Y$ if $Y-X\succeq0$. A function $F\in\mathscr H$ is positive if $F>0$ almost everywhere. For a measurable set $\Omega'\subset\Omega$, we let $P_{\Omega'}$ denote multiplication by $\one_{\Omega'}$. \end{definition} The order $\succeq$ is not the order of quadratic forms; it is the order of positivity-preserving operators. \begin{lemma}\label{lem:uniq-positive} Let $X,Y,X_1,X_2,Y_1,Y_2$ be operators on $\mathscr H$. \begin{enumerate}[label=\textup{(\roman*)}] \item If $0\preceq X\preceq Y$, then $\|X\|\le\|Y\|$ and $r(X)\le r(Y)$. \item If $X\succeq0$, then $X^*\succeq0$. \item If $0\preceq X_i\preceq Y_i$ for $i=1,2$, then $0\preceq X_1X_2\preceq Y_1Y_2$. \item If $X\succeq0$ and $\lambda>r(X)$, then $(\lambda-X)^{-1}=\sum_{n\ge0}\lambda^{-n-1}X^n\succeq0$, where the series converges in norm. If moreover $0\preceq X\preceq Y$ and $\lambda>r(Y)$, then $(\lambda-X)^{-1}\preceq(\lambda-Y)^{-1}$. \item Let $\Omega=\Omega_1\cup\dots\cup\Omega_m$ be a partition into measurable sets and write $P_i=P_{\Omega_i}$. If $X\succeq0$, then $P_iXP_j\succeq0$ for all $i,j$, and $\sum_{(i,j)\in\mathcal S}P_iXP_j\preceq X$ for every set $\mathcal S$ of pairs of indices. \end{enumerate} \end{lemma} \begin{proof} For (i), let $F\in\mathscr H$. Since $X(|F|\pm F)\ge0$, we have $|XF|\le X|F|\le Y|F|$ almost everywhere, which gives $\|XF\|\le\|Y|F|\|\le\|Y\|\|F\|$. By (iii) below, $0\preceq X^n\preceq Y^n$, and the first claim gives $\|X^n\|\le\|Y^n\|$ for every $n$; the spectral radius formula implies that $r(X)\le r(Y)$. For (ii), if $F,H\ge0$, then $\langle X^*F,H\rangle=\langle F,XH\rangle\ge0$; taking $H=\one_{\{X^*F<0\}}\one_{\Omega''}$ for sets $\Omega''$ of finite measure shows that $X^*F\ge0$. For (iii), we have $Y_1Y_2-X_1X_2=(Y_1-X_1)Y_2+X_1(Y_2-X_2)$, and both terms are compositions of positive operators. For (iv), by the spectral radius formula, we have $\sum_n\lambda^{-n-1}\|X^n\|<\infty$, which implies that the Neumann series converges in norm to $(\lambda-X)^{-1}$; each term is positive. If $0\preceq X\preceq Y$, then $X^n\preceq Y^n$ by (iii), and summing the series gives the second claim. For (v), we have $P_iXP_j\succeq0$ by (iii), since $P_i$ and $P_j$ are positive. Since $\sum_iP_i$ is the identity, $X=\sum_{i,j}P_iXP_j$, and $X-\sum_{(i,j)\in\mathcal S}P_iXP_j$ is a sum of positive operators. \end{proof} \begin{lemma}\label{lem:uniq-abba} Let $\mathscr H_1,\mathscr H_2$ be Hilbert spaces and let $X\colon\mathscr H_1\to\mathscr H_2$ and $Y\colon\mathscr H_2\to\mathscr H_1$ be bounded. Then $\sigma(XY)\setminus\{0\}=\sigma(YX)\setminus\{0\}$. If $z\ne0$ is an isolated point of this set, then the Riesz projections of $XY$ and $YX$ at $z$ have the same rank. If $YXv=zv$ with $z\ne0$ and $v\ne0$, then $Xv\ne0$ and $XY(Xv)=zXv$. \end{lemma} \begin{proof} Let $w\ne0$ with $w\notin\sigma(YX)$. A direct multiplication shows that \begin{equation}\label{eq:uniq-abba} (w-XY)^{-1}=w^{-1}\bigl(1+X(w-YX)^{-1}Y\bigr), \end{equation} so $w\notin\sigma(XY)$. By symmetry, the two sets agree away from zero. Let $z\ne0$ be isolated, and let $\gamma$ be a small circle around $z$ that encloses no other point of the spectrum and does not enclose zero. Integrating \eqref{eq:uniq-abba} over $\gamma$, and using that $\oint_\gamma w^{-1}\dd w=0$, we find that the Riesz projection $\Pi_{XY}$ of $XY$ at $z$ equals $X\mathcal RY$, where $\mathcal R=(2\pi\iu)^{-1}\oint_\gamma w^{-1}(w-YX)^{-1}\dd w$. The operator $\mathcal R$ vanishes on the kernel of the Riesz projection $\Pi_{YX}$ of $YX$ at $z$ and maps into its range \cite[Section III.6.4]{Kato1995perturbation}. In particular, the rank of $\Pi_{XY}$ is at most the rank of $\Pi_{YX}$. By symmetry, the ranks agree. For the last claim, $Xv=0$ would give $zv=YXv=0$, and $XY(Xv)=X(YXv)=zXv$. \end{proof} \subsection{Two inequalities for real functions} The next lemma bounds the first two derivatives of a function with small values and polynomially bounded fourth derivative. \begin{lemma}\label{lem:uniq-interpolation} Let $[s_-,s_+]$ be a compact interval, let $f\in C^4([s_-,s_+])$ be real valued, and set \begin{equation*} \epsilon=\sup_{[s_-,s_+]}|f|,\qquad m_4=\sup_{[s_-,s_+]}|f^{(4)}|. \end{equation*} Let $r_0>0$ and let $s\in[s_-+r_0,s_+-r_0]$. Then \begin{equation}\label{eq:uniq-interpolation} |f''(s)|\le2\sqrt{\epsilon m_4}+\frac{8\epsilon}{r_0^2},\qquad |f'(s)|\le2\epsilon^{3/4}m_4^{1/4}+\frac{4\epsilon}{r_0}. \end{equation} \end{lemma} \begin{proof} If $\epsilon=0$, the result is immediate. We henceforth assume $\epsilon>0$ and interpret $h_*=\infty$ whenever $m_4=0$ in both optimizations below. For $|t|\le r_0$, Taylor's formula with Lagrange remainder gives \begin{equation}\label{eq:uniq-taylor4} f(s+t)=f(s)+tf'(s)+\frac{t^2}{2}f''(s)+\frac{t^3}{6}f'''(s)+\frac{t^4}{24}f^{(4)}(s_t) \end{equation} for some $s_t$ between $s$ and $s+t$. We begin with the second derivative. Let $0r_0$, we choose $h=r_0$; then $m_4r_0^4<48\epsilon$, so $m_4r_0^2/12<4\epsilon/r_0^2$ and $|f''(s)|\le8\epsilon/r_0^2$. For the first derivative, let $0r_0/2$, we choose $h=r_0/2$; then $m_4h^4<3\epsilon$, which gives $m_4h^3/6<\epsilon/(2h)$ and $|f'(s)|\le2\epsilon/h=4\epsilon/r_0$. \end{proof} The second inequality is a version of Glaeser's inequality \cite{Glaeser1963racine}, which bounds the derivative of a nonnegative function by the square root of its value. We include its short proof, since we use a form with a boundary term. \begin{lemma}\label{lem:uniq-glaeser} Let $I_1$ be an interval, let $f\in C^2(I_1)$ be nonnegative with $m_2=\sup_{I_1}|f''|<\infty$, and let $s\in I_1$ be a point whose distance to the complement of $I_1$ is at least $r_0>0$. Then \begin{equation}\label{eq:uniq-glaeser} |f'(s)|\le\sqrt{2f(s)m_2}+\frac{2f(s)}{r_0}. \end{equation} \end{lemma} \begin{proof} We may suppose that $f'(s)\ne0$; let $\sigma=-\sgn f'(s)$. For $00$. Set $h_*=(2f(s)/m_2)^{1/2}$ (with $h_*=\infty$ if $m_2=0$). If $h_*\le r_0$, the choice $h=h_*$ gives $|f'(s)|\le\sqrt{2f(s)m_2}$. If $h_*>r_0$, the choice $h=r_0$ gives $|f'(s)|\le f(s)/r_0+r_0m_2/2\le2f(s)/r_0$, where we used that $r_0^2m_2<2f(s)$. \end{proof} \section{A positive rank-two renewal criterion}\label{sec:uniq-renewal} In this subsection we state the abstract form of the derivative argument. The setting is a family of positive operators that differ from a positive operator of small spectral radius by two positive rank-one operators. For a $2\times2$ matrix $T$ with nonnegative entries, we write $r(T)$ for its spectral radius (the Perron--Frobenius eigenvalue). \begin{proposition}\label{prop:uniq-renewal-criterion} Let $\mathscr H=L^2(\Omega,\mathfrak m)$ be a real Hilbert space, let $I\subset I_1$ be open intervals such that every point of $I$ has distance at least $r_0>0$ to the complement of $I_1$, and let $0<\lambda_-<\lambda_+$. For $\xi\in I_1$, let $G_\xi\succeq0$ be an operator on $\mathscr H$, let $h_{1,\xi},h_{2,\xi},\ell_{1,\xi},\ell_{2,\xi}\in\mathscr H$ be nonnegative, and set \begin{equation*} U_\xi=G_\xi+h_{1,\xi}\otimes\ell_{1,\xi}+h_{2,\xi}\otimes\ell_{2,\xi}. \end{equation*} Suppose that $\xi\mapsto G_\xi$ and $\xi\mapsto h_{i,\xi},\ell_{i,\xi}$ are twice continuously differentiable on $I_1$ in the operator norm and in the norm of $\mathscr H$, respectively, and that $r(G_\xi)<\lambda_-$ for every $\xi\in I_1$. For $\xi\in I_1$ and $\lambda>r(G_\xi)$, define the renewal matrix \begin{equation}\label{eq:uniq-abstract-T} T_{ij}(\xi,\lambda)=\lambda\bigl\langle\ell_{i,\xi},(\lambda-G_\xi)^{-1}h_{j,\xi}\bigr\rangle,\qquad i,j\in\{1,2\}. \end{equation} Assume the following. \begin{enumerate}[label=\textup{(\roman*)}] \item For all $\xi\in I_1$ and $\lambda\in[\lambda_-,\lambda_+]$, we have $T_{11}(\xi,\lambda)=T_{22}(\xi,\lambda)$, $T_{12}(\xi,\lambda)>0$ and $T_{21}(\xi,\lambda)>0$. \item There exists $\kappa>0$ such that $\partial_\xi T_{11}(\xi,\lambda)\le-\kappa$ for all $\xi\in I$ and $\lambda\in[\lambda_-,\lambda_+]$, and for every $\lambda\in[\lambda_-,\lambda_+]$, writing $b=T_{12}(\cdot,\lambda)$ and $d=T_{21}(\cdot,\lambda)$ and taking suprema over $I_1$, \begin{equation}\label{eq:uniq-abstract-offdiag} \sqrt{\|d\|_\infty\|\partial_\xi^2b\|_\infty}+\sqrt{\|b\|_\infty\|\partial_\xi^2d\|_\infty}+\frac{2\sqrt{\|b\|_\infty\|d\|_\infty}}{r_0}\le\frac\kappa2. \end{equation} \item For every $\xi\in I$, there exist $\lambda(\xi)\in(\lambda_-,\lambda_+)$ and a nonzero $F_\xi\in\mathscr H$ with $F_\xi\ge0$ such that $U_\xi F_\xi=\lambda(\xi)F_\xi$. \end{enumerate} Then the following hold for every $\xi\in I$. \begin{enumerate}[label=\textup{(\alph*)}] \item The entries of $T(\xi,\lambda)$ are nonincreasing in $\lambda\in[\lambda_-,\lambda_+]$, we have $r(T(\xi,\lambda))=T_{11}+\sqrt{T_{12}T_{21}}$ there, and $\partial_\lambda r(T(\xi,\lambda))\le0$ for $\lambda\in(\lambda_-,\lambda_+)$. \item The number $\lambda(\xi)$ is the unique solution $\lambda\in[\lambda_-,\lambda_+]$ of $\lambda=r(T(\xi,\lambda))$. The numbers $\langle\ell_{1,\xi},F_\xi\rangle$ and $\langle\ell_{2,\xi},F_\xi\rangle$ are positive. \item The function $\xi\mapsto\lambda(\xi)$ is continuously differentiable on $I$, and \begin{equation}\label{eq:uniq-implicit} \lambda'(\xi)=\frac{\partial_\xi r(T(\xi,\lambda))}{1-\partial_\lambda r(T(\xi,\lambda))}\bigg|_{\lambda=\lambda(\xi)}\le-\frac{\kappa}{2\bigl(1-\partial_\lambda r(T(\xi,\lambda(\xi)))\bigr)}<0. \end{equation} \end{enumerate} \end{proposition} The hypotheses involve only the sizes of the off-diagonal entries and of their second derivatives. No lower bound on $T_{12}$ or $T_{21}$ is required, and the proof does not divide by the gap $2\sqrt{T_{12}T_{21}}$ between the two eigenvalues of $T$, nor by a spectral gap of $U_\xi$. \begin{proof} We abbreviate $G=G_\xi$, $h_i=h_{i,\xi}$ and $\ell_i=\ell_{i,\xi}$ when $\xi$ is fixed. \proofstep{1}{The renewal matrix.} We first claim that, for $\xi\in I_1$ and $\lambda>r(G_\xi)$, \begin{equation}\label{eq:uniq-T-series} T_{ij}(\xi,\lambda)=\sum_{n\ge0}\lambda^{-n}\langle\ell_i,G^nh_j\rangle, \end{equation} that $T$ is continuously differentiable in $(\xi,\lambda)$, and that the entries of $T$ are nonincreasing in $\lambda$. By \Cref{lem:uniq-positive}(iv), we have $\lambda(\lambda-G)^{-1}=\sum_{n\ge0}\lambda^{-n}G^n$ in norm, which gives \eqref{eq:uniq-T-series}. Each coefficient $\langle\ell_i,G^nh_j\rangle$ is nonnegative, since $G^n\succeq0$ and $h_j,\ell_i\ge0$. The series \eqref{eq:uniq-T-series} is a power series in $\lambda^{-1}$ whose radius of convergence is at least $1/r(G)$. It may therefore be differentiated termwise, and \begin{equation*} \partial_\lambda T_{ij}(\xi,\lambda)=-\sum_{n\ge1}n\lambda^{-n-1}\langle\ell_i,G^nh_j\rangle\le0. \end{equation*} For the joint differentiability, note that $(\xi,\lambda)\mapsto(\lambda-G_\xi)^{-1}$ is continuous in norm on $\{(\xi,\lambda):\lambda>r(G_\xi)\}$, and that its partial derivatives are $(\lambda-G_\xi)^{-1}(\partial_\xi G_\xi)(\lambda-G_\xi)^{-1}$ and $-(\lambda-G_\xi)^{-2}$, which are continuous as well. The set $\{(\xi,\lambda):\lambda>r(G_\xi)\}$ is open, since the spectral radius is upper semicontinuous in norm \cite[Section IV.3.1]{Kato1995perturbation}. Combined with the differentiability of $h_j$ and $\ell_i$, this shows that $T$ is continuously differentiable, and twice continuously differentiable in $\xi$ for fixed $\lambda$. \proofstep{2}{The Perron--Frobenius eigenvalue.} Let $\xi\in I_1$ and $\lambda\in[\lambda_-,\lambda_+]$, and write $a=T_{11}=T_{22}$, $b=T_{12}$ and $d=T_{21}$. The eigenvalues of $T$ are $a\pm\sqrt{bd}$. Since $a\ge0$, we have $r(T)=a+\sqrt{bd}$. Since $b,d>0$ on $I_1\times[\lambda_-,\lambda_+]$, Step 1 shows that $r(T)$ is continuously differentiable on $I_1\times(\lambda_-,\lambda_+)$. By Step 1, $b$ and $d$ are positive and nonincreasing in $\lambda$. The functions $\sqrt{bd}$ and $a$ are therefore nonincreasing in $\lambda$, and $\partial_\lambda r(T)\le0$. This proves (a). \proofstep{3}{The eigenvector relation.} Let $\xi\in I$, write $\lambda=\lambda(\xi)$ and $F=F_\xi$, and set $u_i=\langle\ell_i,F\rangle\ge0$. The eigenvalue equation $U_\xi F=\lambda F$ reads $(\lambda-G)F=u_1h_1+u_2h_2$. Since $\lambda>r(G)$, we obtain \begin{equation}\label{eq:uniq-F-renewal} F=(\lambda-G)^{-1}(u_1h_1+u_2h_2). \end{equation} If $u_1=u_2=0$, then \eqref{eq:uniq-F-renewal} implies that $F=0$, which is excluded. Pairing \eqref{eq:uniq-F-renewal} with $\ell_i$ and multiplying by $\lambda$, we find that \begin{equation*} \lambda u_i=\sum_{j=1}^2T_{ij}(\xi,\lambda)u_j,\qquad i=1,2, \end{equation*} that is, $u=(u_1,u_2)$ is a nonzero nonnegative eigenvector of $T(\xi,\lambda)$ with eigenvalue $\lambda$. If $u_1=0$, the first equation reads $bu_2=0$, and $b>0$ forces $u_2=0$, which is impossible. Then $u_1>0$, and similarly $u_2>0$. The two equations now read $(\lambda-a)u_1=bu_2$ and $(\lambda-a)u_2=du_1$. Multiplying them, we obtain $(\lambda-a)^2=bd$, and the first one shows that $\lambda-a>0$. We conclude that $\lambda=a+\sqrt{bd}=r(T(\xi,\lambda))$. Let $\Psi(\xi,\lambda)=\lambda-r(T(\xi,\lambda))$. By (a), $\lambda\mapsto r(T(\xi,\lambda))$ is nonincreasing on $[\lambda_-,\lambda_+]$, which implies that $\Psi(\xi,\cdot)$ is strictly increasing there and has at most one zero. This proves (b). \proofstep{4}{The slope in $\xi$.} We claim that $\partial_\xi r(T(\xi,\lambda))\le-\kappa/2$ for $\xi\in I$ and $\lambda\in[\lambda_-,\lambda_+]$. Fix $\lambda$, and let $b,d$ be as in (ii). We have \begin{equation*} \partial_\xi\sqrt{bd}=\frac{\sqrt d}{2\sqrt b}\,\partial_\xi b+\frac{\sqrt b}{2\sqrt d}\,\partial_\xi d. \end{equation*} Since $b>0$ is twice continuously differentiable on $I_1$, we may apply \Cref{lem:uniq-glaeser}, and we obtain, at every $\xi\in I$, \begin{equation*} \frac{|\partial_\xi b|}{2\sqrt b}\le\frac{\sqrt{2b\|\partial_\xi^2b\|_\infty}+2b/r_0}{2\sqrt b}\le\sqrt{\|\partial_\xi^2b\|_\infty}+\frac{\sqrt b}{r_0}, \end{equation*} and similarly for $d$. Inserting these bounds, we obtain \begin{equation*} |\partial_\xi\sqrt{bd}|\le\sqrt{d\|\partial_\xi^2b\|_\infty}+\sqrt{b\|\partial_\xi^2d\|_\infty}+\frac{2\sqrt{bd}}{r_0}\le\frac\kappa2, \end{equation*} where in the last inequality we used \eqref{eq:uniq-abstract-offdiag}. Combined with $\partial_\xi T_{11}\le-\kappa$, this proves the claim. \proofstep{5}{Implicit differentiation.} Fix $\xi_1\in I$. The function $\Psi$ of Step 3 is continuously differentiable on an open neighborhood of $(\xi_1,\lambda(\xi_1))$, it vanishes at this point, and $\partial_\lambda\Psi\ge1$ there. By the implicit function theorem, there exist a neighborhood $I'\subset I$ of $\xi_1$ and a continuously differentiable function $\tilde\lambda$ on $I'$ with values in $(\lambda_-,\lambda_+)$, such that $\tilde\lambda(\xi_1)=\lambda(\xi_1)$ and $\Psi(\xi,\tilde\lambda(\xi))=0$ for $\xi\in I'$. By the uniqueness in (b), $\tilde\lambda=\lambda$ on $I'$. Then $\lambda$ is continuously differentiable on $I$, and differentiating $\Psi(\xi,\lambda(\xi))=0$ gives the identity in \eqref{eq:uniq-implicit}. The numerator is at most $-\kappa/2$ by Step 4, and the denominator is at least one by (a). This proves (c). \end{proof} \section{Parameters and inputs}\label{sec:uniq-inputs} \subsection{Normalization} Throughout \Cref{sec:uniq-inputs,sec:uniq-hilbert,sec:uniq-minorant,sec:uniq-cutoffs,sec:uniq-matrix}, we fix a compact interval $\Xi=[\xi_-,\xi_+]\subset(0,\infty)$ and write $E=\xi\sqrt p$ for $\xi\in\Xi$. We always assume that $p$ is so large that $\xi_-\sqrt p\ge3$. In the notation of \Cref{def:rde-normalizations}, we set \begin{equation}\label{eq:uniq-parameters} g=g(E),\qquad\theta=g^2,\qquad c=1+\theta=Eg,\qquad\tau=\frac{1}{pE^2}=\frac{1}{p^2\xi^2}=\frac{\theta}{pc^2},\qquad\alpha=\sqrt p\,g. \end{equation} All of these are functions of $\xi$ (and $p$). From the formula for $g$, we have $\alpha=1/a_p(\xi)=2/(\xi+(\xi^2-4/p)^{1/2})$ and $g\le2/E$, and we note the identities \begin{equation}\label{eq:uniq-identities} \tau c^2=\frac\theta p,\qquad pc\sqrt\tau=\alpha. \end{equation} Elementary calculus shows that there exists $C(\Xi)$ such that for $0\le j\le5$ and $\xi\in\Xi$, \begin{equation}\label{eq:uniq-parameter-derivatives} |\partial_\xi^j\theta|+|\partial_\xi^jc-\one_{j=0}|+\Bigl|\partial_\xi^j\Bigl(\alpha-\frac1\xi\Bigr)\Bigr|\le\frac{C}{p},\qquad\frac{|\partial_\xi^j\tau|}{\tau}+\frac{|\partial_\xi^j(\tau c)|}{\tau c}\le C. \end{equation} In particular, $\alpha=\xi^{-1}+O(p^{-1})$ and $\partial_\xi\alpha=-\xi^{-2}+O(p^{-1})$, uniformly on $\Xi$. We write $\varpi_\xi(\dd\beta)=\tau^{-1}e^{-\beta/\tau}\dd\beta$ for the law of $\beta=\tau Y$ with $Y\sim\Exp(1)$, and $\gamma(\dd u)=2ue^{-u^2}\dd u$ on $[0,\infty)$ for the law of $\sqrt Y$. We record that \begin{equation}\label{eq:uniq-gamma-moment} \int_0^\infty u\,\gamma(\dd u)=\int_0^\infty2u^2e^{-u^2}\dd u=\frac{\sqrt\pi}{2}=\mustar. \end{equation} \subsection{Inputs from earlier sections} We use the following results, for $\xi\in\Xi$ and $p\ge p_{\mathrm{I}}(\Xi)$, where $p_{\mathrm{I}}(\Xi)$ is at least the thresholds of the cited results for the window $\Xi$; in \Cref{thm:stab-window}, we use the canonical tuple of \Cref{def:stab-class}. \begin{enumerate}[label=\textup{(I\arabic*)}] \item\label{it:uniq-I1} \emph{The principal law.} By \Cref{thm:stab-window}(i),(v), the principal solution $\nu_{p,E}$ (the law of $B$) is defined, and the law of $D=B/g$ is a real solution at $E$. We let $\nu_\xi$ denote the law of $\widetilde B=B/c$ under $\nu_{p,E}$; this is the principal law in the scaling $g=1/E$ of \Cref{def:jets-fixed-law}, denoted $\widetilde\nu_\xi$ there. By \Cref{prop:rde-properties} applied to $D=E\widetilde B$, the law $\nu_\xi$ has an atom of mass $e^{-p}$ at $1$, no other atom (in particular no atom at $0$), full support $\R$, and $\E|\widetilde B|^q<\infty$ for $q\in(0,1)$. \item\label{it:uniq-I2} \emph{Bounds defining the class $\mathfrak S_{p,\theta}$.} By \Cref{thm:stab-window}(iii) and $r_p=p^{-1/4}$, the core bound \begin{equation}\label{eq:uniq-core} \nu_{p,E}\bigl(\{b:|b-1|>p^{-1/4}\}\bigr)\le e^{-a\sqrt p} \end{equation} holds with $a=1$, and it continues to hold if $a$ is decreased. The tail bound of \Cref{thm:stab-window}(iii) gives $\nu_{p,E}(\{b:|b|>L\})\le e^{-p}/(2L)+e^{-pL}/2\le C_p/L$ for all $L\ge3$, with $C_p$ depending only on $p$. (We use the tail bound only at fixed $p$. In \eqref{eq:uniq-core}, any core radius $r_p$ with $r_p\log p\to0$ would do.) \item\label{it:uniq-I3} \emph{The critical transfer operator.} Let $\mathcal Q_1=\mathcal Q_{1,\xi}$ be the transfer operator of \Cref{def:press-operators} at $s=1$, defined with the law $\nu_\xi$ and an independent $\beta\sim\varpi_\xi$, which has the exponential law of rate $1/\tau=pE^2$. This is the scaling $g=1/E$ in \Cref{def:press-pressure}(iii): the law of $gD$ is $\nu_\xi$, and $p/g^2=pE^2$. The Perron--Frobenius eigenvalue of $\mathcal Q_1$ is therefore $\exp\varphi_E(1)=\lambda_p(\xi)$; see also \Cref{prop:jets-critical-analytic}(i). By \Cref{thm:press-perron}(i),(ii), $\mathcal Q_1$ is bounded and positive on $C(\RP)$, and $\lambda_p(\xi)$ is an eigenvalue of $\mathcal Q_1$ on $C(\RP)$ whose Riesz projection has rank one, with an eigenfunction $\psi_\xi\in C(\RP)$ satisfying $\min\psi_\xi>0$. Further, there exists $r_\xi<\lambda_p(\xi)$ such that $|z|\le r_\xi$ for every $z\in\sigma(\mathcal Q_1)\setminus\{\lambda_p(\xi)\}$. \item\label{it:uniq-I4} \emph{Analyticity of the law.} We fix $q=1/2$. By \Cref{thm:jets-analytic}(ii), $\xi\mapsto\nu_\xi$ is real-analytic on a neighborhood of $\Xi$ as a map into the space $X_q$ of \Cref{def:jets-spaces}, whose norm dominates the weighted total variation norm $\|\sigma\|_{q,\TV}$, which in turn dominates the total variation norm $|\sigma|(\R)$. In particular, $\xi\mapsto\nu_\xi$ is four times continuously differentiable in total variation, and $\partial_\xi^j\nu_\xi$ is a signed measure of total mass zero for $j\ge1$. \item\label{it:uniq-I5} \emph{Quantitative derivative bounds.} By \Cref{prop:jets-quantitative}(i), there exists $A\ge1$, depending only on $\Xi$, such that $|\partial_\xi^j\nu_\xi|(\R)\le\|\partial_\xi^j\nu_\xi\|_{q,\TV}\le p^A$ for $1\le j\le4$. \item\label{it:uniq-I6} \emph{Smooth tests.} We apply \Cref{cor:jets-tests} with $J=[1/2,3/2]$, after enlarging $A$ and decreasing $a$ if necessary, and we write $\|\zeta\|_{C^2}=\|\zeta\|_\infty+\|\zeta'\|_\infty+\|\zeta''\|_\infty$ and \begin{equation*} \epsilon_p=\frac Cp+p^Ae^{-a\sqrt p}. \end{equation*} Let $B\sim\nu_{p,\xi\sqrt p}$ and $j\in\{1,2\}$. For every $\zeta\in C^2(\R)$ supported in $[1/2,3/2]$, \begin{equation}\label{eq:uniq-weak-jets} \bigl|\partial_\xi^j\E\zeta(B)\bigr|\le\epsilon_p\|\zeta\|_{C^2}. \end{equation} More generally, if $I_0\supset\Xi$ is an open interval and $\xi\mapsto\zeta_\xi$ is twice continuously differentiable from $I_0$ into the space of $C^2$ functions supported in $[1/2,3/2]$ with the norm $\|\cdot\|_{C^2}$, then \begin{equation}\label{eq:uniq-weak-jets-param} \bigl|\partial_\xi^j\E\zeta_\xi(B)\bigr|\le\epsilon_p\sum_{i=0}^{j-1}\binom ji\|\partial_\xi^i\zeta_\xi\|_{C^2}+\sup|\partial_\xi^j\zeta_\xi|. \end{equation} \end{enumerate} The constants $a$ and $A$ are fixed from now on; they depend only on $\Xi$. The leading asymptotics (\Cref{thm:asym-leading}) and the large-energy bound (\Cref{prop:asym-entry}) are used only in \Cref{sec:uniq-proofs}. \subsection{Order of the parameters}\label{sec:uniq-order} The construction below involves a fixed power $n_0$, a target exponent $N_*$ and a cutoff level $M_*$. They are chosen in the following order. \begin{enumerate}[label=\textup{(\arabic*)}] \item The window $\Xi$ is fixed; then $a$ and $A$ are fixed by \ref{it:uniq-I2}, \ref{it:uniq-I5} and \ref{it:uniq-I6}. \item We fix $n_0=12$. \item We define the exponents \begin{equation}\label{eq:uniq-exponents} \begin{gathered} A_1=2A+2,\qquad A_2=10n_0+8A_1+5,\\ A_{\mathrm j}=2A_1+A_2+1,\qquad A_{\mathrm v}=2A_1+4n_0+2. \end{gathered} \end{equation} They depend only on $A$, so only on $\Xi$. \item We fix an integer $N_*>\max\{2n_0+10,\,A_{\mathrm v}+A_{\mathrm j}+2\}$. \item We fix $M_*=4N_*+20$. \item Finally, we take $p\ge p_1$, where $p_1=p_1(\Xi,N_*,M_*)\ge p_{\mathrm{I}}(\Xi)$ is a threshold that may be enlarged finitely many times below. \end{enumerate} In \Cref{sec:uniq-hilbert,sec:uniq-minorant,sec:uniq-cutoffs,sec:uniq-matrix}, the letters $C$ and $D$ denote positive constants that may depend on $\Xi$, $N_*$ and $M_*$, and may change from line to line, but not on $p$, $\xi$ or $\lambda$. The point of the order above is that every bound of the form $Cp^{e}\log^Dp$ below has an exponent $e$ that does not depend on $M_*$; quantities such as $e^{M_*+1}$ enter only through $C$. In particular, $Cp^{e}\log^Dp\le p^{e+1/2}$ for $p\ge p_1$, which is how such bounds are converted into the pure powers in \eqref{eq:uniq-exponents}. \section{The $L^2$ version of the transfer operator}\label{sec:uniq-hilbert} \subsection{The operator} Let $\kappa_0=p^2\xi_-^2/2$, so that $\kappa_0<1/\tau$ for every $\xi\in\Xi$, and define \begin{equation*} \mu_0(\dd\beta)=\kappa_0e^{-\kappa_0\beta}\dd\beta,\qquad\omega_\xi(\beta)=\Bigl(\frac{1}{\tau\kappa_0}\Bigr)^{1/2}\exp\Bigl(-\frac12\Bigl(\frac1\tau-\kappa_0\Bigr)\beta\Bigr),\qquad\beta>0. \end{equation*} Then $\omega_\xi^2\mu_0=\varpi_\xi$, and in particular $\int\omega_\xi^2\dd\mu_0=1$. \begin{definition}\label{def:uniq-realization} Let $\mathscr L=L^2(\R\times(0,\infty),\dd x\,\mu_0(\dd\beta))$ (real valued), with inner product $\langle\cdot,\cdot\rangle$, and let $L^2(\R)=L^2(\R,\dd x)$. Define $I_\xi\colon L^2(\R)\to\mathscr L$ by $(I_\xi f)(x,\beta)=\omega_\xi(\beta)f(x)$. For a finite signed measure $\sigma$ on $\R$, define $\mathcal C_\sigma$ on $L^2(\R)$ by $(\mathcal C_\sigma f)(x)=\int f(y-x)\,\sigma(\dd y)$. Define $J$ on $\mathscr L$ by \begin{equation*} (JF)(x,\beta)=\frac{\sqrt\beta}{|x|}F\Bigl(\frac\beta x,\beta\Bigr), \end{equation*} and the $L^2$ version of the critical transfer operator \begin{equation}\label{eq:uniq-A} \mathsf A_\xi=pJI_\xi\mathcal C_{\nu_\xi}I_\xi^*. \end{equation} \end{definition} Explicitly, since $(I_\xi^*F)(x)=\int\omega_\xi(\beta')F(x,\beta')\mu_0(\dd\beta')$, we have \begin{equation}\label{eq:uniq-A-explicit} (\mathsf A_\xi F)(x,\beta)=p\,\omega_\xi(\beta)\frac{\sqrt\beta}{|x|}\int\nu_\xi(\dd y)\int\mu_0(\dd\beta')\,\omega_\xi(\beta')F\Bigl(y-\frac\beta x,\beta'\Bigr). \end{equation} The operator $\mathsf A_\xi$ is not the transfer operator $\mathcal A_s$ in the $x$-coordinate of \Cref{def:press-operators}; it is related to the transfer operator $\mathcal Q_1$ in \Cref{lem:uniq-hilbert}. The state $(x,\beta)$ consists of the reciprocal $x$ of the projective coordinate and the rescaled squared modulus $\beta$ used in the next step; the new state is $(y-\beta/x,\beta')$ with a side variable $y\sim\nu_\xi$ and a fresh rescaled squared modulus $\beta'$. The maps $(x,\beta,y)\mapsto y-\beta/x$ do not depend on $\xi$, and $\xi$ enters \eqref{eq:uniq-A-explicit} only through $\nu_\xi$ and $\omega_\xi$. \begin{lemma}\label{lem:uniq-realization} Let $p\ge p_1$. \begin{enumerate}[label=\textup{(\roman*)}] \item The operator $J$ is a positive unitary involution: $J\succeq0$ and $J=J^*=J^{-1}$. For every measurable $\Omega'\subset\R\times(0,\infty)$, we have $JP_{\Omega'}J=P_{J\Omega'}$, where $J\Omega'=\{(\beta/x,\beta):(x,\beta)\in\Omega'\}$. \item The operator $I_\xi$ is an isometry, and for every finite signed measure $\sigma$, $\mathcal C_\sigma$ is self-adjoint with $\|\mathcal C_\sigma\|\le|\sigma|(\R)$. \item We have $\mathsf A_\xi\succeq0$, $\|\mathsf A_\xi\|\le p$, and $\mathsf A_\xi^*=J\mathsf A_\xi J$. \item The map $\xi\mapsto \mathsf A_\xi$ is four times continuously differentiable on $\Xi$ in operator norm, and $\|\partial_\xi^j\mathsf A_\xi\|\le p^{A_1}$ for $0\le j\le4$. \end{enumerate} \end{lemma} \begin{proof} For fixed $\beta>0$, the map $x\mapsto\beta/x$ is an involution of $\R\setminus\{0\}$ with Jacobian $\beta/x^2$. This substitution gives $\int|(JF)(x,\beta)|^2\dd x=\int|F(x,\beta)|^2\dd x$ for every $\beta$, and $J^2F=F$, since $(J^2F)(x,\beta)=(\sqrt\beta/|x|)(\sqrt\beta/|\beta/x|)F(x,\beta)$. A unitary involution is self-adjoint. Positivity is evident, and the formula for $JP_{\Omega'}J$ follows from $\one_{\Omega'}(\beta/x,\beta)=\one_{J\Omega'}(x,\beta)$. This proves (i). For (ii), we have $\|I_\xi f\|^2=\|f\|^2\int\omega_\xi^2\dd\mu_0=\|f\|^2$. By Minkowski's integral inequality, we have $\|\mathcal C_\sigma f\|\le\int\|f(y-\cdot)\|\,|\sigma|(\dd y)=|\sigma|(\R)\|f\|$, and the substitution $x\mapsto y-x$ gives $\langle\mathcal C_\sigma f_1,f_2\rangle=\langle f_1,\mathcal C_\sigma f_2\rangle$. For (iii), positivity is evident from \eqref{eq:uniq-A-explicit}, and (i), (ii) give $\|\mathsf A_\xi\|\le p$. Since $J$ and $\mathcal C_{\nu_\xi}$ are self-adjoint, we have $\mathsf A_\xi^*=pI_\xi\mathcal C_{\nu_\xi}I_\xi^*J=J(pJI_\xi\mathcal C_{\nu_\xi}I_\xi^*)J=J\mathsf A_\xi J$. For (iv), we find by direct computation that $\partial_\xi^j\omega_\xi(\beta)=\omega_\xi(\beta)P_{j,\xi}(\beta/\tau)$ for $j\le5$, where $P_{j,\xi}$ is a polynomial of degree $j$ whose coefficients are bounded on $\Xi$ by \eqref{eq:uniq-parameter-derivatives}. Since $\beta/\tau$ has law $\Exp(1)$ under $\omega_\xi^2\mu_0$, we have $\|\partial_\xi^j\omega_\xi\|_{L^2(\mu_0)}\le C$. Moreover, $\sup_{\xi\in\Xi}|\partial_\xi^j\omega_\xi(\beta)|^2\le C(1+p^2\beta)^{2j}\kappa_0^{-1}p^2\xi_+^2e^{-(p^2\xi_-^2-\kappa_0)\beta}$, which is integrable against $\mu_0$; by dominated convergence, $\xi\mapsto\omega_\xi$ is five times continuously differentiable as a map into $L^2(\mu_0)$. Since $\|\partial_\xi^jI_\xi\|=\|\partial_\xi^j\omega_\xi\|_{L^2(\mu_0)}$, the map $\xi\mapsto I_\xi$ is four times continuously differentiable with $\|\partial_\xi^jI_\xi\|\le C$. By \ref{it:uniq-I4} and (ii), $\xi\mapsto\mathcal C_{\nu_\xi}$ is four times continuously differentiable with $\partial_\xi^j\mathcal C_{\nu_\xi}=\mathcal C_{\partial_\xi^j\nu_\xi}$, and \ref{it:uniq-I5} gives $\|\partial_\xi^j\mathcal C_{\nu_\xi}\|\le p^A$. The Leibniz rule applied to \eqref{eq:uniq-A} yields $\|\partial_\xi^j\mathsf A_\xi\|\le Cp^{A+1}\le p^{A_1}$. \end{proof} \subsection{The spectral radius of the $L^2$ version} We now relate $\mathsf A_\xi$ to the transfer operator $\mathcal Q_1$ of \ref{it:uniq-I3}. We identify $\RP$ with $\R\cup\{\infty\}$ through the reciprocal $x=1/q$ of the projective coordinate $q$ of \Cref{def:press-operators}. For $x\in\R\setminus\{0\}$, $\widetilde B\in\R$ and $\beta>0$, set \begin{equation}\label{eq:uniq-angular-kernel} x'=\widetilde B-\frac\beta x,\qquad w=\frac{\sqrt\beta\,(1+x^2)^{1/2}}{|x|(1+x'^2)^{1/2}}, \end{equation} and extend these by $x'=\infty$, $w=\beta^{-1/2}$ at $x=0$ and by $x'=\widetilde B$, $w=\beta^{1/2}(1+\widetilde B^2)^{-1/2}$ at $x=\infty$. For fixed $\widetilde B$ and $\beta$, the maps $x\mapsto x'$ and $x\mapsto w$ are continuous on $\RP$. Writing $M(\widetilde B,\beta)v_q$ in the coordinates of \Cref{def:press-operators}, we have $M(\widetilde B,\beta)v_q\propto(1,\widetilde B-\beta q)$. The image of $q=1/x$ has reciprocal coordinate $x'$, and $\|M(\widetilde B,\beta)v_q\|^{-1}=w$. We obtain \begin{equation}\label{eq:uniq-Q-reciprocal} (\mathcal Q_1u)(x)=p\E\bigl[w\,u(x')\bigr],\qquad x\in\RP, \end{equation} with $\widetilde B\sim\nu_\xi$ and $\beta\sim\varpi_\xi$ independent; at $x\in\{0,\infty\}$, this agrees with \Cref{def:press-operators} by \eqref{eq:press-norm}. As in \Cref{def:press-operators}, \eqref{eq:uniq-Q-reciprocal} applies to every bounded Borel function $u$ on $\RP$, and we write $\mathcal Q_1(x,\cdot)$ for the finite measure $u\mapsto(\mathcal Q_1u)(x)$, and similarly for powers of $\mathcal Q_1$. \begin{lemma}\label{lem:uniq-hilbert} Let $p\ge p_1$ and $\xi\in\Xi$, and write $\rho_\xi=\lambda_p(\xi)$. \begin{enumerate}[label=\textup{(\roman*)}] \item The spectral radius of $\mathsf A_\xi$ on $\mathscr L$ equals $\rho_\xi$. Further, $\rho_\xi$ is an isolated point of the spectrum of $\mathsf A_\xi$, and its Riesz projection has rank one. \item There exists $F_\xi\in\mathscr L$ with $F_\xi>0$ almost everywhere and $\mathsf A_\xi F_\xi=\rho_\xi F_\xi$. In particular, $\mathsf A_\xi^2F_\xi=\rho_\xi^2F_\xi$. \end{enumerate} \end{lemma} Only part (ii) is used in the proof of \Cref{thm:uniq-window}; part (i) shows that the eigenvalue in (ii) is the spectral radius. \begin{proof} We fix $\xi$ and drop it from the notation; we write $\nu=\nu_\xi$, $\varpi=\varpi_\xi$, $\psi=\psi_\xi$ and $\rho=\rho_\xi$. Let $\varsigma(x)=(1+x^2)^{-1/2}$ and $\tilde{\mathfrak m}(\dd x)=\varsigma(x)^2\dd x$, a finite measure on $\R$ (equivalently, on $\RP$, with $\{\infty\}$ a null set). Let $0<\psi_-\le\psi\le\psi_+$ be the bounds of the positive continuous function $\psi$ on the compact space $\RP$. The proof proceeds by transporting the problem to the transfer operator $\mathcal Q_1$, where the Perron--Frobenius theory of \ref{it:uniq-I3} applies, and then to $\mathsf A_\xi$. \proofstep{1}{The scalar operator and the weight.} Define $\mathcal W$ on $L^2(\R)$ by $(\mathcal Wf)(x)=\int\varpi(\dd\beta)\sqrt\beta|x|^{-1}f(\beta/x)$, and set $\mathsf A^{\mathrm s}=p\mathcal W\mathcal C_\nu$. For each $\beta$, the operator $f\mapsto\sqrt\beta|x|^{-1}f(\beta/x)$ is a unitary involution of $L^2(\R)$, as in the proof of \Cref{lem:uniq-realization}(i). The operator $\mathcal W$ is an average of these involutions, and it is self-adjoint with $\|\mathcal W\|\le1$. We have \begin{equation*} (\mathsf A^{\mathrm s}f)(x)=p\int\nu(\dd y)\int\varpi(\dd\beta)\frac{\sqrt\beta}{|x|}f\Bigl(y-\frac\beta x\Bigr). \end{equation*} By \eqref{eq:uniq-angular-kernel}, $w=\sqrt\beta|x|^{-1}\varsigma(x')/\varsigma(x)$ for $x\ne0$, and \eqref{eq:uniq-Q-reciprocal} gives \begin{equation}\label{eq:uniq-gauge} (\mathcal Q_1u)(x)=\varsigma(x)^{-1}\bigl(\mathsf A^{\mathrm s}(\varsigma u)\bigr)(x)\qquad\text{for almost every }x, \end{equation} for every bounded Borel $u$. The map $\Theta f=f/\varsigma$ is unitary from $L^2(\R)$ onto $L^2(\tilde{\mathfrak m})$, and $\tilde Q=\Theta \mathsf A^{\mathrm s}\Theta^{-1}$ is a bounded operator on $L^2(\tilde{\mathfrak m})$ that acts on bounded functions by the kernel of $\mathcal Q_1$. Set $f=\varsigma\psi$. Then $f\in L^2(\R)$, $f>0$ everywhere, and \eqref{eq:uniq-gauge} with $\mathcal Q_1\psi=\rho\psi$ gives $\mathsf A^{\mathrm s}f=\rho f$. \proofstep{2}{A left eigenvector.} Set $f^*=\mathcal C_\nu f$. Since $\mathcal W$ and $\mathcal C_\nu$ are self-adjoint, $(\mathsf A^{\mathrm s})^*=p\mathcal C_\nu\mathcal W$, and \begin{equation*} (\mathsf A^{\mathrm s})^*f^*=p\,\mathcal C_\nu\mathcal W\mathcal C_\nu f=\mathcal C_\nu \mathsf A^{\mathrm s}f=\rho f^*. \end{equation*} We claim that $\chi=f^*/\varsigma$ satisfies $c_\chi\le\chi\le C_\chi$ for constants $00$ by the full support in \ref{it:uniq-I1}. For the upper bound, if $|x|\le6$ then $\mathcal C_\nu\varsigma(x)\le1\le\sqrt{37}\,\varsigma(x)$. If $|x|>6$, we split the integral according to whether $|y|\le|x|/2$. On the first part, $\varsigma(y-x)\le|y-x|^{-1}\le2/|x|$. The second part is at most $\nu(\{|y|>|x|/2\})=\nu_{p,E}(\{|b|>c|x|/2\})\le2C_p/|x|$, by the tail bound in \ref{it:uniq-I2}. Then $\mathcal C_\nu\varsigma(x)\le(2+2C_p)/|x|\le(4+4C_p)\varsigma(x)$. \proofstep{3}{The Doob transform.} For bounded Borel $u$ on $\RP$, set $\mathcal Pu=(\rho\psi)^{-1}\mathcal Q_1(\psi u)$. Since $\mathcal Q_1\psi=\rho\psi$ holds at every point of $\RP$, $\mathcal P(x,\cdot)$ is a probability measure for every $x\in\RP$. On $L^2(\tilde{\mathfrak m})$, the operator $\mathcal P=\rho^{-1}\psi^{-1}\tilde Q\psi$ is bounded. Let $\pi(\dd x)=Z^{-1}\psi\chi\,\dd\tilde{\mathfrak m}$, with $Z$ chosen so that $\pi$ is a probability measure. By Step 2, the density of $\pi$ with respect to $\tilde{\mathfrak m}$ lies in $[Z^{-1}\psi_-c_\chi,Z^{-1}\psi_+C_\chi]$, and the norms of $L^2(\pi)$ and $L^2(\tilde{\mathfrak m})$ are therefore equivalent. We claim that $\pi$ is invariant: $\int\mathcal Pu\,\dd\pi=\int u\,\dd\pi$ for $u\in L^2(\tilde{\mathfrak m})$, and then for every measurable $u\ge0$ by monotone convergence. Indeed, the adjoint of $\tilde Q$ in $L^2(\tilde{\mathfrak m})$ is $\Theta(\mathsf A^{\mathrm s})^*\Theta^{-1}$. By Step 2, we have $\tilde Q^*\chi=\Theta(\mathsf A^{\mathrm s})^*f^*=\rho\chi$, and \begin{equation*} Z\int\mathcal Pu\,\dd\pi=\rho^{-1}\int\tilde Q(\psi u)\,\chi\,\dd\tilde{\mathfrak m}=\rho^{-1}\int\psi u\,\tilde Q^*\chi\,\dd\tilde{\mathfrak m}=Z\int u\,\dd\pi. \end{equation*} \proofstep{4}{The spectral radius.} Since the rows of $\mathcal P$ are probability measures, we have $|\mathcal Pu|^2\le\mathcal P(|u|^2)$ pointwise by the Cauchy--Schwarz inequality, and Step 3 gives $\int|\mathcal Pu|^2\dd\pi\le\int|u|^2\dd\pi$. It follows that $\|\mathcal P^n\|_{L^2(\pi)}\le1$ for all $n$. Since $\tilde Q^n=\rho^n\psi\mathcal P^n\psi^{-1}$ and the norms are equivalent, we find that $\|\tilde Q^n\|_{L^2(\tilde{\mathfrak m})}\le C'\rho^n$ for a constant $C'$ independent of $n$. By the spectral radius formula, we conclude that $r(\mathsf A^{\mathrm s})=r(\tilde Q)\le\rho$, and Step 1 gives equality. \proofstep{5}{Three-step smoothing.} We claim that for every bounded Borel $u$ on $\RP$, the function $\mathcal P^3u$ is continuous on $\RP$ and depends only on the $\tilde{\mathfrak m}$-equivalence class of $u$. Since $\mathcal P^3u=\rho^{-3}\psi^{-1}\mathcal Q_1^3(\psi u)$, it suffices to show that $x\mapsto\mathcal Q_1^3(x,\cdot)$ is continuous on $\RP$ in total variation and that each $\mathcal Q_1^3(x,\cdot)$ is absolutely continuous with respect to Lebesgue measure on $\R$. Let $x\in\R\setminus\{0\}$. Substituting $z=y-\beta/x$ in \eqref{eq:uniq-Q-reciprocal}, so that $\beta=x(y-z)$ and $\dd\beta=|x|\dd z$, we find that $\mathcal Q_1(x,\dd z)=p\varsigma(x)^{-1}\varsigma(z)(\nu\star k_x)(z)\dd z$, where \begin{equation*} k_x(t)=\tau^{-1}\sqrt{xt}\,e^{-xt/\tau}\one_{\{xt>0\}},\qquad(\nu\star k_x)(z)=\int k_x(y-z)\,\nu(\dd y). \end{equation*} For $t\ne0$, $k_x(t)$ is continuous in $x\in\R\setminus\{0\}$, and $\|k_x\|_{L^1}=\Gamma(3/2)\tau^{1/2}|x|^{-1}$ is continuous as well. By Scheff\'e's lemma \cite{Durrett2019probability}, the map $x\mapsto k_x$ is continuous into $L^1(\R)$. Since $\|\nu\star(k_x-k_{x_0})\|_{L^1}\le\|k_x-k_{x_0}\|_{L^1}$ and $\varsigma\le1$, the map $x\mapsto\mathcal Q_1(x,\cdot)$ is continuous in total variation on $\R\setminus\{0\}$, and each $\mathcal Q_1(x,\cdot)$ with $x\in\R\setminus\{0\}$ is absolutely continuous. Next, $x\mapsto w$ is continuous on $\RP$ for fixed $(\widetilde B,\beta)$, and $\E w=p^{-1}(\mathcal Q_1\one)(x)$ is continuous on $\RP$ since $\mathcal Q_1$ maps $C(\RP)$ into itself. By Scheff\'e's lemma, $\E|w(x)-w(x_0)|\to0$ as $x\to x_0$ in $\RP$, where we indicate the dependence of $w$ and $x'$ on $x$. Let $\mathcal K(y,\cdot)$ be either $\mathcal Q_1(y,\cdot)$ or $\mathcal Q_1^2(y,\cdot)$, and suppose that $y\mapsto\mathcal K(y,\cdot)$ is continuous in total variation at almost every value of $x'(x_0)$. For a Borel function $|u|\le1$, we have $(\mathcal Q_1\mathcal Ku)(x)=p\E[w(x)(\mathcal Ku)(x'(x))]$, so \begin{multline*} \bigl|(\mathcal Q_1\mathcal Ku)(x)-(\mathcal Q_1\mathcal Ku)(x_0)\bigr|\le pm_{\mathcal K}\,\E|w(x)-w(x_0)|\\ +p\E\bigl[w(x_0)\,\|\mathcal K(x'(x),\cdot)-\mathcal K(x'(x_0),\cdot)\|_{\TV}\bigr], \end{multline*} where $m_{\mathcal K}=\sup_y\mathcal K(y,\RP)\le\max\{\|\mathcal Q_1\|,\|\mathcal Q_1\|^2\}$ and $\|\mathcal Q_1\|$ is the norm on $C(\RP)$, which bounds the total mass of every row of $\mathcal Q_1$. The first term tends to zero. In the second term, the integrand tends to zero almost surely, since $x'(x)\to x'(x_0)$, and it is bounded by $2m_{\mathcal K}w(x_0)$, which is integrable; by dominated convergence, it tends to zero. This shows that $\mathcal Q_1\mathcal K$ is continuous in total variation at $x_0$. We apply this twice. If $x_0\in\R\setminus\{0\}$, then $x'(x_0)=\widetilde B-\beta/x_0$ has a density, and if $x_0=\infty$, then $x'(x_0)=\widetilde B$; in both cases $x'(x_0)\in\R\setminus\{0\}$ almost surely, by \ref{it:uniq-I1}. With $\mathcal K=\mathcal Q_1$, we conclude that $\mathcal Q_1^2$ is continuous in total variation on $\RP\setminus\{0\}$. For every $x_0\in\RP$, we have $x'(x_0)\ne0$ almost surely (at $x_0=0$, $x'=\infty$). With $\mathcal K=\mathcal Q_1^2$, we conclude that $\mathcal Q_1^3$ is continuous in total variation on $\RP$. From the same representation, we see that $\mathcal Q_1^2(x,\cdot)=p\E[w\,\mathcal Q_1(x',\cdot)]$ is absolutely continuous for $x\ne0$, since $x'\in\R\setminus\{0\}$ almost surely, and that $\mathcal Q_1^3(x,\cdot)=p\E[w\,\mathcal Q_1^2(x',\cdot)]$ is absolutely continuous for every $x\in\RP$, since $x'\ne0$ almost surely. This proves the claim. Since the rows of $\mathcal P^3$ are probability measures, we also have $\sup|\mathcal P^3u|\le\|u\|_{L^\infty(\pi)}$. \proofstep{6}{Decay in $C(\RP)$ and in $L^\infty(\pi)$.} On $C(\RP)$, the operator $\mathcal P$ is similar to $\rho^{-1}\mathcal Q_1$ through multiplication by $\psi$. By \ref{it:uniq-I3}, $1$ is an algebraically simple eigenvalue of $\mathcal P$ on $C(\RP)$, with eigenfunction $\one$, and the rest of the spectrum lies in a disc of radius $\theta_1<1$. Let $\Pi_C$ be the Riesz projection at $1$. It has rank one and range spanned by $\one$; we write $\Pi_Cv=\phi(v)\one$ with a bounded linear functional $\phi$. Further, for $\theta_2\in(\theta_1,1)$ there exists $C_0$ with $\sup|\mathcal P^nv-\phi(v)|\le C_0\theta_2^n\sup|v|$ for $v\in C(\RP)$. Integrating against $\pi$ and using the invariance of $\pi$, we find that $\int v\,\dd\pi=\int\mathcal P^nv\,\dd\pi\to\phi(v)$, which shows that $\phi(v)=\int v\,\dd\pi$. Now let $u\in L^\infty(\pi)$ and $n\ge3$. The function $v=\mathcal P^3u$ lies in $C(\RP)$ by Step 5, and $\int v\,\dd\pi=\int u\,\dd\pi$ by invariance. Applying the bound in $C(\RP)$ to $v$, we find that \begin{equation}\label{eq:uniq-Linfty-decay} \Bigl\|\mathcal P^nu-\int u\,\dd\pi\Bigr\|_{L^\infty(\pi)}=\Bigl\|\mathcal P^{n-3}v-\int v\,\dd\pi\Bigr\|_{\infty}\le C_0\theta_2^{n-3}\|u\|_{L^\infty(\pi)}. \end{equation} \proofstep{7}{The gap in $L^2(\pi)$.} Let $\Pi u=(\int u\,\dd\pi)\one$ and $\mathcal P_0=\mathcal P-\Pi$. Since $\mathcal P\one=\one$ and $\pi$ is invariant, we have $\Pi\mathcal P=\mathcal P\Pi=\Pi^2=\Pi$. Then $\Pi\mathcal P_0=\mathcal P_0\Pi=0$ and $\mathcal P_0^n=\mathcal P^n-\Pi$. On $L^1(\pi)$, we have $|\mathcal Pu|\le\mathcal P|u|$ and $\int\mathcal P|u|\,\dd\pi=\int|u|\,\dd\pi$, which give $\|\mathcal P_0^n\|_{L^1(\pi)}\le2$. Applying \eqref{eq:uniq-Linfty-decay} to real and imaginary parts, $\|\mathcal P_0^n\|_{L^\infty(\pi)}\le2C_0\theta_2^{n-3}$ on complex functions. The Riesz--Thorin interpolation theorem \cite[Section IX.4]{ReedSimon1975methods} gives $\|\mathcal P_0^n\|_{L^2(\pi)}\le2(C_0\theta_2^{n-3})^{1/2}$, and the spectral radius formula yields $r(\mathcal P_0)\le\theta_2^{1/2}<1$ on $L^2(\pi)$. For $z\ne1$ with $|z|>\theta_2^{1/2}$, a direct computation using $\Pi\mathcal P_0=\mathcal P_0\Pi=0$ shows that \begin{equation*} (z-\mathcal P)^{-1}=(z-1)^{-1}\Pi+(z-\mathcal P_0)^{-1}(1-\Pi). \end{equation*} Therefore, $1$ is an isolated point of the spectrum of $\mathcal P$ on $L^2(\pi)$, with Riesz projection $\Pi$ of rank one. Since $\tilde Q=\rho\psi\mathcal P\psi^{-1}$ on the space $L^2(\tilde{\mathfrak m})=L^2(\pi)$ (with equivalent norms), and $\mathsf A^{\mathrm s}=\Theta^{-1}\tilde Q\Theta$, the same holds for $\rho$ and $\mathsf A^{\mathrm s}$. \proofstep{8}{The augmented operator.} Let $\mathscr L'=L^2(\R\times(0,\infty),\dd x\,\varpi(\dd\beta))$ and define $V\colon L^2(\R)\to\mathscr L'$ and $\Pi'\colon\mathscr L'\to L^2(\R)$ by \begin{equation*} (Vf)(x,\beta)=p\frac{\sqrt\beta}{|x|}(\mathcal C_\nu f)\Bigl(\frac\beta x\Bigr),\qquad(\Pi'F)(x)=\int\varpi(\dd\beta)F(x,\beta). \end{equation*} For fixed $\beta$, the substitution $x\mapsto\beta/x$ shows that $\|Vf\|^2=p^2\|\mathcal C_\nu f\|^2\le p^2\|f\|^2$, and by Jensen's inequality, we have $\|\Pi'\|\le1$. Further, $\Pi'V=p\mathcal W\mathcal C_\nu=\mathsf A^{\mathrm s}$. The map $\mathcal UF=\omega_\xi F$ is unitary from $\mathscr L'$ onto $\mathscr L$, since $\omega_\xi^2\mu_0=\varpi$, and comparing with \eqref{eq:uniq-A-explicit} shows that $\mathsf A_\xi=\mathcal UV\Pi'\mathcal U^{-1}$. By \Cref{lem:uniq-abba} with $X=V$ and $Y=\Pi'$, the operators $\mathsf A_\xi$ and $\mathsf A^{\mathrm s}$ have the same nonzero spectrum, and their Riesz projections at $\rho$ have the same rank. With Steps 4 and 7, this proves (i). For (ii), set $F_\xi=\mathcal UVf$. Then $\mathsf A_\xi F_\xi=\mathcal UV\Pi'Vf=\mathcal UV\mathsf A^{\mathrm s}f=\rho F_\xi$. Since $f>0$ everywhere and $\nu$ is a probability measure, $\mathcal C_\nu f>0$ everywhere, so $F_\xi(x,\beta)>0$ for all $x\ne0$ and $\beta>0$. \end{proof} \section{A positive decomposition of the square}\label{sec:uniq-minorant} In this subsection we construct the decomposition $\mathsf A_\xi^2=G_\xi+h_{\mathrm c}\otimes\ell_{\mathrm c}+h_{\mathrm t}\otimes\ell_{\mathrm t}$. We assume $p\ge p_1$ and $\xi\in\Xi$ throughout, and use the conventions of \Cref{sec:uniq-order}. \subsection{Regions and cutoffs} \begin{definition}\label{def:uniq-regions} Set \begin{equation*} r=\frac{1}{\log p},\qquad W=\bigl((M_*+1)\log p\bigr)^{1/2},\qquad\beta_{\max}=\frac{4W^2}{p^2\xi_-^2}. \end{equation*} The \emph{core region} is $\Omega_{\mathrm c}=\{(x,\beta)\in\R\times(0,\infty):|x|>\sqrt\beta\}$, and the \emph{tail region} is its complement $\Omega_{\mathrm t}$. The \emph{ordinary set} is $\mathsf O=[1-r,1+r]\times(0,\beta_{\max}]$. We also set $\mathsf U_{\mathrm c}=\Omega_{\mathrm c}\setminus\mathsf O$ and $\mathsf U_{\mathrm t}=J\mathsf U_{\mathrm c}$, with the notation $J\Omega'$ of \Cref{lem:uniq-realization}(i). \end{definition} The map $(x,\beta)\mapsto(\beta/x,\beta)$ sends $\Omega_{\mathrm c}$ onto $\{0<|x|<\sqrt\beta\}$. Then $J\Omega_{\mathrm c}$ and $\Omega_{\mathrm t}$ differ by a null set, and $JP_{\Omega_{\mathrm c}}J=P_{\Omega_{\mathrm t}}$ as operators. For $p\ge p_1$, we have $\sqrt{\beta_{\max}}<1-r$, which implies that $\mathsf O\subset\Omega_{\mathrm c}$ and $J\mathsf O\subset\Omega_{\mathrm t}$ up to a null set. Up to null sets, $\R\times(0,\infty)$ is the disjoint union of the four sets \begin{equation}\label{eq:uniq-blocks} X_1=J\mathsf O,\qquad X_2=\mathsf U_{\mathrm t},\qquad X_3=\mathsf U_{\mathrm c},\qquad X_4=\mathsf O, \end{equation} with $\Omega_{\mathrm t}=X_1\cup X_2$ and $\Omega_{\mathrm c}=X_3\cup X_4$. We fix, once and for all, smooth functions $\chi_0,\psi_0\colon\R\to[0,1]$ with $\chi_0=1$ on $[-1/8,1/8]$, $\supp\chi_0\subset(-1/4,1/4)$, $\psi_0=1$ on $(-\infty,0]$ and $\psi_0=0$ on $[1,\infty)$. Define \begin{equation*} \chi(b)=\chi_0\Bigl(\frac{b-1}{r}\Bigr),\qquad\psi(u)=\psi_0\Bigl(\frac{u^2}{\log p}-M_*\Bigr),\qquad u\ge0. \end{equation*} Then $\chi(b)=1$ for $|b-1|\le r/8$, $\chi(b)=0$ for $|b-1|\ge r/4$, $\psi(u)=1$ for $u^2\le M_*\log p$, and $\psi(u)=0$ for $u\ge W$. For $l\le5$, we have \begin{equation}\label{eq:uniq-cutoff-derivatives} |\chi^{(l)}|\le C\log^lp,\qquad|(u\partial_u)^l\psi(u)|\le C, \end{equation} where the second bound holds because $(u\partial_u)^l\psi(u)$ is a combination of $(u^2/\log p)^i\psi_0^{(i)}(u^2/\log p-M_*)$, $i\le l$, and $u^2/\log p\le M_*+1$ on the support of $\psi_0'$. For $x>0$ and $\beta>0$, define \begin{equation*} w_\xi(x,\beta)=\Bigl(\frac{\beta}{\tau cx}\Bigr)^{1/2}, \end{equation*} and define the multipliers \begin{equation*} \Psi^{\mathrm{out}}_\xi(x,\beta)=\one_{\mathsf O}(x,\beta)\,\psi(w_\xi(x,\beta)),\qquad\Psi^{\mathrm{in}}_\xi(x,\beta)=\one_{\mathsf O}(x,\beta)\,\chi(cx)\,\psi(w_\xi(x,\beta)). \end{equation*} The \emph{retained operator} is $B_\xi=\Psi^{\mathrm{out}}_\xi \mathsf A_\xi\Psi^{\mathrm{in}}_\xi$, where the multipliers act as multiplication operators. \begin{lemma}\label{lem:uniq-multipliers} \begin{enumerate}[label=\textup{(\roman*)}] \item If $x>0$, $\beta>0$ and $\chi(cx)\psi(w_\xi(x,\beta))\ne0$, then $(x,\beta)\in\mathsf O$ and $|cx-1|W$ (where $h_\xi$ vanishes anyway). Let $\Upsilon(b,u)=\sqrt b\,e^{-(b-1)u^2}\chi(b)$ for $b>0$ and $\Upsilon(b,u)=0$ for $b\le0$; this is a smooth function, since $\chi$ vanishes for $b\le1/2$. For $w,u\in[0,W]$, define \begin{equation}\label{eq:uniq-Upsilon} \Upsilon_\xi(w,u)=\E\,\Upsilon\Bigl(B-\frac{\theta w^2}{p},u\Bigr),\qquad\Upsilon^0_\xi(u)=\E\,\Upsilon(B,u), \end{equation} where $B\sim\nu_{p,\xi\sqrt p}$, that is, $B=c\widetilde B$ with $\widetilde B\sim\nu_\xi$. \begin{lemma}\label{lem:uniq-factorization} The operators $R_\xi$ and $S_\xi$ are positive, $\|R_\xi\|\le C$, $\|S_\xi\|\le Cp$, and $B_\xi=R_\xi S_\xi$. The operator $N_\xi=S_\xi R_\xi$ on $\mathscr V$ is the integral operator \begin{equation}\label{eq:uniq-N} (N_\xi\phi)(w)=\alpha\int_0^W\gamma(\dd u)\,u\,\psi(u)^2\,\Upsilon_\xi(w,u)\,\phi(u). \end{equation} Moreover, $R_\xi\one=h_\xi$. \end{lemma} \begin{proof} Positivity is evident. To bound $R_\xi$, we substitute $\beta=\tau cxu^2$ for fixed $x\in[1-r,1+r]$; then $\omega_\xi(\beta)^2\mu_0(\dd\beta)=\varpi_\xi(\dd\beta)=e^{-cxu^2}2cxu\,\dd u$ and $p^2\beta x^{-2}=p^2\tau cu^2/x$, and we have \begin{equation*} \|R_\xi\phi\|^2\le\int_{1-r}^{1+r}\dd x\int_0^W\gamma(\dd u)\,p^2\tau c^2u^2e^{-(cx-1)u^2}|\phi(u)|^2\le\frac{c^2}{\xi^2}\cdot2r\cdot W^2e^{2rW^2}\|\phi\|^2. \end{equation*} Here we used $p^2\tau=\xi^{-2}$, $|cx-1|\le2r$ and $u\le W$. Since $rW^2=M_*+1$, this gives $\|R_\xi\|\le C$. To bound $S_\xi$, we use the Cauchy--Schwarz inequality in $\beta'$ with $\int\omega_\xi^2\dd\mu_0=1$, Jensen's inequality in $y$, and $\Psi^{\mathrm{in}}_\xi\le1$, which give $|(S_\xi F)(w)|^2\le\int\nu_\xi(\dd y)\int\mu_0(\dd\beta')|F(y-\tau cw^2,\beta')|^2$. For fixed $y$, the substitution $x'=y-\tau cw^2$ has $\dd x'=2\tau cw\,\dd w$, which gives \begin{equation*} \|S_\xi F\|^2\le\int\nu_\xi(\dd y)\int\mu_0(\dd\beta')\int_0^\infty2we^{-w^2}|F(y-\tau cw^2,\beta')|^2\dd w\le\frac{1}{\tau c}\|F\|^2\le Cp^2\|F\|^2. \end{equation*} For the factorization, let $(x,\beta)$ with $\Psi^{\mathrm{out}}_\xi(x,\beta)\ne0$. Then $(x,\beta)\in\mathsf O$ and $x>0$, and $\beta/x=\tau cw^2$ with $w=w_\xi(x,\beta)0$. By \Cref{lem:uniq-multipliers}(i), $\Psi^{\mathrm{in}}_\xi(x',\beta')h_\xi(x',\beta')=\one_{\{x'>0\}}\chi(cx')\psi(w')^2p\,\omega_\xi(\beta')\sqrt{\beta'}/x'$, and therefore \begin{equation*} (S_\xi R_\xi\phi)(w)=\int\nu_\xi(\dd y)\,\one_{\{x'>0\}}\chi(cx')\int\varpi_\xi(\dd\beta')\,\psi(w')^2\,\frac{p\sqrt{\beta'}}{x'}\,\phi(w'). \end{equation*} We substitute $\beta'=\tau cx'u^2$ in the inner integral. Then $w'=u$, $\varpi_\xi(\dd\beta')=2cx'ue^{-cx'u^2}\dd u$ and $p\sqrt{\beta'}/x'=p(\tau c)^{1/2}u\,x'^{-1/2}$, so the inner integral equals \begin{align*} &\int_0^\infty\gamma(\dd u)\,u\,\psi(u)^2\,p\tau^{1/2}c^{3/2}x'^{1/2}e^{-(cx'-1)u^2}\phi(u)\\ &\qquad=\alpha\int_0^W\gamma(\dd u)\,u\,\psi(u)^2(cx')^{1/2}e^{-(cx'-1)u^2}\phi(u), \end{align*} where we used $p\tau^{1/2}c=\alpha$ from \eqref{eq:uniq-identities}. Since $cx'=cy-\tau c^2w^2=B-\theta w^2/p$ with $B=cy$, again by \eqref{eq:uniq-identities}, we obtain \eqref{eq:uniq-N} from Fubini's theorem. Finally, $R_\xi\one=h_\xi$ since $h_\xi$ vanishes where $w_\xi>W$. \end{proof} \subsection{The rank-one minorant} Let $\eta_p=C_\eta\log^3p/p^2$, where $C_\eta$ is the constant of \Cref{lem:uniq-compression}(ii) below, and define \begin{equation*} m_\xi(u)=\Upsilon^0_\xi(u)-\eta_p,\qquad\hat k_\xi(u)=\alpha\,u\,\psi(u)^2m_\xi(u),\qquad k_\xi(\phi)=\int_0^W\hat k_\xi(u)\phi(u)\,\gamma(\dd u), \end{equation*} and the operators $N^0_\xi=\one\otimes\hat k_\xi$ (so that $N^0_\xi\phi=k_\xi(\phi)\one$) and $\Delta_\xi=N_\xi-N^0_\xi$ on $\mathscr V$. We also set $\sigma_\xi=k_\xi(\one)$. \begin{lemma}\label{lem:uniq-compression} For $u,w\in[0,W]$ and $j\in\{1,2\}$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item $|\Upsilon^0_\xi(u)-1|\le Cp^{-1/4}\log p$; in particular, $m_\xi(u)\ge1/2$. \item There exists $C_\eta$ such that $|\Upsilon_\xi(w,u)-\Upsilon^0_\xi(u)|\le\eta_p$, so that $0\le\Upsilon_\xi(w,u)-m_\xi(u)\le2\eta_p$. Further, $|\partial_\xi^j(\Upsilon_\xi(w,u)-\Upsilon^0_\xi(u))|\le Cp^{-2}\log^Dp$. \item $|\partial_\xi^j\Upsilon^0_\xi(u)|\le C\epsilon_p\log^4p$, with $\epsilon_p$ as in \eqref{eq:uniq-weak-jets}. \item $\Delta_\xi\succeq0$, $N^0_\xi\succeq0$, and for $j\le2$, $\|\partial_\xi^j\Delta_\xi\|\le Cp^{-2}\log^Dp$, $\|\partial_\xi^jN_\xi\|\le C$ and $\|\partial_\xi^j\hat k_\xi\|_{\mathscr V}\le C$. \end{enumerate} \end{lemma} \begin{proof} We first record properties of $\Upsilon$. It vanishes unless $|b-1|0\}}. \end{equation} This is a smoothly truncated density of Gamma type whose threshold $t=0$ does not depend on $\xi$. \begin{lemma}\label{lem:uniq-rank-one} \begin{enumerate}[label=\textup{(\roman*)}] \item For almost every $(x,\beta)$, \begin{equation}\label{eq:uniq-ell} \ell_{\mathrm c}(x,\beta)=\Psi^{\mathrm{in}}_\xi(x,\beta)\,\omega_\xi(\beta)\int\nu_\xi(\dd y)\,q_\xi(y-x). \end{equation} In particular, $h_{\mathrm c}$ and $\ell_{\mathrm c}$ are nonnegative and vanish outside $\mathsf O$. \item The maps $\xi\mapsto h_{\mathrm c},\ell_{\mathrm c}$ are four times continuously differentiable into $\mathscr L$, and for $j\le4$, \begin{equation*} \|\partial_\xi^jh_{\mathrm c}\|\le C,\qquad\|\partial_\xi^j\ell_{\mathrm c}\|\le Cp^{2A+1}\log^Dp\le p^{A_1},\qquad\|\partial_\xi^jL_{\mathrm c}\|\le p^{A_1+1}. \end{equation*} \item There exist a nonempty open set $V_0\subset\mathsf O$ and $c_0>0$, both possibly depending on $p$ and $\xi$, such that $h_{\mathrm c}\ge c_0$ and $\ell_{\mathrm c}\ge c_0$ on $V_0$. \end{enumerate} \end{lemma} \begin{proof} For (i), let $F\in\mathscr L$. By the definition of $\ell_{\mathrm c}$ and \eqref{eq:uniq-S}, \begin{equation*} \langle\ell_{\mathrm c},F\rangle=\int\nu_\xi(\dd y)\int\mu_0(\dd\beta')\,\omega_\xi(\beta')\int_0^W\gamma(\dd w)\,\hat k_\xi(w)\,(\Psi^{\mathrm{in}}_\xi F)(y-\tau cw^2,\beta'). \end{equation*} For fixed $y$, we substitute $x=y-\tau cw^2$ in the inner integral. Then $w=((y-x)/(\tau c))^{1/2}$ and $\gamma(\dd w)=(\tau c)^{-1}e^{-(y-x)/(\tau c)}\dd x$ on $\{x0$, so that $q_\xi(t)=\alpha s_\xi^{-1}Q_\xi(t/s_\xi)\one_{\{t>0\}}$. We claim that for $i\le4$, $l\le4$ and $u\in[0,W]$, \begin{equation}\label{eq:uniq-m-derivatives} \bigl|(u\partial_u)^l\partial_\xi^im_\xi(u)\bigr|\le Cp^A\log^Dp. \end{equation} For $i=l=0$ this follows from (i) of \Cref{lem:uniq-compression}. Otherwise, the constant $\eta_p$ does not contribute, and $(u\partial_u)^l\partial_\xi^im_\xi(u)=\partial_\xi^i\int f(c(\xi)y)\,\nu_\xi(\dd y)$ with $f=(u\partial_u)^l\Upsilon(\cdot,u)$. By the Leibniz rule, this is a sum of terms $\int\partial_\xi^{i_2}[f(c(\xi)y)]\,(\partial_\xi^{i_1}\nu_\xi)(\dd y)$ with $i_1+i_2=i$. The function $f(c(\xi)y)$ vanishes unless $|y|\le2$, and on this set its $\xi$-derivatives of order $i_2$ are bounded by $C\log^Dp$, by \eqref{eq:uniq-Upsilon-derivatives} and \eqref{eq:uniq-parameter-derivatives}. Using $|\partial_\xi^{i_1}\nu_\xi|(\R)\le p^A$ from \ref{it:uniq-I5} (and $\nu_\xi(\R)=1$ for $i_1=0$), we find \eqref{eq:uniq-m-derivatives}. By induction on $i$, $\partial_\xi^iq_\xi(t)$ is $\alpha s_\xi^{-1}$ times a combination of the functions $((s\partial_s)^l\partial_\xi^{i'}Q_\xi)(t/s_\xi)$ with $l+i'\le i$, with coefficients that are polynomials in $\partial_\xi^{i''}\alpha/\alpha$ and $\partial_\xi^{i''}s_\xi/s_\xi$, which are bounded by \eqref{eq:uniq-parameter-derivatives}. Since $(s\partial_s)$ acts on $s^{1/2}e^{-s}$ by multiplication with $1/2-s$, and on a function of $s^{1/2}$ as $(u\partial_u)/2$ with $u=s^{1/2}$, the bounds \eqref{eq:uniq-cutoff-derivatives} and \eqref{eq:uniq-m-derivatives} give \begin{equation*} |\partial_\xi^iq_\xi(t)|\le Cp^A\log^Dp\,\frac{\alpha}{s_\xi}\Bigl(\frac{t}{s_\xi}\Bigr)^{1/2}e^{-t/s_\xi}\Bigl(1+\frac{t}{s_\xi}\Bigr)^4\one_{\{t>0\}},\qquad i\le4. \end{equation*} Integrating the square, we obtain $\|\partial_\xi^iq_\xi\|_{L^2(\R)}^2\le Cp^{2A}\log^{2D}p\,\alpha^2s_\xi^{-1}\le Cp^{2A+2}\log^{2D}p$, since $s_\xi^{-1}\le p^2\xi_+^2$. The supremum over $\xi\in\Xi$ of the right-hand side is square integrable, and dominated convergence shows that $\xi\mapsto q_\xi$ is four times continuously differentiable into $L^2(\R)$. Now let $\bar q_\xi=\nu_\xi\star q_\xi$, that is, $\bar q_\xi(x)=\int\nu_\xi(\dd y)q_\xi(y-x)$. As in \Cref{lem:uniq-realization}(ii), $\|\sigma\star f\|_{L^2}\le|\sigma|(\R)\|f\|_{L^2}$ for a finite signed measure $\sigma$. By \ref{it:uniq-I4} and the Leibniz rule, $\xi\mapsto \bar q_\xi$ is four times continuously differentiable into $L^2(\R)$, and by \ref{it:uniq-I5}, \begin{equation*} \|\partial_\xi^i\bar q_\xi\|_{L^2}\le\sum_{i_1+i_2=i}\binom{i}{i_1}|\partial_\xi^{i_1}\nu_\xi|(\R)\,\|\partial_\xi^{i_2}q_\xi\|_{L^2}\le Cp^{2A+1}\log^Dp,\qquad i\le4. \end{equation*} By \eqref{eq:uniq-ell}, $\ell_{\mathrm c}(x,\beta)=\Psi^{\mathrm{in}}_\xi(x,\beta)\omega_\xi(\beta)\bar q_\xi(x)$. By \Cref{lem:uniq-multipliers}(ii) and the proof of \Cref{lem:uniq-realization}(iv), $|\partial_\xi^i[\Psi^{\mathrm{in}}_\xi(x,\beta)\omega_\xi(\beta)]|\le C\log^4p\,\omega_\xi(\beta)(1+\beta/\tau)^i$ for $i\le4$. It follows that, for every $f\in L^2(\R)$, the function $\partial_\xi^i[\Psi^{\mathrm{in}}_\xi\omega_\xi]\,f$ has norm at most $C\log^4p\,\|f\|_{L^2}$ in $\mathscr L$. The Leibniz rule now gives the differentiability of $\xi\mapsto\ell_{\mathrm c}$ and the bound $\|\partial_\xi^j\ell_{\mathrm c}\|\le Cp^{2A+1}\log^Dp\le p^{A_1}$. Finally, $\partial_\xi^jL_{\mathrm c}=\sum_i\binom ji\partial_\xi^ih_{\mathrm c}\otimes\partial_\xi^{j-i}\ell_{\mathrm c}$ and $\|f\otimes g\|=\|f\|\|g\|$ give $\|\partial_\xi^jL_{\mathrm c}\|\le16Cp^{A_1}\le p^{A_1+1}$. For (iii), let $x_1=1-3\tau c/2$, and let $V_0=(x_1-\tau c/4,x_1+\tau c/4)\times(\beta_1/2,\beta_1)$ with $\beta_1=\tau$. For $(x,\beta)\in V_0$ and $p\ge p_1$, we have $|cx-1|\le\theta+2\tau c^2\le r/8$ and $w_\xi(x,\beta)^2\le2\le M_*\log p$. Then $\Psi^{\mathrm{in}}_\xi(x,\beta)=1$ and $\psi(w_\xi(x,\beta))=1$; also $(x,\beta)\in\mathsf O$. On $V_0$, the function $h_{\mathrm c}=p\omega_\xi(\beta)\sqrt\beta/x$ is bounded below by a positive constant. Further, the atom of $\nu_\xi$ at $1$ gives $\bar q_\xi(x)\ge e^{-p}q_\xi(1-x)$, and for $(x,\beta)\in V_0$ we have $(1-x)/(\tau c)\in[5/4,7/4]$, where $q_\xi(1-x)\ge\alpha(\tau c)^{-1}e^{-2}/2$ by \Cref{lem:uniq-compression}(i). We conclude that $\ell_{\mathrm c}$ is bounded below by a positive constant on $V_0$ as well. \end{proof} \begin{lemma}\label{lem:uniq-square} Define $G^{\mathrm{ord}}_\xi=B_\xi^2-L_{\mathrm c}$, the operator $L_{\mathrm t}=JL_{\mathrm c}^*J$, the vectors $h_{\mathrm t}=J\ell_{\mathrm c}$ and $\ell_{\mathrm t}=Jh_{\mathrm c}$, and \begin{equation}\label{eq:uniq-G} G_\xi=\mathsf A_\xi^2-L_{\mathrm c}-L_{\mathrm t}. \end{equation} \begin{enumerate}[label=\textup{(\roman*)}] \item We have $0\preceq L_{\mathrm c}\preceq B_\xi^2\preceq(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2\preceq P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}$. \item We have $G^{\mathrm{ord}}_\xi=R_\xi\Delta_\xi S_\xi\succeq0$, $G^{\mathrm{ord}}_\xi=P_{\mathsf O}G^{\mathrm{ord}}_\xi P_{\mathsf O}$, and $\|G^{\mathrm{ord}}_\xi\|\le Cp^{-1}\log^Dp$. The map $\xi\mapsto G^{\mathrm{ord}}_\xi$ is four times continuously differentiable, with $\|\partial_\xi^jG^{\mathrm{ord}}_\xi\|\le p^{2A_1+1}$ for $j\le4$. \item We have $L_{\mathrm t}=h_{\mathrm t}\otimes\ell_{\mathrm t}$, the vectors $h_{\mathrm t},\ell_{\mathrm t}$ are nonnegative and vanish outside $J\mathsf O$, and $0\preceq L_{\mathrm t}\preceq P_{J\mathsf O}\mathsf A_\xi^2P_{J\mathsf O}$. \item We have $G_\xi\succeq0$, $G_\xi^*=JG_\xi J$ and $\|G_\xi\|\le p^2$. The map $\xi\mapsto G_\xi$ is four times continuously differentiable, with $\|\partial_\xi^jG_\xi\|\le p^{2A_1+1}$ for $j\le4$. Finally, \begin{equation}\label{eq:uniq-square-decomposition} \mathsf A_\xi^2=G_\xi+h_{\mathrm c}\otimes\ell_{\mathrm c}+h_{\mathrm t}\otimes\ell_{\mathrm t}. \end{equation} \end{enumerate} \end{lemma} \begin{proof} For (i) and (ii), by \Cref{lem:uniq-factorization}, we have $B_\xi^2=R_\xi N_\xi S_\xi$ and $L_{\mathrm c}=R_\xi N^0_\xi S_\xi$. Then $G^{\mathrm{ord}}_\xi=R_\xi\Delta_\xi S_\xi$. Since $R_\xi,S_\xi,\Delta_\xi,N^0_\xi$ are positive (\Cref{lem:uniq-factorization,lem:uniq-compression}), we have $L_{\mathrm c}\succeq0$ and $G^{\mathrm{ord}}_\xi\succeq0$. By \Cref{lem:uniq-multipliers}(iii) and \Cref{lem:uniq-positive}(iii), $B_\xi^2\preceq(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2$, and $(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2=P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}\preceq P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}$ since $P_{\mathsf O}\preceq1$. The range of $R_\xi$ consists of functions vanishing outside $\mathsf O$, and $S_\xi F$ depends only on $F\one_{\mathsf O}$ since $\Psi^{\mathrm{in}}_\xi\le\one_{\mathsf O}$; this gives $G^{\mathrm{ord}}_\xi=P_{\mathsf O}G^{\mathrm{ord}}_\xi P_{\mathsf O}$. The norm bound follows from \Cref{lem:uniq-factorization} and \Cref{lem:uniq-compression}(iv): $\|R_\xi\Delta_\xi S_\xi\|\le C\cdot Cp^{-2}\log^Dp\cdot Cp$. For the derivatives, we do not differentiate $R_\xi$ and $S_\xi$ separately; instead, $G^{\mathrm{ord}}_\xi=B_\xi^2-L_{\mathrm c}$, and \Cref{lem:uniq-multipliers}(iii) and \Cref{lem:uniq-rank-one}(ii) give $\|\partial_\xi^jG^{\mathrm{ord}}_\xi\|\le Cp^{2A_1}\log^{16}p+p^{A_1+1}\le p^{2A_1+1}$. For (iii), we have $L_{\mathrm c}^*=\ell_{\mathrm c}\otimes h_{\mathrm c}$ and $L_{\mathrm t}F=\langle h_{\mathrm c},JF\rangle J\ell_{\mathrm c}=\langle Jh_{\mathrm c},F\rangle J\ell_{\mathrm c}$, which gives $L_{\mathrm t}=h_{\mathrm t}\otimes\ell_{\mathrm t}$. The support claim follows from \Cref{lem:uniq-rank-one}(i) and $JP_{\mathsf O}J=P_{J\mathsf O}$. Since $J\succeq0$, conjugation by $J$ and taking adjoints preserve the order $\preceq$ (\Cref{lem:uniq-positive}(ii),(iii)). Applying them to (i) and using $JP_{\mathsf O}J=P_{J\mathsf O}$ and $J(\mathsf A_\xi^*)^2J=(J\mathsf A_\xi^*J)^2=\mathsf A_\xi^2$ (\Cref{lem:uniq-realization}(iii)), we find that $L_{\mathrm t}\preceq JP_{\mathsf O}(\mathsf A_\xi^*)^2P_{\mathsf O}J=P_{J\mathsf O}\mathsf A_\xi^2P_{J\mathsf O}$. For (iv), \eqref{eq:uniq-square-decomposition} is the definition \eqref{eq:uniq-G}. Since the sets $\mathsf O$ and $J\mathsf O$ are disjoint up to a null set, \Cref{lem:uniq-positive}(v), applied to the partition \eqref{eq:uniq-blocks}, implies that $P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}+P_{J\mathsf O}\mathsf A_\xi^2P_{J\mathsf O}\preceq \mathsf A_\xi^2$. With (i) and (iii), we find that \begin{equation*} G_\xi=\bigl(P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}-L_{\mathrm c}\bigr)+\bigl(P_{J\mathsf O}\mathsf A_\xi^2P_{J\mathsf O}-L_{\mathrm t}\bigr)+\bigl(\mathsf A_\xi^2-P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}-P_{J\mathsf O}\mathsf A_\xi^2P_{J\mathsf O}\bigr)\succeq0. \end{equation*} Since $J\mathsf A_\xi^2J=(\mathsf A_\xi^*)^2$, $JL_{\mathrm c}J=L_{\mathrm t}^*$ and $JL_{\mathrm t}J=L_{\mathrm c}^*$, we have $JG_\xi J=G_\xi^*$. Since $0\preceq G_\xi\preceq \mathsf A_\xi^2$, \Cref{lem:uniq-positive}(i) gives $\|G_\xi\|\le\|\mathsf A_\xi\|^2\le p^2$. The derivative bound follows from \Cref{lem:uniq-realization}(iv), \Cref{lem:uniq-rank-one}(ii) and $\|\partial_\xi^jL_{\mathrm t}\|=\|\partial_\xi^jL_{\mathrm c}\|$: we have $\|\partial_\xi^jG_\xi\|\le16p^{2A_1}+2p^{A_1+1}\le p^{2A_1+1}$. \end{proof} \section{Cutoff bounds and a polynomial inverse}\label{sec:uniq-cutoffs} We continue to assume $p\ge p_1$ and $\xi\in\Xi$. The next lemma shows that, apart from transitions of size $p^{-N_*}$ or $e^{-a\sqrt p/2}$, the core region $\Omega_{\mathrm c}$ feeds only into the ordinary set $\mathsf O$, and the retained operator captures $P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}$. In the notation $P_X\mathsf A_\xi P_Y$, the set $X$ constrains the output state $(x,\beta)$ and $Y$ constrains the input state $(y-\beta/x,\beta')$ in \eqref{eq:uniq-A-explicit}. \begin{lemma}\label{lem:uniq-cuts} For $p\ge p_1$ and $\xi\in\Xi$, \begin{align} \|P_{\Omega_{\mathrm c}}\mathsf A_\xi P_{\mathsf U_{\mathrm c}}\|&\le p^{-N_*-2},\label{eq:uniq-cut1}\\ \|P_{\Omega_{\mathrm c}}\mathsf A_\xi P_{\Omega_{\mathrm t}}\|&\le e^{-a\sqrt p/2},\label{eq:uniq-cut2}\\ \|P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}-B_\xi\|\le p^{-N_*-3},\qquad\|(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2-B_\xi^2\|&\le p^{-N_*-2}.\label{eq:uniq-cut3} \end{align} \end{lemma} \begin{proof} We begin with a general bound. Let $\mathcal C=\{y:|cy-1|\le p^{-1/4}\}$; by \eqref{eq:uniq-core}, we have $\nu_\xi(\R\setminus\mathcal C)\le e^{-a\sqrt p}$. For thresholds $b_1,b_2>0$, let $P_{\{\beta>b\}}$ denote multiplication by $\one_{\{\beta>b\}}$, and let $\mathsf A_\xi^{\mathrm{exc}}=pJI_\xi\mathcal C_{\sigma}I_\xi^*$ with $\sigma$ the restriction of $\nu_\xi$ to $\R\setminus\mathcal C$. Since $J$ commutes with $P_{\{\beta>b\}}$ and $\|P_{\{\beta>b\}}I_\xi\|^2=\int_{\{\beta>b\}}\omega_\xi^2\dd\mu_0=e^{-b/\tau}$, we have \begin{equation}\label{eq:uniq-tails} \|P_{\{\beta>b_1\}}\mathsf A_\xi\|\le pe^{-b_1/(2\tau)},\qquad\|\mathsf A_\xi P_{\{\beta>b_2\}}\|\le pe^{-b_2/(2\tau)},\qquad\|\mathsf A_\xi^{\mathrm{exc}}\|\le pe^{-a\sqrt p}. \end{equation} Let $X,Y$ be measurable sets, and suppose that $(y-\beta/x,\beta')\notin Y$ whenever $(x,\beta)\in X$, $\beta\le b_1$, $y\in\mathcal C$ and $\beta'\le b_2$. Then \begin{equation*} \one_X(x,\beta)\one_Y(y-\beta/x,\beta')\le\one_{\{\beta>b_1\}}+\one_{\{\beta'>b_2\}}+\one_{\{y\notin\mathcal C\}}, \end{equation*} and \eqref{eq:uniq-A-explicit} gives $P_X\mathsf A_\xi P_Y\preceq P_{\{\beta>b_1\}}\mathsf A_\xi+\mathsf A_\xi P_{\{\beta>b_2\}}+\mathsf A_\xi^{\mathrm{exc}}$. By \Cref{lem:uniq-positive}(i) and \eqref{eq:uniq-tails}, \begin{equation}\label{eq:uniq-domination} \|P_X\mathsf A_\xi P_Y\|\le p\bigl(e^{-b_1/(2\tau)}+e^{-b_2/(2\tau)}+e^{-a\sqrt p}\bigr). \end{equation} For \eqref{eq:uniq-cut1}, take $X=\Omega_{\mathrm c}$, $Y=\mathsf U_{\mathrm c}$, $b_1=r^2/16$ and $b_2=\beta_{\max}$. If $(x,\beta)\in\Omega_{\mathrm c}$ and $\beta\le b_1$, then $|\beta/x|<\sqrt\beta\le r/4$. If $y\in\mathcal C$, then $|y-1|\le p^{-1/4}+\theta\le r/4$. Then $|y-\beta/x-1|\le r/2$, and if $\beta'\le\beta_{\max}$, the point $(y-\beta/x,\beta')$ lies in $\mathsf O$, which is disjoint from $\mathsf U_{\mathrm c}$. Since $1/\tau=p^2\xi^2$, we have $b_1/(2\tau)\ge p^2\xi_-^2/(32\log^2p)$ and $b_2/(2\tau)\ge2W^2=2(M_*+1)\log p$. Inserting these bounds into \eqref{eq:uniq-domination}, we obtain $\|P_{\Omega_{\mathrm c}}\mathsf A_\xi P_{\mathsf U_{\mathrm c}}\|\le2p^{-2M_*-1}\le p^{-N_*-2}$, for $p\ge p_1$. For \eqref{eq:uniq-cut2}, take $X=\Omega_{\mathrm c}$, $Y=\Omega_{\mathrm t}$ and $b_1=b_2=1/64$. If $(x,\beta)\in\Omega_{\mathrm c}$ and $\beta\le1/64$, then $|\beta/x|<1/8$; if $y\in\mathcal C$, then $y\ge3/4$; so $y-\beta/x\ge5/8>\sqrt{\beta'}$ for $\beta'\le1/64$, and $(y-\beta/x,\beta')\in\Omega_{\mathrm c}$. Since $1/(128\tau)=p^2\xi^2/128$, the bound \eqref{eq:uniq-cut2} follows from \eqref{eq:uniq-domination} for $p\ge p_1$. For \eqref{eq:uniq-cut3}, write $D_1=P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}-B_\xi=(P_{\mathsf O}-\Psi^{\mathrm{out}}_\xi)\mathsf A_\xi P_{\mathsf O}+\Psi^{\mathrm{out}}_\xi \mathsf A_\xi(P_{\mathsf O}-\Psi^{\mathrm{in}}_\xi)$, a sum of positive operators. Since $x\ge1/2$ and $c\ge1$ on $\mathsf O$, we have $1-\psi(w_\xi(x,\beta))\le\one_{\{\beta>\tau M_*\log p/2\}}$. Hence $(P_{\mathsf O}-\Psi^{\mathrm{out}}_\xi)\mathsf A_\xi P_{\mathsf O}\preceq P_{\{\beta>\tau M_*\log p/2\}}\mathsf A_\xi$, which has norm at most $p^{1-M_*/4}$ by \eqref{eq:uniq-tails}. For the second term, $\one_{\mathsf O}(1-\chi(cx')\psi(w'))\le\one_{\mathsf O}(1-\chi(cx'))+(1-\psi(w'))\one_{\mathsf O}$ at the input state $(x',\beta')$, and the second summand is at most $\one_{\{\beta'>\tau M_*\log p/2\}}$. If the output state $(x,\beta)$ lies in $\mathsf O$ and $y\in\mathcal C$, then $c\beta/x\le4\beta_{\max}$ and $|cx'-1|\le|cy-1|+c\beta/x\le p^{-1/4}+4\beta_{\max}\le r/8$, and therefore $\chi(cx')=1$. This shows that $\Psi^{\mathrm{out}}_\xi \mathsf A_\xi(P_{\mathsf O}-\Psi^{\mathrm{in}}_\xi)\preceq \mathsf A_\xi P_{\{\beta>\tau M_*\log p/2\}}+\mathsf A_\xi^{\mathrm{exc}}$. Combining these bounds with \eqref{eq:uniq-tails} and \Cref{lem:uniq-positive}(i), we obtain $\|D_1\|\le2p^{1-M_*/4}+pe^{-a\sqrt p}=2p^{-N_*-4}+pe^{-a\sqrt p}\le p^{-N_*-3}/2$ for $p\ge p_1$, since $M_*/4=N_*+5$. Finally, $(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2-B_\xi^2=D_1P_{\mathsf O}\mathsf A_\xi P_{\mathsf O}+B_\xi D_1$, and $\|B_\xi\|\le\|\mathsf A_\xi\|\le p$, so its norm is at most $2p\|D_1\|\le p^{-N_*-2}$. \end{proof} We now bound the blocks of $G_\xi$ with respect to the partition \eqref{eq:uniq-blocks}. We write $G_{ij}=P_{X_i}G_\xi P_{X_j}$ for $i,j\in\{1,2,3,4\}$, and set \begin{equation*} E_{\mathsf O}=G_{44}-G^{\mathrm{ord}}_\xi,\qquad d_p=\|G_{44}\|. \end{equation*} \begin{lemma}\label{lem:uniq-block-bounds} For $p\ge p_1$ and $\xi\in\Xi$, the following hold. \begin{enumerate}[label=\textup{(\roman*)}] \item $\|G_{ij}\|\le p^2$ for all $i,j$. \item $\|G_{43}\|+\|G_{33}\|+\|G_{22}\|+\|G_{21}\|\le8p^{-N_*-1}$. \item $\|G_{ij}\|\le2pe^{-a\sqrt p/2}$ for $i\in\{3,4\}$ and $j\in\{1,2\}$. \item $E_{\mathsf O}\succeq0$, $\|E_{\mathsf O}\|\le3p^{-N_*-1}$, $d_p\le Cp^{-1}\log^Dp$, and $G_{11}=JG_{44}^*J$, so that $\|G_{11}\|=d_p$. \end{enumerate} \end{lemma} \begin{proof} Part (i) follows from $\|G_\xi\|\le p^2$. The rank-one operators $L_{\mathrm c}$ and $L_{\mathrm t}$ have nonzero blocks only at $(4,4)$ and $(1,1)$, respectively. Then $G_{ij}=P_{X_i}\mathsf A_\xi^2P_{X_j}$ for $(i,j)\notin\{(1,1),(4,4)\}$. For any sets $X,Y$, we have $P_X\mathsf A_\xi^2P_Y=P_X\mathsf A_\xi P_{\Omega_{\mathrm c}}\mathsf A_\xi P_Y+P_X\mathsf A_\xi P_{\Omega_{\mathrm t}}\mathsf A_\xi P_Y$. For (ii), let $X\subset\Omega_{\mathrm c}$. By \eqref{eq:uniq-cut1} and \eqref{eq:uniq-cut2}, we have $\|P_X\mathsf A_\xi^2P_{\mathsf U_{\mathrm c}}\|\le\|\mathsf A_\xi\|\,\|P_{\Omega_{\mathrm c}}\mathsf A_\xi P_{\mathsf U_{\mathrm c}}\|+\|P_{\Omega_{\mathrm c}}\mathsf A_\xi P_{\Omega_{\mathrm t}}\|\,\|\mathsf A_\xi\|\le2p^{-N_*-1}$. This bounds $G_{43}$ and $G_{33}$. For the other two blocks, we use the symmetry $G_\xi^*=JG_\xi J$: for any sets $X,Y$, \begin{equation}\label{eq:uniq-block-symmetry} P_{JX}G_\xi P_{JY}=JP_XJG_\xi JP_YJ=JP_XG_\xi^*P_YJ=J(P_YG_\xi P_X)^*J. \end{equation} Since $X_2=JX_3$ and $X_1=JX_4$, this gives $\|G_{22}\|=\|G_{33}\|$ and $\|G_{21}\|=\|G_{43}\|$. For (iii), let $X\subset\Omega_{\mathrm c}$ and $Y\subset\Omega_{\mathrm t}$. In the decomposition of $P_X\mathsf A_\xi^2P_Y$ above, the first term contains $P_{\Omega_{\mathrm c}}\mathsf A_\xi P_Y$ and the second contains $P_X\mathsf A_\xi P_{\Omega_{\mathrm t}}$. With \eqref{eq:uniq-cut2} and $\|\mathsf A_\xi\|\le p$, we obtain $\|P_X\mathsf A_\xi^2P_Y\|\le2pe^{-a\sqrt p/2}$. For (iv), we have \begin{equation*} G_{44}=P_{\mathsf O}\mathsf A_\xi^2P_{\mathsf O}-L_{\mathrm c}=(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2+P_{\mathsf O}\mathsf A_\xi P_{\mathsf U_{\mathrm c}}\mathsf A_\xi P_{\mathsf O}+P_{\mathsf O}\mathsf A_\xi P_{\Omega_{\mathrm t}}\mathsf A_\xi P_{\mathsf O}-L_{\mathrm c}, \end{equation*} so $E_{\mathsf O}=[(P_{\mathsf O}\mathsf A_\xi P_{\mathsf O})^2-B_\xi^2]+P_{\mathsf O}\mathsf A_\xi P_{\mathsf U_{\mathrm c}}\mathsf A_\xi P_{\mathsf O}+P_{\mathsf O}\mathsf A_\xi P_{\Omega_{\mathrm t}}\mathsf A_\xi P_{\mathsf O}$. Each term is positive (the first by \Cref{lem:uniq-square}(i)), and \Cref{lem:uniq-cuts} bounds the norms by $p^{-N_*-2}$, $p\cdot p^{-N_*-2}$ and $pe^{-a\sqrt p/2}$, respectively. With \Cref{lem:uniq-square}(ii), this gives $d_p\le\|G^{\mathrm{ord}}_\xi\|+\|E_{\mathsf O}\|\le Cp^{-1}\log^Dp$. The identity $G_{11}=JG_{44}^*J$ is \eqref{eq:uniq-block-symmetry} with $X=Y=\mathsf O$. \end{proof} \begin{lemma}\label{lem:uniq-resolvent} For $p\ge p_1$ and $\xi\in\Xi$, we have $\|G_\xi^{n_0}\|\le2^{-2n_0}$; in particular, $r(G_\xi)\le1/4$. For $\lambda\in[1/2,2]$, the operator $(\lambda-G_\xi)^{-1}$ is positive, \begin{equation}\label{eq:uniq-resolvent} \|(\lambda-G_\xi)^{-1}\|\le p^{2n_0},\qquad\|\partial_\xi^j(\lambda-G_\xi)^{-1}\|\le p^{A_2}\quad(j\le4), \end{equation} and $\|(\lambda-G')^{-1}\|\le p^{2n_0}$ for every operator $G'$ with $0\preceq G'\preceq G_\xi$. \end{lemma} \begin{proof} Order the blocks as $X_1,X_2,X_3,X_4$ (that is, $J\mathsf O$, $\mathsf U_{\mathrm t}$, $\mathsf U_{\mathrm c}$, $\mathsf O$), and decompose $G_\xi=G^{\mathrm{fw}}+G^{\mathrm{dg}}+G^{\mathrm{sm}}$, where \begin{equation*} G^{\mathrm{fw}}=\sum_{ij}G_{ij}+G_{22}+G_{33}. \end{equation*} The blocks with $i>j$ are $(2,1)$, $(4,3)$ and the four blocks of \Cref{lem:uniq-block-bounds}(iii). By \Cref{lem:uniq-block-bounds}, we have \begin{equation*} \|G^{\mathrm{fw}}\|\le6p^2,\qquad\|G^{\mathrm{dg}}\|\le d_p,\qquad\|G^{\mathrm{sm}}\|\le8p^{-N_*-1}+8pe^{-a\sqrt p/2}\le p^{-N_*}. \end{equation*} We expand $G_\xi^{n_0}$ as a sum of $3^{n_0}$ products of $n_0$ factors, each factor being $G^{\mathrm{fw}}$, $G^{\mathrm{dg}}$ or $G^{\mathrm{sm}}$. A product that contains at least one factor $G^{\mathrm{sm}}$ has norm at most $(7p^2)^{n_0-1}p^{-N_*}$. Consider a product of factors $G^{\mathrm{fw}}$ and $G^{\mathrm{dg}}$ only, and expand each factor into its blocks. A product of blocks $G_{i_1j_1}G_{i_2j_2}\cdots G_{i_{n_0}j_{n_0}}$ vanishes unless $j_k=i_{k+1}$ for each $k$. For the blocks of $G^{\mathrm{fw}}$ and $G^{\mathrm{dg}}$, we have $i_k\le j_k$, with strict inequality for the blocks of $G^{\mathrm{fw}}$. Then the indices $i_1\le j_1=i_2\le\dots$ are nondecreasing in $\{1,2,3,4\}$ and increase strictly at each factor from $G^{\mathrm{fw}}$. A nonvanishing product therefore contains at most three such factors. In other words, a product of factors $G^{\mathrm{fw}}$ and $G^{\mathrm{dg}}$ with at least four factors $G^{\mathrm{fw}}$ vanishes, and every other such product has norm at most $(6p^2)^3d_p^{n_0-3}$, using $d_p\le1$. Summing over the $3^{n_0}$ products, we obtain \begin{equation*} \|G_\xi^{n_0}\|\le3^{n_0}\bigl(216p^6d_p^{n_0-3}+(7p^2)^{n_0-1}p^{-N_*}\bigr)\le C\bigl(p^{6-9}\log^{9D}p+p^{2n_0-N_*}\bigr)\le2^{-2n_0}, \end{equation*} for $p\ge p_1$, where we used $n_0=12$, \Cref{lem:uniq-block-bounds}(iv) and $N_*>2n_0+10$. By the spectral radius formula, we have $r(G_\xi)\le\|G_\xi^{n_0}\|^{1/n_0}\le1/4$. Let $\lambda\in[1/2,2]$. By \Cref{lem:uniq-positive}(iv), we have $(\lambda-G_\xi)^{-1}\succeq0$. Since $\|G_\xi^{n_0}\|\le\lambda^{n_0}/2$, the operator $\lambda^{n_0}-G_\xi^{n_0}$ is invertible with $\|(\lambda^{n_0}-G_\xi^{n_0})^{-1}\|\le2\lambda^{-n_0}\le2^{n_0+1}$, and the identity \begin{equation*} (\lambda-G_\xi)^{-1}=\bigl(\lambda^{n_0-1}+\lambda^{n_0-2}G_\xi+\dots+G_\xi^{n_0-1}\bigr)(\lambda^{n_0}-G_\xi^{n_0})^{-1} \end{equation*} and $\|G_\xi\|\le p^2$ give $\|(\lambda-G_\xi)^{-1}\|\le n_02^{2n_0+1}p^{2n_0-2}\le p^{2n_0}$. The map $\xi\mapsto(\lambda-G_\xi)^{-1}$ is four times continuously differentiable, since $\xi\mapsto G_\xi$ is (\Cref{lem:uniq-square}(iv)). Its $j$th derivative is a sum of at most $C$ products of at most $j+1$ factors $(\lambda-G_\xi)^{-1}$ and at most $j$ factors $\partial_\xi^{i}G_\xi$ with total order $j$; for instance, $\partial_\xi(\lambda-G_\xi)^{-1}=(\lambda-G_\xi)^{-1}(\partial_\xi G_\xi)(\lambda-G_\xi)^{-1}$. With \Cref{lem:uniq-square}(iv), we find that $\|\partial_\xi^j(\lambda-G_\xi)^{-1}\|\le Cp^{2n_0(j+1)+j(2A_1+1)}\le p^{A_2}$ for $j\le4$. Finally, if $0\preceq G'\preceq G_\xi$, then $r(G')\le r(G_\xi)$ and $0\preceq(\lambda-G')^{-1}\preceq(\lambda-G_\xi)^{-1}$ by \Cref{lem:uniq-positive}(i),(iv), which implies that $\|(\lambda-G')^{-1}\|\le p^{2n_0}$. \end{proof} \section{The renewal matrix}\label{sec:uniq-matrix} For $\xi\in\Xi$ and $\lambda\in[1/2,2]$, we define the renewal matrix \begin{equation}\label{eq:uniq-T} T_{ab}(\xi,\lambda)=\lambda\bigl\langle\ell_a,(\lambda-G_\xi)^{-1}h_b\bigr\rangle,\qquad a,b\in\{\mathrm c,\mathrm t\}, \end{equation} with $h_{\mathrm c},\ell_{\mathrm c}$ from \Cref{lem:uniq-rank-one} and $h_{\mathrm t},\ell_{\mathrm t}$ from \Cref{lem:uniq-square}. This is the matrix \eqref{eq:uniq-abstract-T} for the decomposition \eqref{eq:uniq-square-decomposition}, with the index $1$ for the core component $\mathrm c$ and $2$ for the tail component $\mathrm t$. \begin{lemma}\label{lem:uniq-matrix} Let $p\ge p_1$, $\xi\in\Xi$ and $\lambda\in[1/2,2]$. \begin{enumerate}[label=\textup{(\roman*)}] \item $T_{\mathrm{cc}}(\xi,\lambda)=T_{\mathrm{tt}}(\xi,\lambda)$. \item $T_{\mathrm{ct}}(\xi,\lambda)>0$ and $T_{\mathrm{tc}}(\xi,\lambda)>0$. \item The entries of $T$ are four times continuously differentiable in $\xi$, and $|\partial_\xi^jT_{ab}|\le p^{A_{\mathrm j}}$ for $j\le4$ and all $a,b$. \item $T_{\mathrm{ct}}(\xi,\lambda)\le e^{-a\sqrt p/4}$. \end{enumerate} \end{lemma} \begin{proof} For (i), we use $J\ell_{\mathrm c}=h_{\mathrm t}$, $Jh_{\mathrm c}=\ell_{\mathrm t}$ and $J(\lambda-G_\xi)^{-1}J=(\lambda-JG_\xi J)^{-1}=(\lambda-G_\xi^*)^{-1}$, which give \begin{equation*} T_{\mathrm{tt}}=\lambda\langle Jh_{\mathrm c},(\lambda-G_\xi)^{-1}J\ell_{\mathrm c}\rangle=\lambda\langle h_{\mathrm c},(\lambda-G_\xi^*)^{-1}\ell_{\mathrm c}\rangle=\lambda\langle(\lambda-G_\xi)^{-1}h_{\mathrm c},\ell_{\mathrm c}\rangle=T_{\mathrm{cc}}. \end{equation*} For (ii), we first claim that if $V\subset(\R\setminus\{0\})\times(0,\infty)$ is a nonempty bounded open set, then $\mathsf A_\xi\one_V$ has a representative that is lower semicontinuous and positive at every point $(x,\beta)$ with $x\ne0$; in particular, $\mathsf A_\xi\one_V\ge c_1\one_{V'}$ for some nonempty bounded open set $V'$ and some $c_1>0$. Indeed, by \eqref{eq:uniq-A-explicit}, \begin{equation*} (\mathsf A_\xi\one_V)(x,\beta)=p\,\omega_\xi(\beta)\frac{\sqrt\beta}{|x|}\int\nu_\xi(\dd y)\int\mu_0(\dd\beta')\,\omega_\xi(\beta')\one_V\Bigl(y-\frac\beta x,\beta'\Bigr). \end{equation*} For fixed $(x,\beta)$, the set of $(y,\beta')$ with $(y-\beta/x,\beta')\in V$ is open and nonempty. It has positive $\nu_\xi\otimes\mu_0$ measure, since $\nu_\xi$ has full support by \ref{it:uniq-I1} and $\mu_0$ has a positive density. For fixed $(y,\beta')$, the function $(x,\beta)\mapsto\one_V(y-\beta/x,\beta')$ is lower semicontinuous on $\{x\ne0\}$, and Fatou's lemma implies that the integral is lower semicontinuous, and the prefactor is continuous and positive. A positive lower semicontinuous function is bounded below by a positive constant on a neighborhood of each point, which proves the claim. Now let $V_0$ and $c_0$ be as in \Cref{lem:uniq-rank-one}(iii). Since $(x,\beta)\mapsto(\beta/x,\beta)$ is a homeomorphism of $(\R\setminus\{0\})\times(0,\infty)$, the set $JV_0$ is open, and $J\ell_{\mathrm c}\ge c_0J\one_{V_0}\ge c_2\one_{V_1}$ for a nonempty open set $V_1\subset JV_0$ and some $c_2>0$. By the claim applied to $V_1$, there exist a nonempty open set $V'$ and $c_1>0$ with $\mathsf A_\xi\one_{V_1}\ge c_1\one_{V'}$. Then $\mathsf A_\xi^2J\ell_{\mathrm c}\ge c_2c_1\mathsf A_\xi\one_{V'}$, and $\mathsf A_\xi\one_{V'}>0$ almost everywhere by the claim applied to $V'$. Since $\ell_{\mathrm c}\ge c_0\one_{V_0}$ and $V_0$ has positive measure, we obtain $\langle\ell_{\mathrm c},\mathsf A_\xi^2J\ell_{\mathrm c}\rangle>0$. Since $\ell_{\mathrm c}$ vanishes outside $\mathsf O$ and $J\ell_{\mathrm c}$ vanishes outside $J\mathsf O$, we have $\langle\ell_{\mathrm c},J\ell_{\mathrm c}\rangle=0$, and so \begin{equation*} \langle\ell_{\mathrm c},L_{\mathrm c}J\ell_{\mathrm c}\rangle=\langle\ell_{\mathrm c},J\ell_{\mathrm c}\rangle\langle\ell_{\mathrm c},h_{\mathrm c}\rangle=0,\qquad\langle\ell_{\mathrm c},L_{\mathrm t}J\ell_{\mathrm c}\rangle=\langle\ell_{\mathrm c},h_{\mathrm t}\rangle\langle Jh_{\mathrm c},J\ell_{\mathrm c}\rangle=\langle\ell_{\mathrm c},J\ell_{\mathrm c}\rangle\langle h_{\mathrm c},\ell_{\mathrm c}\rangle=0. \end{equation*} Then $\langle\ell_{\mathrm c},G_\xi h_{\mathrm t}\rangle=\langle\ell_{\mathrm c},\mathsf A_\xi^2J\ell_{\mathrm c}\rangle>0$. By \eqref{eq:uniq-T-series}, every term of the series for $T_{\mathrm{ct}}$ is nonnegative, and its term with $n=1$ is $\lambda^{-1}\langle\ell_{\mathrm c},G_\xi h_{\mathrm t}\rangle>0$. The argument for $T_{\mathrm{tc}}$ is the same, with $\langle\ell_{\mathrm t},G_\xi h_{\mathrm c}\rangle=\langle Jh_{\mathrm c},\mathsf A_\xi^2h_{\mathrm c}\rangle$ and the lower bound $h_{\mathrm c}\ge c_0\one_{V_0}$. For (iii), the entries are compositions of the four times continuously differentiable maps $\xi\mapsto\ell_a,h_b,(\lambda-G_\xi)^{-1}$. By the Leibniz rule, \Cref{lem:uniq-rank-one}(ii), $\|\partial_\xi^jh_{\mathrm t}\|=\|\partial_\xi^j\ell_{\mathrm c}\|$, $\|\partial_\xi^j\ell_{\mathrm t}\|=\|\partial_\xi^jh_{\mathrm c}\|$ and \eqref{eq:uniq-resolvent}, we have $|\partial_\xi^jT_{ab}|\le Cp^{2A_1+A_2}\le p^{A_{\mathrm j}}$. For (iv), since $\ell_{\mathrm c}$ vanishes outside $\Omega_{\mathrm c}$ and $h_{\mathrm t}$ vanishes outside $\Omega_{\mathrm t}$, we have $T_{\mathrm{ct}}=\lambda\langle\ell_{\mathrm c},P_{\Omega_{\mathrm c}}(\lambda-G_\xi)^{-1}P_{\Omega_{\mathrm t}}h_{\mathrm t}\rangle$. Let $G'=G_\xi-P_{\Omega_{\mathrm c}}G_\xi P_{\Omega_{\mathrm t}}$. By \Cref{lem:uniq-positive}(v), $0\preceq G'\preceq G_\xi$. Further, $P_{\Omega_{\mathrm c}}G'P_{\Omega_{\mathrm t}}=0$, which gives $P_{\Omega_{\mathrm c}}G'=P_{\Omega_{\mathrm c}}G'P_{\Omega_{\mathrm c}}$; by induction, $P_{\Omega_{\mathrm c}}(G')^nP_{\Omega_{\mathrm t}}=0$ for all $n\ge0$; the Neumann series gives $P_{\Omega_{\mathrm c}}(\lambda-G')^{-1}P_{\Omega_{\mathrm t}}=0$. By the resolvent identity, we then have \begin{equation*} P_{\Omega_{\mathrm c}}(\lambda-G_\xi)^{-1}P_{\Omega_{\mathrm t}}=P_{\Omega_{\mathrm c}}(\lambda-G')^{-1}\bigl(P_{\Omega_{\mathrm c}}G_\xi P_{\Omega_{\mathrm t}}\bigr)(\lambda-G_\xi)^{-1}P_{\Omega_{\mathrm t}}. \end{equation*} By \Cref{lem:uniq-block-bounds}(iii), $\|P_{\Omega_{\mathrm c}}G_\xi P_{\Omega_{\mathrm t}}\|\le8pe^{-a\sqrt p/2}$, and \Cref{lem:uniq-resolvent} bounds both resolvents by $p^{2n_0}$. With $\|\ell_{\mathrm c}\|=\|h_{\mathrm t}\|\le p^{A_1}$, we obtain $T_{\mathrm{ct}}\le16p^{2A_1+4n_0+1}e^{-a\sqrt p/2}\le e^{-a\sqrt p/4}$ for $p\ge p_1$. \end{proof} \subsection{The slope of the diagonal} Let $\Xi_1\subset\Xi$ be a compact interval whose distance to $\R\setminus\Xi$ is at least $\delta_0>0$. In the following two lemmas, the threshold $p_1$ may also depend on $\delta_0$. \begin{lemma}\label{lem:uniq-diagonal} We have \begin{equation*} \lim_{p\to\infty}\sup_{\xi\in\Xi_1,\,\lambda\in[1/2,2]}\Bigl|\partial_\xi T_{\mathrm{cc}}(\xi,\lambda)+\frac{2\mustar^2}{\xi^3}\Bigr|=0. \end{equation*} \end{lemma} \begin{proof} We compare $T_{\mathrm{cc}}$ with the entry obtained by keeping only the ordinary part $G^{\mathrm{ord}}_\xi$ of the remainder, \begin{equation*} a^{\mathrm{ord}}(\xi,\lambda)=\lambda\bigl\langle\ell_{\mathrm c},(\lambda-G^{\mathrm{ord}}_\xi)^{-1}h_{\mathrm c}\bigr\rangle, \end{equation*} which is well defined since $\|G^{\mathrm{ord}}_\xi\|\le1/4$ for $p\ge p_1$. The slope of $a^{\mathrm{ord}}$ is computed from the factorization of \Cref{lem:uniq-factorization}, and the difference $T_{\mathrm{cc}}-a^{\mathrm{ord}}$ is small together with its first derivative. \proofstep{1}{A series for $a^{\mathrm{ord}}$.} We claim that \begin{equation}\label{eq:uniq-aord-series} a^{\mathrm{ord}}(\xi,\lambda)=\sum_{n\ge0}\lambda^{-n}k_\xi\bigl(N_\xi(\Delta_\xi N_\xi)^n\one\bigr). \end{equation} Indeed, $\lambda(\lambda-G^{\mathrm{ord}}_\xi)^{-1}=\sum_{n\ge0}\lambda^{-n}(R_\xi\Delta_\xi S_\xi)^n$ in norm, and since $\langle\ell_{\mathrm c},F\rangle=k_\xi(S_\xi F)$ and $h_{\mathrm c}=R_\xi\one$, we have \begin{equation*} \bigl\langle\ell_{\mathrm c},(R_\xi\Delta_\xi S_\xi)^nh_{\mathrm c}\bigr\rangle=k_\xi\bigl(S_\xi R_\xi(\Delta_\xi S_\xi R_\xi)^n\one\bigr)=k_\xi\bigl(N_\xi(\Delta_\xi N_\xi)^n\one\bigr). \end{equation*} The series \eqref{eq:uniq-aord-series} converges together with its first $\xi$-derivative, since $\|\Delta_\xi N_\xi\|\le Cp^{-2}\log^Dp$ by \Cref{lem:uniq-compression}(iv). This representation involves only the operators $N_\xi$, $\Delta_\xi$ and the functional $k_\xi$ on $\mathscr V$, whose first two derivatives are controlled by \Cref{lem:uniq-compression}; it does not require differentiating $R_\xi$ or $S_\xi$. \proofstep{2}{The slope of $a^{\mathrm{ord}}$.} We claim that $\partial_\xi a^{\mathrm{ord}}(\xi,\lambda)=-2\mustar^2\xi^{-3}+o(1)$ uniformly in $\xi\in\Xi$ and $\lambda\in[1/2,2]$. Since $N_\xi=N^0_\xi+\Delta_\xi$ and $N^0_\xi\one=\sigma_\xi\one$, the term with $n=0$ in \eqref{eq:uniq-aord-series} is $k_\xi(N_\xi\one)=\sigma_\xi^2+k_\xi(\Delta_\xi\one)$. Every other term, and $k_\xi(\Delta_\xi\one)$, contains at least one factor $\Delta_\xi$. Differentiating such a product of $2n+2$ factors by the Leibniz rule, and using \Cref{lem:uniq-compression}(iv) and $\lambda^{-n}\le2^n$, we find that \begin{equation*} \bigl|\partial_\xi\bigl(a^{\mathrm{ord}}-\sigma_\xi^2\bigr)\bigr|\le\sum_{n\ge0}(2n+2)2^nC^{2n+2}\bigl(Cp^{-2}\log^Dp\bigr)^{\max\{n,1\}}\le Cp^{-2}\log^Dp. \end{equation*} It remains to differentiate $\sigma_\xi^2$. By definition, $\sigma_\xi=\alpha\int\gamma(\dd u)\,u\,\psi(u)^2m_\xi(u)$. By \eqref{eq:uniq-gamma-moment}, \Cref{lem:uniq-compression}(i), and $1-\psi^2\le\one_{\{u^2>M_*\log p\}}$, \begin{equation*} |\sigma_\xi-\alpha\mustar|\le\alpha\Bigl(\mustar\sup_{u\le W}|m_\xi(u)-1|+\int_{u^2>M_*\log p}u\,\gamma(\dd u)\Bigr)\le Cp^{-1/4}\log p. \end{equation*} Similarly, $\partial_\xi\sigma_\xi=(\partial_\xi\alpha)\int\gamma(\dd u)u\psi^2m_\xi+\alpha\int\gamma(\dd u)u\psi^2\partial_\xi\Upsilon^0_\xi$, and \Cref{lem:uniq-compression}(iii) implies that $|\partial_\xi\sigma_\xi-\mustar\partial_\xi\alpha|\le C(p^{-1/4}\log p+\epsilon_p\log^4p)$. By \eqref{eq:uniq-parameter-derivatives}, $\alpha=\xi^{-1}+O(p^{-1})$ and $\partial_\xi\alpha=-\xi^{-2}+O(p^{-1})$. Then \begin{equation*} \partial_\xi(\sigma_\xi^2)=2\sigma_\xi\partial_\xi\sigma_\xi=2\mustar^2\alpha\,\partial_\xi\alpha+o(1)=-\frac{2\mustar^2}{\xi^3}+o(1), \end{equation*} since $\epsilon_p\log^4p\to0$. This proves the claim. \proofstep{3}{The size of $T_{\mathrm{cc}}-a^{\mathrm{ord}}$.} We claim that $|T_{\mathrm{cc}}-a^{\mathrm{ord}}|\le p^{A_{\mathrm v}-N_*}$ on $\Xi\times[1/2,2]$. Since $\ell_{\mathrm c}$ and $h_{\mathrm c}$ vanish outside $\mathsf O$, we have $T_{\mathrm{cc}}=\lambda\langle\ell_{\mathrm c},P_{\mathsf O}(\lambda-G_\xi)^{-1}P_{\mathsf O}h_{\mathrm c}\rangle$. Let $G_1=G_\xi-P_{\mathsf O}G_\xi P_{\mathsf O^c}$, where $\mathsf O^c$ is the complement of $\mathsf O$. As in the proof of \Cref{lem:uniq-matrix}(iv), $0\preceq G_1\preceq G_\xi$ and $P_{\mathsf O}G_1=G_{44}P_{\mathsf O}$. By the Neumann series, $P_{\mathsf O}(\lambda-G_1)^{-1}P_{\mathsf O}=(\lambda-G_{44})^{-1}P_{\mathsf O}$, where $(\lambda-G_{44})^{-1}$ exists since $\|G_{44}\|=d_p\le1/4$. By the resolvent identity, we have \begin{equation*} P_{\mathsf O}\bigl[(\lambda-G_\xi)^{-1}-(\lambda-G_1)^{-1}\bigr]P_{\mathsf O}=P_{\mathsf O}(\lambda-G_1)^{-1}\bigl(P_{\mathsf O}G_\xi P_{\mathsf O^c}\bigr)(\lambda-G_\xi)^{-1}P_{\mathsf O}. \end{equation*} By \Cref{lem:uniq-block-bounds}(ii),(iii), $\|P_{\mathsf O}G_\xi P_{\mathsf O^c}\|\le\|G_{43}\|+\|G_{42}\|+\|G_{41}\|\le9p^{-N_*-1}$, and \Cref{lem:uniq-resolvent} bounds the two resolvents by $p^{2n_0}$. Further, since $G_{44}=G^{\mathrm{ord}}_\xi+E_{\mathsf O}$, the difference \begin{equation*} (\lambda-G_{44})^{-1}-(\lambda-G^{\mathrm{ord}}_\xi)^{-1}=(\lambda-G_{44})^{-1}E_{\mathsf O}(\lambda-G^{\mathrm{ord}}_\xi)^{-1} \end{equation*} has norm at most $16\|E_{\mathsf O}\|\le48p^{-N_*-1}$, by \Cref{lem:uniq-block-bounds}(iv) and since both resolvents have norm at most $4$. Combining these bounds with $\|\ell_{\mathrm c}\|\,\|h_{\mathrm c}\|\le Cp^{A_1}$, we conclude that $|T_{\mathrm{cc}}-a^{\mathrm{ord}}|\le Cp^{A_1}(9p^{4n_0-N_*-1}+48p^{-N_*-1})\le p^{A_{\mathrm v}-N_*}$. \proofstep{4}{Interpolation.} Fix $\lambda$ and let $f(\xi)=T_{\mathrm{cc}}(\xi,\lambda)-a^{\mathrm{ord}}(\xi,\lambda)$. By Step 3, $\sup_\Xi|f|\le p^{A_{\mathrm v}-N_*}$. By \Cref{lem:uniq-matrix}(iii), $|\partial_\xi^4T_{\mathrm{cc}}|\le p^{A_{\mathrm j}}$. The same bound holds for $a^{\mathrm{ord}}$: by \Cref{lem:uniq-square}(ii), the derivatives of $(\lambda-G^{\mathrm{ord}}_\xi)^{-1}$ of order $j\le4$ are bounded by $Cp^{4(2A_1+1)}$, and with \Cref{lem:uniq-rank-one}(ii), we have $|\partial_\xi^4a^{\mathrm{ord}}|\le Cp^{10A_1+4}\le p^{A_{\mathrm j}}$. Applying \Cref{lem:uniq-interpolation} with $r_0=\delta_0$, $\epsilon=p^{A_{\mathrm v}-N_*}$ and $m_4=2p^{A_{\mathrm j}}$, we find that for $\xi\in\Xi_1$, \begin{equation*} |f'(\xi)|\le4p^{(3A_{\mathrm v}+A_{\mathrm j}-3N_*)/4}+\frac{4p^{A_{\mathrm v}-N_*}}{\delta_0}\to0, \end{equation*} since $N_*>A_{\mathrm v}+A_{\mathrm j}$. Combined with Step 2, this proves the lemma. \end{proof} \begin{lemma}\label{lem:uniq-rare} For $p\ge p_1$ and $\lambda\in[1/2,2]$, write $b=T_{\mathrm{ct}}(\cdot,\lambda)$ and $d=T_{\mathrm{tc}}(\cdot,\lambda)$, and take suprema over $\Xi_1$. Then \begin{equation*} \|b\|_\infty\le e^{-a\sqrt p/4},\qquad\|\partial_\xi^2b\|_\infty\le e^{-a\sqrt p/16},\qquad\|d\|_\infty+\|\partial_\xi^2d\|_\infty\le2p^{A_{\mathrm j}}. \end{equation*} In particular, \begin{equation*} \sup_{\lambda\in[1/2,2]}\Bigl(\sqrt{\|d\|_\infty\|\partial_\xi^2b\|_\infty}+\sqrt{\|b\|_\infty\|\partial_\xi^2d\|_\infty}+\frac{2\sqrt{\|b\|_\infty\|d\|_\infty}}{\delta_0}\Bigr)\to0\qquad\text{as }p\to\infty. \end{equation*} \end{lemma} \begin{proof} The bounds on $b$, $d$ and $\partial_\xi^2d$ follow from \Cref{lem:uniq-matrix}(iii),(iv). For $\partial_\xi^2b$, we apply \Cref{lem:uniq-interpolation} on $\Xi$ with $r_0=\delta_0$, $\epsilon=e^{-a\sqrt p/4}$ and $m_4=p^{A_{\mathrm j}}$, which gives $|\partial_\xi^2b|\le2p^{A_{\mathrm j}/2}e^{-a\sqrt p/8}+8\delta_0^{-2}e^{-a\sqrt p/4}\le e^{-a\sqrt p/16}$ on $\Xi_1$ for $p\ge p_1$. The last claim follows by inserting these bounds. \end{proof} The bound on $\partial_\xi^2b$ is an absolute bound; it is not a bound relative to the value of $b$, which may be much smaller than $e^{-a\sqrt p}$. \section{Proofs of the main results}\label{sec:uniq-proofs} \begin{proof}[Proof of \Cref{thm:uniq-window}] Let $\Xi_0$ be given, and fix $\delta_0\in(0,\mustar/8]$ such that $[\mustar-4\delta_0,\mustar+4\delta_0]\subset\Xi_0$. Set \begin{equation*} \Xi=[\mustar-3\delta_0,\mustar+3\delta_0],\qquad\Xi_1=[\mustar-2\delta_0,\mustar+2\delta_0],\qquad\Xi_2=[\mustar-\delta_0,\mustar+\delta_0]. \end{equation*} We apply the constructions of \Cref{sec:uniq-inputs,sec:uniq-hilbert,sec:uniq-minorant,sec:uniq-cutoffs,sec:uniq-matrix} on the window $\Xi$, with the parameters chosen in the order of \Cref{sec:uniq-order}, and we take $p\ge p_1$, with $p_1$ also depending on $\delta_0$ and enlarged finitely many times below. We also take $p_1$ at least the threshold $p_0(\min\Xi_0)$ of \Cref{cor:stab-halfline}, so that $\lambda_p$ is defined on $\Xi_0$; by the uniqueness in \Cref{thm:stab-window}(i), it agrees on $\Xi$ with the function used in the constructions. \proofstep{1}{Monotonicity on $\Xi_2$.} We apply \Cref{prop:uniq-renewal-criterion} with $\mathscr H=\mathscr L$, $I=\operatorname{int}\Xi_2$, $I_1=\operatorname{int}\Xi_1$, $r_0=\delta_0$, $\lambda_-=1/2$, $\lambda_+=2$, the operators $G_\xi$ of \eqref{eq:uniq-G}, and $h_{1,\xi}=h_{\mathrm c}$, $h_{2,\xi}=h_{\mathrm t}$, $\ell_{1,\xi}=\ell_{\mathrm c}$, $\ell_{2,\xi}=\ell_{\mathrm t}$. By \eqref{eq:uniq-square-decomposition}, $U_\xi=\mathsf A_\xi^2$. We verify the hypotheses. \begin{itemize} \item The maps $\xi\mapsto G_\xi,h_{\mathrm c},\ell_{\mathrm c}$ are four times continuously differentiable by \Cref{lem:uniq-rank-one,lem:uniq-square}, and so are $h_{\mathrm t}=J\ell_{\mathrm c}$ and $\ell_{\mathrm t}=Jh_{\mathrm c}$ since $J$ does not depend on $\xi$. By \Cref{lem:uniq-resolvent}, $r(G_\xi)\le1/4<\lambda_-$. \item Hypothesis (i) holds by \Cref{lem:uniq-matrix}(i),(ii). \item Let $\kappa=\mustar^2(\mustar+2\delta_0)^{-3}$. By \Cref{lem:uniq-diagonal}, $\partial_\xi T_{\mathrm{cc}}\le-2\mustar^2\xi^{-3}+\kappa\le-\kappa$ on $\Xi_1\times[1/2,2]$ for $p\ge p_1$, and by \Cref{lem:uniq-rare}, \eqref{eq:uniq-abstract-offdiag} holds for $p\ge p_1$. So hypothesis (ii) holds. \item By \Cref{thm:asym-leading}, for $p\ge p_1$ we have $\sup_{\Xi_0}|\lambda_p(\xi)-\mustar/\xi|\le\delta_0/(4\mustar)$. For $\xi\in\Xi_2$, we have $\mustar/\xi\in[8/9,8/7]$ since $\delta_0\le\mustar/8$, and so $\lambda_p(\xi)^2\in(1/2,2)$. By \Cref{lem:uniq-hilbert}(ii), $F_\xi>0$ satisfies $\mathsf A_\xi^2F_\xi=\lambda_p(\xi)^2F_\xi$. So hypothesis (iii) holds with $\lambda(\xi)=\lambda_p(\xi)^2$. \end{itemize} \Cref{prop:uniq-renewal-criterion}(c) shows that $\lambda_p^2$ is continuously differentiable on $\operatorname{int}\Xi_2$ with negative derivative. Since $\lambda_p>0$, the same holds for $\lambda_p$, with $\partial_\xi\lambda_p=\partial_\xi(\lambda_p^2)/(2\lambda_p)<0$. \proofstep{2}{Signs on $\Xi_0$.} For $\xi\le\mustar-\delta_0/2$, we have $\mustar/\xi\ge1+\delta_0/(2\mustar)$, and for $\xi\ge\mustar+\delta_0/2$, we have $\mustar/\xi\le1-\delta_0/(2\mustar+\delta_0)\le1-\delta_0/(3\mustar)$, since $\delta_0\le\mustar$. With the bound from \Cref{thm:asym-leading} in Step 1, we obtain $\lambda_p>1$ on $\Xi_0\cap(-\infty,\mustar-\delta_0/2]$ and $\lambda_p<1$ on $\Xi_0\cap[\mustar+\delta_0/2,\infty)$. Since $\lambda_p$ is strictly decreasing on $\operatorname{int}\Xi_2\supset[\mustar-\delta_0/2,\mustar+\delta_0/2]$ by Step 1, and $\lambda_p(\mustar-\delta_0/2)>1>\lambda_p(\mustar+\delta_0/2)$, it takes the value one at exactly one point $\xi_c(p)\in(\mustar-\delta_0/2,\mustar+\delta_0/2)$, and $\lambda_p-1$ is positive to its left and negative to its right within $\operatorname{int}\Xi_2$. Combined with the signs outside $[\mustar-\delta_0/2,\mustar+\delta_0/2]$, this shows that $\xi_c(p)$ is the only solution of $\lambda_p(\xi)=1$ in $\Xi_0$, with $\lambda_p>1$ to its left and $\lambda_p<1$ to its right in $\Xi_0$. By Step 1, $\lambda_p$ is continuously differentiable near $\xi_c(p)$ and $\partial_\xi\lambda_p(\xi_c(p))<0$. \proofstep{3}{Convergence.} Let $\delta\in(0,\delta_0/2]$. By \Cref{thm:asym-leading}, there exists $p(\delta)$ such that $\sup_{\Xi_0}|\lambda_p(\xi)-\mustar/\xi|\le\delta/(4\mustar)$ for $p\ge p(\delta)$; then $\lambda_p>1$ on $\Xi_0\cap(-\infty,\mustar-\delta]$ and $\lambda_p<1$ on $\Xi_0\cap[\mustar+\delta,\infty)$, by the argument of Step 2 with $\delta_0/2$ replaced by $\delta$. Then $|\xi_c(p)-\mustar|<\delta$ for $p\ge\max\{p_1,p(\delta)\}$, which proves $\xi_c(p)\to\mustar$. \end{proof} \begin{proof}[Proof of \Cref{cor:uniq-halfline}] Let $M_{\mathrm{ext}}\ge3$ and $p_{\mathrm{ext}}$ be as in \Cref{prop:asym-entry}, and let $\Xi_0=[\xi_0,M_{\mathrm{ext}}+1]$; since $\mustar<10$ is numerical. The bound holds for all real solutions, not only for the principal solution of \Cref{sec:stab}. In \Cref{sec:main}, it is combined with \Cref{thm:deloc-criterion} to show that the boundary values at these energies are not real. The proof uses the elementary lower bound $\lambda(1)\ge p\,\E\|M_g\|^{-1}$ on the critical eigenvalue, valid for every scaling $g$, and two choices of scaling. With the scaling $g=1$, the mean $p\,\E\|M_1\|^{-1}$ is large if $F=\E(1+|D|)^{-1}$ is at least a large multiple of $p^{-1/2}$. With the scaling $g=\sqrt p$, the mean $p\,\E\|M_{\sqrt p}\|^{-1}$ is large if $q=\P(|D|\le p^{-1/2})$ is at least a large multiple of $p^{-1}$. If both tests fail, then with probability bounded below no child denominator is small, and the cavity equation shows that $|D|$ is at most $E$ plus a bounded quantity; this gives a lower bound on $F$ of order $(1+E)^{-1}$, which contradicts the failure of the first test when $E\le\xi_{\mathrm{in}}\sqrt p$ and $\xi_{\mathrm{in}}$ is small. \section{The critical transfer operator with two scalings}\label{sec:inner-gauge} Throughout this section, $\pi$ is a real solution at energy $E>0$ (\Cref{def:rde-real}), $D\sim\pi$, and $Y\sim\Exp(1)$ is independent of $D$. For a scaling $g>0$, we set \[ M_g=M\Big(gD,\frac{g^2Y}p\Big)=\Big(\frac{g^2Y}p\Big)^{-1/2}\begin{pmatrix}0&1\\-g^2Y/p&gD\end{pmatrix}, \] as in \Cref{def:rde-normalizations} and \eqref{eq:press-M}. The variable $\beta=g^2Y/p$ has the exponential law of rate $K_g=p/g^2$, and by \Cref{prop:rde-properties} the law $\nu_g$ of $gD$ satisfies the hypotheses \eqref{eq:press-hyp} of \Cref{sec:press}. We write $\mathcal Q^{(g)}_1$ for the transfer operator of \Cref{def:press-operators} at $s=1$, defined with $\nu_g$ and $K_g$: \[ (\mathcal Q^{(g)}_1\chi)(x)=p\,\E\big[\|M_gv_x\|^{-1}\chi(M_g\cdot x)\big],\qquad\chi\in C(\RP),\ x\in\RP. \] By \Cref{thm:press-perron}(i), this operator is bounded and positive on $C(\RP)$, and we write $\lambda^{(g)}(1)$ for its spectral radius. By \Cref{def:press-pressure}(iii), the free energy of $\pi$ at $s=1$ is $\varphi_\pi(1)=\log\lambda^{(g)}(1)$. We use the two scalings $g=1$ and $g=\sqrt p$, and we recall why they give the same critical eigenvalue. Let $S=\operatorname{diag}(p^{-1/4},p^{1/4})$. Conjugation by $S$ multiplies the $(1,2)$ entry of a matrix by $p^{-1/2}$ and the $(2,1)$ entry by $p^{1/2}$, and leaves the diagonal entries unchanged. Since $M_1=(Y/p)^{-1/2}\begin{pmatrix}0&1\\-Y/p&D\end{pmatrix}$, we have \[ SM_1S^{-1}=\Big(\frac Yp\Big)^{-1/2}\begin{pmatrix}0&p^{-1/2}\\-Y/\sqrt p&D\end{pmatrix} =Y^{-1/2}\begin{pmatrix}0&1\\-Y&\sqrt pD\end{pmatrix}=M_{\sqrt p}, \] which is \eqref{eq:press-gauge-conj} with $\gamma=\sqrt p$. This identity between the random matrices of the two scalings, which holds for every realization of $(D,Y)$, induces the similarity $\mathcal Q^{(\sqrt p)}_1=\Theta^{-1}\mathcal Q^{(1)}_1\Theta$ of \Cref{lem:press-gauge}(ii), where $(\Theta\chi)(x)=\|Sv_x\|^{-1}\chi(S\cdot x)$ is a bounded operator on $C(\RP)$ with bounded inverse. Similar operators have the same spectrum, which gives \begin{equation}\label{eq:inner-gauge} \lambda^{(1)}(1)=\lambda^{(\sqrt p)}(1)=e^{\varphi_\pi(1)}. \end{equation} \begin{lemma}\label{lem:inner-lower} For every $g>0$, we have $\lambda^{(g)}(1)\ge p\,\E\|M_g\|^{-1}$. \end{lemma} \begin{proof} Set $m=p\,\E\|M_g\|^{-1}$ and let $\one$ be the constant function one on $\RP$. For every $x\in\RP$, the vector $v_x$ is a unit vector. Then $\|M_gv_x\|\le\|M_g\|$, and \[ (\mathcal Q^{(g)}_1\one)(x)=p\,\E\|M_gv_x\|^{-1}\ge m. \] The operator $\mathcal Q^{(g)}_1$ is positive: if $\chi\ge\psi$ pointwise, then $\mathcal Q^{(g)}_1\chi\ge\mathcal Q^{(g)}_1\psi$ pointwise. By induction on $n$, we have $(\mathcal Q^{(g)}_1)^n\one\ge m^n\one$ for every $n\ge1$; indeed, if this holds for $n-1$, then \[ (\mathcal Q^{(g)}_1)^n\one=\mathcal Q^{(g)}_1\big((\mathcal Q^{(g)}_1)^{n-1}\one\big) \ge m^{n-1}\mathcal Q^{(g)}_1\one\ge m^n\one. \] Since $\|\one\|_\infty=1$, the operator norm on $C(\RP)$ satisfies $\|(\mathcal Q^{(g)}_1)^n\|\ge\|(\mathcal Q^{(g)}_1)^n\one\|_\infty\ge m^n$. By Gelfand's formula for the spectral radius \cite[Theorem~VI.6]{ReedSimon1980methods}, this implies that $\lambda^{(g)}(1)=\lim_{n\to\infty}\|(\mathcal Q^{(g)}_1)^n\|^{1/n}\ge m$. \end{proof} Since $M_g$ has determinant one, we have $\|M_g\|\ge1$, and the bound of \Cref{lem:inner-lower} is at most $p$. This is consistent with the bound $\varphi_\pi(1)<\log p$ of \Cref{prop:press-properties}. \section{Proof of the lower bound}\label{sec:inner-proof} Let $r_0=\P(1\le Y\le2)=e^{-1}-e^{-2}$ for $Y\sim\Exp(1)$. We choose the constants in the following order: \begin{equation}\label{eq:inner-constants} A=\frac6{r_0},\qquad H=\frac4{r_0},\qquad\xi_{\mathrm{in}}=\frac{e^{-H}}{8A},\qquad p_{\mathrm{in}}=\Big(\frac{1+4A}{\xi_{\mathrm{in}}}\Big)^2,\qquad\delta=\log2. \end{equation} These are numerical constants. Numerically, $r_0\approx0.2325$, $A\approx25.8$, $H\approx17.2$ and $\xi_{\mathrm{in}}\approx1.6\cdot10^{-10}$; we make no attempt to optimize them. \begin{proposition}\label{prop:inner-pressure} Let $\xi_{\mathrm{in}}$, $p_{\mathrm{in}}$ and $\delta$ be the numerical constants in \eqref{eq:inner-constants}. For every $p\ge p_{\mathrm{in}}$, every $E\in(0,\xi_{\mathrm{in}}\sqrt p]$, and every real solution $\pi$ at energy $E$, we have $\varphi_\pi(1)\ge\delta$. \end{proposition} \begin{proof} Fix $p\ge p_{\mathrm{in}}$, $E\in(0,\xi_{\mathrm{in}}\sqrt p]$ and a real solution $\pi$ at energy $E$. Let $D\sim\pi$ and $Y\sim\Exp(1)$ be independent, and set \[ F=\E\frac1{1+|D|},\qquad q=\P\big(|D|\le p^{-1/2}\big). \] Note that $p\ge p_{\mathrm{in}}>1$. We show that $F\ge A/\sqrt p$ or $pq\ge H$, and that each of these alternatives gives $\lambda(1)\ge2$, where $\lambda(1)=e^{\varphi_\pi(1)}$ equals $\lambda^{(1)}(1)$ and $\lambda^{(\sqrt p)}(1)$ by \eqref{eq:inner-gauge}. \proofstep{1}{The scaling $g=1$.} We claim that if $F\ge A/\sqrt p$, then $\lambda(1)\ge2$. On the event $\{1\le Y\le2\}$, we bound the operator norm of $M_1=(Y/p)^{-1/2}\begin{pmatrix}0&1\\-Y/p&D\end{pmatrix}$ by its Frobenius norm and obtain \[ \|M_1\|\le\Big(\frac pY\Big)^{1/2}\Big(1+\frac{Y^2}{p^2}+D^2\Big)^{1/2} \le\sqrt p\Big(1+\frac Yp+|D|\Big)\le3\sqrt p\,(1+|D|). \] In the second inequality, we used $Y\ge1$ and $(a^2+b^2+c^2)^{1/2}\le a+b+c$ for $a,b,c\ge0$, and in the third, $Y/p\le2$. Since $Y$ is independent of $D$, we find \[ p\,\E\|M_1\|^{-1}\ge p\,\E\Big[\one_{\{1\le Y\le2\}}\frac1{3\sqrt p\,(1+|D|)}\Big]=\frac{r_0\sqrt p}3F. \] If $F\ge A/\sqrt p$, the right side is at least $r_0A/3=2$, and \Cref{lem:inner-lower} gives $\lambda(1)\ge2$. \proofstep{2}{The scaling $g=\sqrt p$.} We claim that if $pq\ge H$, then $\lambda(1)\ge2$. For $g=\sqrt p$, we have $gD=\sqrt pD$ and $g^2Y/p=Y$. Then \[ M_{\sqrt p}=Y^{-1/2}\begin{pmatrix}0&1\\-Y&\sqrt pD\end{pmatrix},\qquad \|M_{\sqrt p}\|^2\le\frac1Y+Y+\frac{pD^2}Y. \] On the event $\{1\le Y\le2\}\cap\{|D|\le p^{-1/2}\}$, the right side is at most $1+2+1=4$, and $\|M_{\sqrt p}\|\le2$. Since $Y$ is independent of $D$, we find \[ p\,\E\|M_{\sqrt p}\|^{-1}\ge p\,\P\big(1\le Y\le2,\ |D|\le p^{-1/2}\big)\cdot\frac12=\frac{r_0}2\,pq. \] If $pq\ge H$, the right side is at least $r_0H/2=2$, and \Cref{lem:inner-lower} gives $\lambda(1)\ge2$. \proofstep{3}{Both tests cannot fail.} Suppose, for contradiction, that $Fp^{-1/2}}\frac{Y_j}{|D_j|}. \] The point process $\sum_{j\le N}\delta_{(Y_j,D_j)}$ is a Poisson process on $(0,\infty)\times\R$ with intensity measure $p$ times the law of $(Y_1,D_1)$, since $N\sim\Pois(p)$ is independent of the independent and identically distributed pairs $(Y_j,D_j)$. Its restrictions to the disjoint sets $(0,\infty)\times[-p^{-1/2},p^{-1/2}]$ and $(0,\infty)\times(\R\setminus[-p^{-1/2},p^{-1/2}])$ are therefore independent Poisson processes \cite[Chapter~5]{LastPenrose2018lectures}. The variable $Z$ is the total mass of the first restriction, which has law $\Pois(pq)$, and $\Sigma$ is a function of the second. In particular, $Z$ and $\Sigma$ are independent, and \[ \P(Z=0)=e^{-pq}>e^{-H}. \] We next bound the mean of $\Sigma$. On the event $\{|D|>p^{-1/2}\}$, we have $1+|D|\le(1+\sqrt p)|D|$, that is, $|D|^{-1}\le(1+\sqrt p)(1+|D|)^{-1}$. Using the independence of $N$, $(Y_j)$ and $(D_j)$, we find \[ \E\Sigma=\frac1p\,\E N\,\E Y_1\,\E\big[|D|^{-1}\one_{\{|D|>p^{-1/2}\}}\big] \le(1+\sqrt p)F<\frac{(1+\sqrt p)A}{\sqrt p}\le2A, \] where in the last inequality we used $p\ge1$. By Markov's inequality, $\P(\Sigma\ge4A)<1/2$. On the event $\{Z=0\}\cap\{\Sigma<4A\}$, all indices are large, and $|E-p^{-1}\sum_{j\le N}Y_j/D_j|\le E+\Sigmae^{-H}/2$, and we conclude that \[ F\ge\frac{e^{-H}}2\cdot\frac1{1+E+4A}. \] Since $p\ge p_{\mathrm{in}}$, we have $1+4A\le\xi_{\mathrm{in}}\sqrt p$, and since $E\le\xi_{\mathrm{in}}\sqrt p$, we have $1+E+4A\le2\xi_{\mathrm{in}}\sqrt p$. By the choice of $\xi_{\mathrm{in}}$ in \eqref{eq:inner-constants}, we have \[ F\ge\frac{e^{-H}}{4\xi_{\mathrm{in}}\sqrt p}=\frac{2A}{\sqrt p}, \] which contradicts $F1$ and $E\ge\xi_0\sqrt p$ in the range of \Cref{cor:stab-halfline}, we write $\varphi_E(1)$ for the free energy at $s=1$ of the principal solution: by \Cref{cor:stab-halfline}, if $B$ has law $\nu_{p,E}$, then the law of $B/g(E)$ is a real solution at energy $E$, and $\varphi_E(1)$ is the value at $s=1$ of its free energy (\Cref{def:press-pressure}). This is the quantity $\varphi_E(1)$ of \Cref{thm:critical}, and in the notation of \Cref{sec:jets,sec:asym,sec:uniq}, we have $\lambda_p(\xi)=\exp\varphi_{\xi\sqrt p}(1)$. \section{Choice of constants}\label{sec:main-constants} Let $\xi_{\mathrm{in}}>0$, $\delta>0$, and $p_{\mathrm{in}}$ be the numerical constants of \Cref{prop:inner-pressure}. Since the conclusion of \Cref{prop:inner-pressure} is preserved if $\xi_{\mathrm{in}}$ is decreased, we may and do assume that $\xi_{\mathrm{in}}<\mustar/2$. Set \[ \xi_0=\frac{\xi_{\mathrm{in}}}{2}. \] Let $p_*$ be the numerical constant of \Cref{thm:deloc-criterion}, and let $M_{\mathrm{ext}}$ and $p_{\mathrm{ext}}$ be the numerical constants of the large-energy estimate in \Cref{prop:asym-entry}. Let $\Xi=[\xi_0,M_{\mathrm{ext}}+1]$, and let $p_{\mathrm{phys}}(\Xi)$ be the threshold of \Cref{prop:stab-selection} for this window. By \Cref{cor:uniq-halfline}, there exists $p(\xi_0)$ such that for every $p\ge p(\xi_0)$ there is a unique $E_c(p)\ge\xi_0\sqrt p$ with \begin{equation}\label{eq:main-sign} \varphi_E(1)>0\quad\text{for }\xi_0\sqrt p\le EE_c(p), \end{equation} and such that $E_c(p)/\sqrt p\to\mustar$ as $p\to\infty$. Since $\xi_{\mathrm{in}}<\mustar/2$, there exists $p'$ such that $E_c(p)>\xi_{\mathrm{in}}\sqrt p$ for every $p\ge p'$. We define $p_0$ to be the maximum of $p_*$, $p_{\mathrm{in}}$, $p_{\mathrm{ext}}$, $p_{\mathrm{phys}}(\Xi)$, the threshold $p_0(\xi_0)$ of \Cref{cor:stab-halfline}, $p(\xi_0)$, and $p'$. This is the maximum of finitely many finite numbers. Fix $p\ge p_0$ for the remainder of this section. We record three consequences of the choice of $p_0$. First, $\xi_{\mathrm{in}}\sqrt p0$ for $E0$. Since $p\ge p_*$, this contradicts \Cref{thm:deloc-criterion}. Therefore $\P^\infty(\Im\Gamma_o(E+\iu0)=0)<1$ for every $E\in\mathcal E_p\cap(0,\xi_{\mathrm{in}}\sqrt p]$. Since $\mathcal E_p$ has full measure, \Cref{cor:deloc-ac} implies that $\P^\infty$-almost surely, $\mathcal H$ has purely absolutely continuous spectrum in $(0,\xi_{\mathrm{in}}\sqrt p)$. \proofstep{2}{Absolute continuity at energies of order $\sqrt p$ below $E_c(p)$.} Let $E\in\mathcal E_p\cap(\xi_0\sqrt p,E_c(p))$, and suppose that $\P^\infty(\Im\Gamma_o(E+\iu0)=0)=1$. Since $E_c(p)0$ to its left and $\varphi_E(1)<0$ to its right. This zero coincides with the energy $E_c(p)$ of \Cref{sec:main-constants}. Indeed, the half-lines $[\xi_0\sqrt p,\infty)$ and $[\xi_0'\sqrt p,\infty)$ are nested, and $E\mapsto\varphi_E(1)$ has exactly one zero on each of them. The zero on the smaller half-line is also a zero on the larger one, so the two zeros coincide. Finally, let $\Xi_{\mathrm c}\subset(0,\infty)$ be a compact interval whose interior contains $E_c(p)/\sqrt p$ for all $p\ge p_1(\xi_0')$; for instance $\Xi_{\mathrm c}=[\mustar/2,2\mustar]$, after increasing $p_1(\xi_0')$. Increasing $p_1(\xi_0')$ further, we may assume that $p\ge p_{\mathrm{cr}}(\Xi_{\mathrm c})$, the threshold of \Cref{prop:jets-critical-analytic}. By that proposition, $\xi\mapsto\lambda_p(\xi)$ is real-analytic on an open interval containing $\Xi_{\mathrm c}$, so $E\mapsto\varphi_E(1)=\log\lambda_p(E/\sqrt p)$ is real-analytic in a neighborhood of $E_c(p)$, and its derivative at $E_c(p)$ is negative by \Cref{cor:uniq-halfline}. This proves part (ii). Part (iii) is \Cref{thm:asym-leading}, applied to a compact set $\Xi\subset[\xi_0',\infty)$. \chapter{The giant component}\label{sec:giant} In this section, we transfer the spectral dichotomy of \Cref{thm:main} to the giant component of the finite graph and prove \Cref{thm:giant}. The section has three parts. The first part, \Cref{sec:giant-graphs,sec:giant-local}, proves \Cref{prop:giant-local}: the giant component, seen from finitely many uniformly chosen vertices, converges locally to independent copies of the marked $\PGW(p)$ tree conditioned on survival, and this convergence holds in probability for averages over the giant component. Membership in the giant component is not determined by a bounded neighborhood of a vertex. We therefore first prove the statement with the giant component replaced by the set of vertices whose component has at least $K$ vertices, which is a local condition, and then let $K\to\infty$. Since the tree operator is unbounded, local convergence alone does not give convergence of resolvents; this step uses the essential self-adjointness of the tree operator from \Cref{thm:tree-esa}, as in \cite{BordenaveLelarge2010resolvent}. The second part, \Cref{sec:giant-dos}, proves \Cref{prop:giant-dos}: on every compact interval separated from zero, the averaged root spectral measure of the survival-conditioned tree has a density bounded above and below. The upper bound comes from a Wegner-type estimate for an auxiliary bipartite random graph, whose squared operator is a sum of rank-one projections with independent Gamma-distributed weights. The lower bound comes from a configuration at the root (one surviving child and one leaf child) in which the weight of the leaf edge enters the root resolvent through a Poisson kernel. The third part, \Cref{sec:giant-transfer}, combines the first two to prove \Cref{prop:giant-counts,prop:giant-transfer}, and then \Cref{thm:giant}. \section{Setting and notation}\label{sec:giant-setting} Throughout this section, $p>1$ is fixed, and constants may depend on $p$ without further mention. We use the finite model of \Cref{def:tree-model}. The graph $G_n$ is an Erd\H{o}s--R\'enyi graph $G(n,p/n)$ on $[n]=\{1,\dots,n\}$. Conditionally on $G_n$, the variables $w_{ij}$, $1\le i0$ for which there exists an isomorphism $\varphi$ from $B_{\lfloor r\rfloor}(x)$ to $B_{\lfloor r\rfloor}(x')$ as $k$-rooted graphs (ignoring marks) such that $|h'_{\varphi(u)\varphi(v)}-h_{uv}|\le r^{-1}$ for every edge $uv$ of $B_{\lfloor r\rfloor}(x)$, with $R_*=0$ if there is no such $r$. We set \[ d_{\mathrm{loc}}(x,x')=\frac{1}{1+R_*(x,x')}. \] \end{definition} Local weak convergence was introduced by Benjamini and Schramm \cite{BenjaminiSchramm2001recurrence} and by Aldous and Steele \cite{AldousSteele2004objective}. The distance $d_{\mathrm{loc}}$ is the metric of \cite[Section~2]{AldousLyons2007processes}; see also \cite[Chapter~2 and Appendix~A]{vanderHofstad2024random}. By these references, $(\mathscr G_k,d_{\mathrm{loc}})$ is a complete separable metric space. (These references treat $k=1$ and marks in a general complete separable metric space; the case of $k$ roots is identical.) A sequence $x_n$ converges to $x$ if and only if, for every $R\ge1$, there exists $n_R$ such that for all $n\ge n_R$ there exists an isomorphism of $B_R(x)$ onto $B_R(x_n)$ as $k$-rooted graphs with all mark differences at most $R^{-1}$. Since a locally finite graph has finite balls, every $B_R(x)$ is finite, and $d_{\mathrm{loc}}(x,B_R(x))\le(1+R)^{-1}$. A function on $\mathscr G_k$ that depends only on $B_R(x)$ as a $k$-rooted graph (ignoring marks) is locally constant, hence continuous. Examples are $x\mapsto\one\{\deg(o_1)\le L\}$, $x\mapsto\one\{\operatorname{dist}(o_a,o_b)\le R\}$ for $R\ge0$, and \begin{equation}\label{eq:giant-AK} \chi_{K,a}(x)=\one\{\text{the component of $o_a$ has at least $K$ vertices}\}, \qquad K\ge1. \end{equation} For $\chi_{K,a}$, this holds with $R=K-1$, since a component with at least $K$ vertices contains a connected set of $K$ vertices containing $o_a$, and such a set lies within distance $K-1$ of $o_a$. The set of finite elements of $\mathscr G_k$ is Borel, since it is the union over $R$ of the sets $\{|B_R(x)|=|B_{R+1}(x)|\}$, which are open and closed. For $1\le k'\le k$ and an injective map $\sigma\colon[k']\to[k]$, the \emph{restriction} $r_\sigma\colon\mathscr G_k\to\mathscr G_{k'}$ keeps the roots $o_{\sigma(1)},\dots,o_{\sigma(k')}$ and the union of their components. It satisfies $d_{\mathrm{loc}}(r_\sigma x,r_\sigma x')\le d_{\mathrm{loc}}(x,x')$, because an isomorphism of $B_R(x)$ onto $B_R(x')$ restricts to an isomorphism of $B_R(r_\sigma x)$ onto $B_R(r_\sigma x')$. Here we used that rooted isomorphisms of $R$-balls preserve the distance to the roots, since for a vertex of $B_R(x)$ this distance is attained by a path inside $B_R(x)$. We use two kinds of elements of $\mathscr G_k$. For distinct vertices $v_1,\dots,v_k$ of $G_n$, we write $[G_n,v]$ for the $k$-rooted marked graph formed by the union of the components of $v_1,\dots,v_k$ in $G_n$, with the marks $h_{ij}$ and roots $v=(v_1,\dots,v_k)$. For $k$ rooted marked trees $(T_1,o_1),\dots,(T_k,o_k)$, we consider their disjoint union, rooted at $(o_1,\dots,o_k)$, as an element of $\mathscr G_k$. We write $\mathfrak F_k$ for the disjoint union of $k$ independent marked $\PGW(p)$ trees with law $\P$, and $\mathfrak F_k^\infty$ for the disjoint union of $k$ independent trees with law $\P^\infty$. We write $\E^{\infty,\otimes k}$ for the expectation with respect to the law of $\mathfrak F_k^\infty$. \subsection{Local convergence of the uniformly rooted graph} We need the local limit for two graph models. The second one is used only in \Cref{sec:giant-dos}. \begin{enumerate}[label=\textup{(\alph*)}] \item The model of \Cref{sec:giant-setting}: $G_n=G(n,p/n)$ with vertex set $\mathsf V_n=[n]$ and marks $h_{ij}$. \item The bipartite model: $\mathsf V_n=[n]\times\{0,1\}$, and each pair $\{(i,0),(j,1)\}$ is an edge independently with probability $p/n$; there are no other edges. Conditionally on the graph, the edges carry independent standard complex Gaussians $w$, and the edge $\{(i,0),(j,1)\}$ has marks $h_{(j,1),(i,0)}=w/\sqrt p$ and $h_{(i,0),(j,1)}=\overline w/\sqrt p$. In this model, we also write $G_n$ for the marked graph. \end{enumerate} In both models, the marks are independent of the graph. In the marked tree, the mark $h_{uv}$ on the edge from a parent $u$ to its child $v$ is $w_{uv}/\sqrt p$. Since the standard complex Gaussian law is invariant under complex conjugation, the choice of orientation used to attach the Gaussian variable to an edge does not affect the law of a marked graph: in both models, conditionally on the graph, the marks $h_{uv}$, for any choice of one orientation $(u,v)$ per edge, are independent with the law of $w/\sqrt p$. \begin{lemma}\label{lem:giant-explore} Consider either model \textup{(a)} or model \textup{(b)}. Fix $k\ge1$, and let $\mathbf V=(V_1,\dots,V_k)$ be uniformly distributed among the $k$-tuples of distinct elements of $\mathsf V_n$, independent of the marked graph. Then for every $R\ge1$, the law of $B_R([G_n,\mathbf V])$ converges in total variation to the law of $B_R(\mathfrak F_k)$ as $n\to\infty$. In particular, $[G_n,\mathbf V]\to\mathfrak F_k$ in distribution in $\mathscr G_k$. Further, in model \textup{(a)}, $\E[\deg(V_1)^2]\le p+p^2$ for all $n$. \end{lemma} This is the standard exploration argument (see \cite[Chapter~2]{vanderHofstad2024random} for local convergence of $G(n,p/n)$); we include the proof because we need $k$ roots and marks. \begin{proof} We treat model (a) and indicate the changes for model (b) at the end. Fix $R\ge1$ and $\eps>0$. Since the balls of radius $R$ in $\PGW(p)$ trees are almost surely finite, there exists $M\ge k$ such that $\P(\sum_{a=1}^k|B_R(T_a)|>M)\le\eps$, where $T_1,\dots,T_k$ are independent $\PGW(p)$ trees. We explore the balls of radius $R$ around $V_1,\dots,V_k$ in $G_n$, one root at a time. Each vertex is first \emph{discovered} and later possibly \emph{processed}. When the exploration of the root $V_a$ starts, $V_a$ is discovered (if it was discovered earlier, we say that a \emph{collision} occurs). Then, as long as there is an unprocessed vertex discovered during the exploration of $V_a$ at exploration depth at most $R-1$, we process the earliest discovered such vertex $x$. Processing $x$ reveals, for every vertex $y$ not yet processed, whether $xy$ is an edge. Every such $y$ that was not yet discovered becomes discovered, as a child of $x$, at depth one more than $x$. When all roots have been explored, we reveal all pairs of discovered vertices that have not been revealed. Each pair of vertices is revealed at most once. Since the roots are independent of the graph, conditionally on everything revealed before a pair is revealed, its indicator is a Bernoulli variable with parameter $p/n$. Let $\mathcal C$ be the event that no collision occurs and that every revealed pair $\{x,y\}$ of vertices that were both discovered before it was revealed is not an edge. On $\mathcal C$, the induced subgraph on the discovered vertices is the union of $k$ disjoint trees, the exploration trees of $V_1,\dots,V_k$. Further, the vertices of the $a$-th tree are exactly the vertices at distance at most $R$ from $V_a$. Then, on $\mathcal C$, the $k$-rooted graph $B_R([G_n,\mathbf V])$ (ignoring marks) is the disjoint union of the exploration trees. Its isomorphism class is determined by the sequence of numbers of children of the processed vertices, in the order of processing. We stop the exploration as soon as more than $M$ vertices are discovered. Before stopping, there are at most $M$ processing steps and at most $(M+1)^2$ revealed pairs of the type excluded in $\mathcal C$, each of which is an edge with conditional probability $p/n$. Further, conditionally on the exploration of $V_1,\dots,V_{a-1}$, the root $V_a$ is uniform on $[n]\setminus\{V_1,\dots,V_{a-1}\}$, and the probability of a collision at $V_a$ is at most $(M+1)/(n-k)$. When a vertex $x$ is processed and $m\le M$ vertices are discovered, the number of its children is, conditionally on the past, binomial with parameters $n-m$ and $p/n$. By Le Cam's inequality and the bound $d_{\TV}(\Pois(\lambda),\Pois(\lambda'))\le|\lambda-\lambda'|$, we have \[ d_{\TV}\bigl(\operatorname{Bin}(n-m,p/n),\Pois(p)\bigr)\le(n-m)\frac{p^2}{n^2}+\frac{mp}{n}\le\frac{p^2+Mp}{n}. \] Explore $T_1,\dots,T_k$ up to depth $R$ in the same breadth-first order. Using at each processing step a maximal coupling of the two conditional offspring laws, we obtain a coupling under which, except on an event of probability at most \[ \eps+M\frac{p^2+Mp}{n}+(M+1)^2\frac pn+\frac{k(M+1)}{n-k}, \] the event $\mathcal C$ holds and all offspring numbers agree. On this event, $B_R([G_n,\mathbf V])$ and $B_R(T_1)\sqcup\dots\sqcup B_R(T_k)$ are isomorphic as $k$-rooted graphs through the exploration. We transport the marks of $G_n$ on the edges of the ball to the corresponding tree edges, and give the remaining tree edges independent marks. The marks are independent of the graph and of $\mathbf V$, and the exploration only reveals the graph. Hence, by the conjugation invariance noted above, the transported marks are independent with the law of $w/\sqrt p$, and independent of the trees. This produces marked trees with the law of $\mathfrak F_k$, and the marked balls agree on the event above. Since $\eps>0$ was arbitrary, the total variation distance tends to zero. For the convergence in distribution, let $F\colon\mathscr G_k\to\R$ be bounded and uniformly continuous, with modulus of continuity $\omega_F$. Since $d_{\mathrm{loc}}(x,B_R(x))\le(1+R)^{-1}$, we have \[ \bigl|\E F([G_n,\mathbf V])-\E F(\mathfrak F_k)\bigr| \le2\omega_F\bigl((1+R)^{-1}\bigr)+2\|F\|_\infty\,d_{\TV}\bigl(\Law B_R([G_n,\mathbf V]),\Law B_R(\mathfrak F_k)\bigr). \] Letting $n\to\infty$ and then $R\to\infty$, we find $\E F([G_n,\mathbf V])\to\E F(\mathfrak F_k)$. By the portmanteau theorem \cite[Theorem~2.1]{Billingsley1999convergence}, this proves the convergence in distribution. Finally, since the degree of $V_1$ is binomial with parameters $n-1$ and $p/n$, we have $\E[\deg(V_1)^2]\le p+p^2$. In model (b), the vertex $x$ processed in the class $\mathsf V_n\cap([n]\times\{c\})$ only has potential neighbors in the other class, and the number of its children is binomial with parameters $n-m'$ and $p/n$, where $m'\le M$ is the number of discovered vertices in the other class. The rest of the argument is unchanged. \end{proof} \subsection{Stieltjes transforms} For a finite Borel measure $m$ on $\R$, we write $s_m(z)=\int m(\dd\lambda)/(\lambda-z)$, $z\in\Hup$, for its Stieltjes transform. \begin{lemma}\label{lem:giant-stieltjes} Let $D\subset\Hup$ be a set with an accumulation point in $\Hup$. \begin{enumerate}[label=\textup{(\roman*)}] \item Let $m_n$ and $m$ be probability measures on $\R$ such that $s_{m_n}(z)\to s_m(z)$ for every $z\in D$. Then $m_n\to m$ weakly. \item Suppose that $D$ is countable. Let $m_n$ be random probability measures on $\R$ and $m$ a deterministic probability measure such that $s_{m_n}(z)\to s_m(z)$ in probability for every $z\in D$. Then $m_n\to m$ weakly in probability. \end{enumerate} \end{lemma} \begin{proof} For (i), it suffices to show that every subsequence of $(m_n)$ has a further subsequence converging weakly to $m$. By Helly's selection theorem, every subsequence has a further subsequence that converges vaguely to a measure $m'$ with $m'(\R)\le1$. Since, for $z\in\Hup$, the function $\lambda\mapsto(\lambda-z)^{-1}$ is continuous and vanishes at infinity, we have $s_{m_n}(z)\to s_{m'}(z)$ along this subsequence. Then $s_{m'}=s_m$ on $D$. Both functions are analytic on $\Hup$, and the identity theorem implies that they agree on $\Hup$. By the Stieltjes inversion formula, $\pi^{-1}\Im s_{m'}(E+\iu\eps)\dd E$ converges vaguely to $m'$ as $\eps\downarrow0$; the same holds for $m$, and we conclude that $m'=m$. Since $m(\R)=1$, the vague convergence to $m$ is weak convergence \cite[Section~3.2]{Durrett2019probability}. For (ii), fix a bounded continuous $f$. It suffices to show that every subsequence has a further subsequence along which $\int f\dd m_n\to\int f\dd m$ almost surely. Since $D$ is countable, by a diagonal argument there exists a further subsequence along which, almost surely, $s_{m_n}(z)\to s_m(z)$ for all $z\in D$ simultaneously. Along it, part (i) applies pathwise. \end{proof} \subsection{Resolvents on rooted marked graphs} For $x\in\mathscr G_k$ with vertex set $V$, let $H_{x,0}$ be the weighted adjacency operator on finitely supported functions $f\colon V\to\C$, given by $(H_{x,0}f)(u)=\sum_{v\sim u}h_{uv}f(v)$. It is a symmetric operator on $\ell^2(V)$. For $R\ge0$, let $H_x^{(R)}$ be the Hermitian matrix of the finite marked graph $B_R(x)$, acting on functions on the vertex set of $B_R(x)$. For $z\in\C\setminus\R$ and $a,b\in[k]$, set \[ G^{(R)}_{ab}(x,z)=\bigl\langle\delta_{o_a},(H_x^{(R)}-z)^{-1}\delta_{o_b}\bigr\rangle, \] and let $\mu^{(R)}_{x,a}$ be the spectral measure of $\delta_{o_a}$ for $H_x^{(R)}$. \begin{definition}\label{def:giant-functions} For $x\in\mathscr G_k$, $z\in\C\setminus\R$ and $a,b\in[k]$, let $G_{ab}(x,z)=\lim_{R\to\infty}G^{(R)}_{ab}(x,z)$ if this limit exists, and $G_{ab}(x,z)=0$ otherwise. Let $\mu_{x,a}$ be the weak limit of $\mu^{(R)}_{x,a}$ as $R\to\infty$ if it exists, and $\mu_{x,a}=\delta_0$ otherwise. Let $\mathscr G_k^{\mathrm{esa}}$ be the set of $x\in\mathscr G_k$ such that $H_{x,0}$ is essentially self-adjoint; for such $x$, let $H_x$ denote the closure of $H_{x,0}$. \end{definition} If $x$ is finite, then $B_R(x)=x$ for large $R$; in this case, $G_{ab}(x,z)$ and $\mu_{x,a}$ are the resolvent entries and the root spectral measures of the matrix $H_{x,0}$. In particular, if $v_1,\dots,v_k$ are distinct vertices of $\gc$, then \eqref{eq:giant-block} implies that $G_{ab}([G_n,v],z)=\langle\delta_{v_a},(H_n^{\gc}-z)^{-1}\delta_{v_b}\rangle$ and $\mu_{[G_n,v],a}=\mu_{n,v_a}$. \begin{lemma}\label{lem:giant-borel} For every $z\in\C\setminus\R$ and $a,b\in[k]$, the maps $x\mapsto G_{ab}(x,z)$ and $x\mapsto\mu_{x,a}$ are Borel, and $|G_{ab}(x,z)|\le|\Im z|^{-1}$. The set $\mathscr G_k^{\mathrm{esa}}$ is a Borel subset of $\mathscr G_k$. \end{lemma} \begin{proof} For fixed $R$, the quantities $G^{(R)}_{ab}(x,z)$ and $\mu^{(R)}_{x,a}$ depend only on the marked graph $B_R(x)$ and are invariant under its rooted isomorphisms. If $d_{\mathrm{loc}}(x,x')<(1+R')^{-1}$ with $R'\ge R$, then there exists an isomorphism of $B_{R'}(x)$ onto $B_{R'}(x')$ with mark differences at most $1/R'$, which restricts to $B_R(x)$; after relabeling, $H_{x'}^{(R)}$ is a Hermitian matrix with the same pattern as $H_x^{(R)}$ and entries within $1/R'$. The resolvent at a fixed nonreal $z$ and the spectral measure of a fixed basis vector (in the weak topology) depend continuously on the entries of a Hermitian matrix. Then $x\mapsto G^{(R)}_{ab}(x,z)$ and $x\mapsto\mu^{(R)}_{x,a}$ are continuous, the latter as a map into the space $\mathcal P(\R)$ of probability measures with the weak topology. Since $\mathcal P(\R)$ with the L\'evy--Prokhorov metric is a complete separable metric space \cite[Theorem~6.8]{Billingsley1999convergence}, the set where a sequence of continuous maps into $\C$ or into $\mathcal P(\R)$ is Cauchy is Borel, and the limit is a Borel map on it. This proves the first claim. The bound on $G_{ab}$ holds for each $G^{(R)}_{ab}$, hence for the limit. For the second claim, recall that a symmetric operator $A$ is essentially self-adjoint if and only if $\Ran(A+\iu)$ and $\Ran(A-\iu)$ are dense \cite[Section~VIII.2]{ReedSimon1980methods}. For $R'\ge R$ and a vertex $v$ of $B_R(x)$, let $d^\pm_{R'}(x,v)$ be the distance from $\delta_v$ to the subspace of vectors $(H_{x,0}\pm\iu)f$ with $f$ supported in the vertex set of $B_{R'}(x)$. These subspaces increase with $R'$, and their union is $\Ran(H_{x,0}\pm\iu)$. Since the vectors $\delta_v$ span a dense subspace, we have $x\in\mathscr G_k^{\mathrm{esa}}$ if and only if, for every $R$, we have $\max_{v\in B_R(x)}d^\pm_{R'}(x,v)\to0$ as $R'\to\infty$ for both signs. For fixed $R\le R'$, the function $x\mapsto\max_{v\in B_R(x)}d^\pm_{R'}(x,v)$ depends only on $B_{R'+1}(x)$, since $(H_{x,0}\pm\iu)f$ is supported in $B_{R'+1}(x)$ when $f$ is supported in $B_{R'}(x)$. Let $M_x$ denote the matrix of $f\mapsto(H_{x,0}\pm\iu)f$ from functions on $B_{R'}(x)$ to functions on $B_{R'+1}(x)$. Since $H_{x,0}$ is symmetric, we have $\|M_xf\|^2=\|H_{x,0}f\|^2+\|f\|^2\ge\|f\|^2$. Then $M_x^*M_x\ge1$, and the orthogonal projection $M_x(M_x^*M_x)^{-1}M_x^*$ onto $\Ran M_x$ depends continuously on the entries of $M_x$. As in the first part, it follows that $x\mapsto\max_{v\in B_R(x)}d^\pm_{R'}(x,v)$ is continuous. Then \[ \mathscr G_k^{\mathrm{esa}}=\bigcap_{\pm}\bigcap_{R\ge1}\bigcap_{j\ge1}\bigcup_{R'\ge R}\Bigl\{x:\max_{v\in B_R(x)}d^\pm_{R'}(x,v)r+1$, the vertices of $x_n$ adjacent to $\varphi_n(\supp f)$ lie in $B_{r+1}(x_n)=\varphi_n(B_{r+1}(x))$, and the adjacency relations between them correspond under $\varphi_n$. Then $H_{x_n,0}J_nf-J_nH_{x,0}f$ is supported in $\varphi_n(B_{r+1}(x))$, its entries are bounded by $\eps_n\sum_v|f(v)|$, and we obtain \[ \|H_{x_n,0}J_nf-J_nH_{x,0}f\|\le\eps_n\,|B_{r+1}(x)|^{1/2}\sum_{v}|f(v)|\longrightarrow0. \] Since $(H_{x,0}-\overline z)f$ is finitely supported and $J_n^*u_n\to u$ pointwise along the subsequence, we obtain \begin{align*} \bigl\langle(H_{x,0}-\overline z)f,u\bigr\rangle &=\lim_{n}\bigl\langle J_n(H_{x,0}-\overline z)f,u_n\bigr\rangle =\lim_{n}\bigl\langle(H_{x_n,0}-\overline z)J_nf,u_n\bigr\rangle\\ &=\lim_{n}\bigl\langle J_nf,(H_{x_n,0}-z)u_n\bigr\rangle =\lim_{n}\bigl\langle J_nf,\delta_{o^n_b}\bigr\rangle =\langle f,\delta_{o_b}\rangle. \end{align*} In the second equality, we used the previous display and $\|u_n\|\le|\Im z|^{-1}$; in the third, that $H_{x_n,0}$ is a Hermitian matrix; in the last, that $\varphi_n(o_b)=o^n_b$. Since this holds for all finitely supported $f$, the vector $u$ lies in the domain of $H_{x,0}^*$, and $H_{x,0}^*u=\delta_{o_b}+zu$. Since $H_{x,0}$ is essentially self-adjoint, $H_{x,0}^*=H_x$, and we conclude that $u=(H_x-z)^{-1}\delta_{o_b}$. Then $G_{ab}(x_n,z)=u_n(o^n_a)=(J_n^*u_n)(o_a)\to u(o_a)$ along the subsequence. The limit does not depend on the subsequence, which proves (i). For (ii), we apply (i) to the finite graphs $x_R=B_R(x)$, which converge to $x$ as $R\to\infty$. Since $B_{R'}(x_R)=B_R(x)$ for $R'\ge R$, we have $G_{ab}(x_R,z)=G^{(R)}_{ab}(x,z)$. By (i), $G^{(R)}_{ab}(x,z)$ converges to $\langle\delta_{o_a},(H_x-z)^{-1}\delta_{o_b}\rangle$, which is therefore equal to $G_{ab}(x,z)$. In particular, the Stieltjes transforms of $\mu^{(R)}_{x,a}$ converge on $\Hup$ to that of the spectral measure of $\delta_{o_a}$ for $H_x$, and \Cref{lem:giant-stieltjes}(i) shows that $\mu^{(R)}_{x,a}$ converges weakly to this measure. This proves (ii). Part (iii) follows from (i), (ii) and \Cref{lem:giant-stieltjes}(i), since $s_{\mu_{x_n,a}}(z)=G_{aa}(x_n,z)$. \end{proof} By \Cref{thm:tree-esa}, the operator of each tree of $\mathfrak F_k$ is almost surely essentially self-adjoint, and a direct sum of finitely many essentially self-adjoint operators is essentially self-adjoint. We conclude that $\P(\mathfrak F_k\in\mathscr G_k^{\mathrm{esa}})=1$; this is an event by \Cref{lem:giant-borel}. On this event, \Cref{lem:giant-continuity}(ii) shows that $G_{ab}(\mathfrak F_k,z)$ is the resolvent entry of the closed tree operator, which vanishes for $a\ne b$, and that $\mu_{\mathfrak F_k,a}$ is the root spectral measure of the $a$-th tree. In particular, $\mu_{\mathfrak F_1,1}$ has the law of $\mu_o$ under $\P$, and $\mu_{\mathfrak F_1^\infty,1}$ has the law of $\mu_o$ under $\P^\infty$. \subsection{Admissible functions} \begin{definition}\label{def:giant-admissible} A bounded Borel function $F\colon\mathscr G_k\to\C$ is \emph{admissible} if there exists a Borel set $\mathscr A_F\subset\mathscr G_k$ with $\P(\mathfrak F_k\in\mathscr A_F)=1$ such that $F(x_n)\to F(x)$ for every $x\in\mathscr A_F$ and every sequence of finite $x_n\in\mathscr G_k$ with $x_n\to x$. \end{definition} \begin{lemma}\label{lem:giant-admissible} The following functions on $\mathscr G_k$ are admissible. \begin{enumerate}[label=\textup{(\roman*)}] \item Every bounded continuous function. \item For $z_1,\dots,z_m\in\C\setminus\R$ and a bounded continuous $\phi\colon\C^{k\times k\times m}\to\C$, the function $x\mapsto\phi\bigl((G_{ab}(x,z_l))_{a,b\le k,\,l\le m}\bigr)$. \item For intervals $I_1,\dots,I_m\subset\R$ whose endpoints $c$ satisfy $\nu_p(\{c\})=0$, and a bounded continuous $\phi\colon[0,1]^{k\times m}\to\C$, the function $x\mapsto\phi\bigl((\mu_{x,a}(I_l))_{a\le k,\,l\le m}\bigr)$. \item Linear combinations, products and complex conjugates of admissible functions; and $F\circ r_\sigma$ for every admissible $F$ on $\mathscr G_{k'}$ and every injective $\sigma\colon[k']\to[k]$. \end{enumerate} \end{lemma} \begin{proof} For (i), take $\mathscr A_F=\mathscr G_k$. For (ii), the function is Borel by \Cref{lem:giant-borel}, and we take $\mathscr A_F=\mathscr G_k^{\mathrm{esa}}$; the convergence follows from \Cref{lem:giant-continuity}(i),(ii). For (iii), the function is Borel since $x\mapsto\mu_{x,a}$ is Borel and $\mu\mapsto\mu(I)$ is Borel on $\mathcal P(\R)$. Let $\mathscr A_F$ be the set of $x\in\mathscr G_k^{\mathrm{esa}}$ with $\mu_{x,a}(\{c\})=0$ for all $a$ and all endpoints $c$ of the intervals $I_l$; it is Borel since $\mu\mapsto\mu(\{c\})$ is upper semicontinuous. By the discussion after \Cref{lem:giant-continuity}, $\mu_{\mathfrak F_k,a}$ has the law of $\mu_o$ under $\P$ for each $a$, and $\E\mu_o(\{c\})=\nu_p(\{c\})=0$; hence $\P(\mathfrak F_k\in\mathscr A_F)=1$. For $x\in\mathscr A_F$ and finite $x_n\to x$, \Cref{lem:giant-continuity}(iii) and the portmanteau theorem give $\mu_{x_n,a}(I_l)\to\mu_{x,a}(I_l)$, since the boundary of $I_l$ is $\mu_{x,a}$-null. For (iv), the claim about linear combinations, products and conjugates follows by intersecting the sets $\mathscr A_F$. For $F\circ r_\sigma$, take $\mathscr A=r_\sigma^{-1}(\mathscr A_F)$, which is Borel since $r_\sigma$ is continuous. Since the image $r_\sigma(\mathfrak F_k)$ is the disjoint union of the trees with indices $\sigma(1),\dots,\sigma(k')$, it has the law of $\mathfrak F_{k'}$, and $\P(\mathfrak F_k\in\mathscr A)=\P(\mathfrak F_{k'}\in\mathscr A_F)=1$. If $x\in\mathscr A$ and $x_n\to x$ are finite, then the finite graphs $r_\sigma x_n$ converge to $r_\sigma x\in\mathscr A_F$, and $F(r_\sigma x_n)\to F(r_\sigma x)$. \end{proof} \begin{lemma}\label{lem:giant-annealed} Consider either model \textup{(a)} or \textup{(b)}, and let $\mathbf V$ be as in \Cref{lem:giant-explore}. For every admissible $F\colon\mathscr G_k\to\C$, we have $\E F([G_n,\mathbf V])\to\E F(\mathfrak F_k)$. \end{lemma} \begin{proof} By \Cref{lem:giant-explore}, $[G_n,\mathbf V]\to\mathfrak F_k$ in distribution in the complete separable metric space $\mathscr G_k$. By Skorokhod's representation theorem \cite[Theorem~6.7]{Billingsley1999convergence}, there exist random elements $X_n$ and $X$ of $\mathscr G_k$ on a common probability space such that $X_n$ has the law of $[G_n,\mathbf V]$, $X$ has the law of $\mathfrak F_k$, and $X_n\to X$ almost surely. Since the set of finite elements and $\mathscr A_F$ are Borel, almost surely every $X_n$ is finite and $X\in\mathscr A_F$. Then $F(X_n)\to F(X)$ almost surely, and we conclude by the bounded convergence theorem that $\E F(X_n)\to\E F(X)$. \end{proof} \section{Local convergence on the giant component}\label{sec:giant-local} For the rest of this section, except in \Cref{sec:giant-dos}, $G_n$ is the graph of model (a). For a bounded function $F$ on $\mathscr G_k$, define the averages over ordered $k$-tuples of distinct vertices \begin{equation}\label{eq:giant-averages} S_n(F)=\frac{1}{(n)_k}\sum_{\substack{v\in[n]^k\\\text{distinct}}}F([G_n,v]), \qquad S_n^{\gc}(F)=\frac{1}{(N_n)_k}\sum_{\substack{v\in\gc^k\\\text{distinct}}}F([G_n,v]), \end{equation} where $(m)_k=m(m-1)\cdots(m-k+1)$. Thus $S_n(F)$ and $S_n^{\gc}(F)$ are the conditional expectations of $F([G_n,\mathbf V])$ given the marked graph, when $\mathbf V$ is a uniform $k$-tuple of distinct vertices of $[n]$ or of $\gc$, respectively. \begin{lemma}\label{lem:giant-lln} For every admissible $F\colon\mathscr G_k\to\C$, we have $S_n(F)\to\E F(\mathfrak F_k)$ in $L^2$. \end{lemma} \begin{proof} By \Cref{lem:giant-annealed}, we have $\E S_n(F)=\E F([G_n,\mathbf V])\to\E F(\mathfrak F_k)$. It remains to show that $\E|S_n(F)|^2\to|\E F(\mathfrak F_k)|^2$. We have \[ \E|S_n(F)|^2=\frac{1}{(n)_k^2}\sum_{v,v'}\E\Bigl[F([G_n,v])\overline{F([G_n,v'])}\Bigr], \] where the sum is over pairs of $k$-tuples of distinct vertices. The number of pairs $(v,v')$ such that some entry of $v$ equals some entry of $v'$ is at most $k^2(n)_kn^{k-1}$, and their contribution is at most $k^2\|F\|_\infty^2n^{k-1}/(n)_k=O(n^{-1})$. The remaining $(n)_{2k}$ pairs correspond to $2k$-tuples $(v,v')$ of distinct vertices. Let $F^{\otimes}\colon\mathscr G_{2k}\to\C$ be given by $F^\otimes(y)=F(r_1y)\overline{F(r_2y)}$, where $r_1$ and $r_2$ are the restrictions to the first $k$ and the last $k$ roots. Then $F([G_n,v])\overline{F([G_n,v'])}=F^\otimes([G_n,(v,v')])$, and $F^\otimes$ is admissible by \Cref{lem:giant-admissible}(iv). Applying \Cref{lem:giant-annealed} with $2k$ roots, we obtain \[ \E|S_n(F)|^2=\frac{(n)_{2k}}{(n)_k^2}\,\E F^\otimes([G_n,\mathbf V'])+O(n^{-1})\longrightarrow\E F^\otimes(\mathfrak F_{2k})=|\E F(\mathfrak F_k)|^2, \] where $\mathbf V'$ is a uniform $2k$-tuple of distinct vertices. In the last equality, we used that $r_1\mathfrak F_{2k}$ and $r_2\mathfrak F_{2k}$ are independent copies of $\mathfrak F_k$. Expanding the square, we conclude that $\E|S_n(F)-\E F(\mathfrak F_k)|^2\to0$. \end{proof} Next, we compare membership in $\gc$ with the local event that the component of a vertex has at least $K$ vertices. For $K\ge1$, set \begin{align*} a_K&=\P(|\mathcal T|\ge K),\\ Z_{n,K}&=\#\bigl\{v\in[n]\setminus\gc:\ \text{the component of $v$ has at least $K$ vertices}\bigr\}. \end{align*} Since $\{|\mathcal T|\ge K\}$ decreases to $\{|\mathcal T|=\infty\}=\mathcal S$ as $K\to\infty$, we have $a_K\downarrow\varrho_p$. \begin{lemma}\label{lem:giant-membership} For every $K\ge1$, we have $Z_{n,K}/n\to a_K-\varrho_p$ in probability. \end{lemma} \begin{proof} Let $\chi_K=\chi_{K,1}$ be the function from \eqref{eq:giant-AK} with $k=1$. It is continuous, and $\E\chi_K(\mathfrak F_1)=a_K$. By \Cref{lem:giant-lln} with $k=1$, \[ \frac1n\#\{v\in[n]:\ \text{the component of $v$ has at least $K$ vertices}\}=S_n(\chi_K)\longrightarrow a_K \] in probability. On the event $\{N_n\ge K\}$, every vertex of $\gc$ belongs to a component with at least $K$ vertices, which gives $Z_{n,K}=nS_n(\chi_K)-N_n$. Since $\P(N_n\ge K)\to1$ and $N_n/n\to\varrho_p$ by \eqref{eq:giant-size}, the claim follows. \end{proof} \begin{lemma}\label{lem:giant-quenched} For every admissible $F\colon\mathscr G_k\to\C$, we have $S_n^{\gc}(F)\to\E^{\infty,\otimes k}F(\mathfrak F_k^\infty)$ in probability. \end{lemma} \begin{proof} We first replace membership in $\gc$ by the local event $\{\chi_{K,a}=1\}$ for each root, then compute the limit of the resulting average for fixed $K$, and finally let $K\to\infty$. \proofstep{1}{The average with the local event.} Fix $K\ge1$ and set $\Phi_K(x)=F(x)\prod_{a=1}^k\chi_{K,a}(x)$. By \Cref{lem:giant-admissible}(i),(iv), $\Phi_K$ is admissible. Since, in $\mathfrak F_k$, the component of the $a$-th root is the $a$-th tree $T_a$ and the trees are independent, \Cref{lem:giant-lln} implies that \begin{equation}\label{eq:giant-step1} S_n(\Phi_K)\longrightarrow\E\Bigl[F(\mathfrak F_k)\prod_{a=1}^k\one\{|T_a|\ge K\}\Bigr] =a_K^k\,\E\Bigl[F(\mathfrak F_k)\Bigm|\ |T_a|\ge K\text{ for all }a\Bigr] \end{equation} in $L^2$. The factor $a_K^k$ is the probability that each of the $k$ roots lies in a component with at least $K$ vertices. \proofstep{2}{Comparison with the giant component.} On the event $\{N_n\ge\max(k,K)\}$, every $k$-tuple of distinct vertices of $\gc$ satisfies $\prod_a\chi_{K,a}([G_n,v])=1$. A $k$-tuple of distinct vertices with $\prod_a\chi_{K,a}([G_n,v])=1$ that is not contained in $\gc$ has an entry outside $\gc$ whose component has at least $K$ vertices. There are at most $kZ_{n,K}n^{k-1}$ such $k$-tuples, and on this event we have \[ \Bigl|(N_n)_k\,S_n^{\gc}(F)-(n)_k\,S_n(\Phi_K)\Bigr|\le k\|F\|_\infty Z_{n,K}n^{k-1}. \] Dividing by $(N_n)_k$, we find \begin{equation}\label{eq:giant-step2} \Bigl|S_n^{\gc}(F)-\frac{(n)_k}{(N_n)_k}S_n(\Phi_K)\Bigr|\le k\|F\|_\infty\,\frac{Z_{n,K}}{n}\cdot\frac{n^k}{(N_n)_k}. \end{equation} \proofstep{3}{The limit $n\to\infty$ for fixed $K$.} By \eqref{eq:giant-size}, we have $(N_n)_k/n^k\to\varrho_p^k$ and $(N_n)_k/(n)_k\to\varrho_p^k$ in probability. Combining this with \eqref{eq:giant-step1}, \eqref{eq:giant-step2} and \Cref{lem:giant-membership}, we find that for every $\eta>0$, \begin{equation}\label{eq:giant-step3} \lim_{n\to\infty}\P\Bigl(\bigl|S_n^{\gc}(F)-L_K\bigr|>\eps_K+\eta\Bigr)=0, \end{equation} where \[ L_K=\varrho_p^{-k}\,\E\Bigl[F(\mathfrak F_k)\prod_{a=1}^k\one\{|T_a|\ge K\}\Bigr], \qquad \eps_K=k\|F\|_\infty\varrho_p^{-k}(a_K-\varrho_p). \] In other words, after normalization by the size of the giant component, the average over $\gc$ is close to $(a_K/\varrho_p)^k\E[F(\mathfrak F_k)\mid|T_a|\ge K\ \forall a]$. \proofstep{4}{The limit $K\to\infty$.} As $K\to\infty$, the indicator $\prod_a\one\{|T_a|\ge K\}$ decreases to $\prod_a\one_{\mathcal S_a}$, where $\mathcal S_a$ is the survival event of $T_a$. Since $F$ is bounded, by dominated convergence we have \[ L_K\longrightarrow\varrho_p^{-k}\,\E\Bigl[F(\mathfrak F_k)\prod_{a=1}^k\one_{\mathcal S_a}\Bigr]=\E^{\infty,\otimes k}F(\mathfrak F_k^\infty). \] In the last equality, we used that the trees are independent, so that $\P(\bigcap_a\mathcal S_a)=\varrho_p^k$ and, conditionally on $\bigcap_a\mathcal S_a$, the trees are independent with law $\P^\infty$. Further, $\eps_K\to0$ since $a_K\to\varrho_p$. Given $\eta>0$, we choose $K$ such that $|L_K-\E^{\infty,\otimes k}F(\mathfrak F_k^\infty)|+\eps_K<\eta$. Then \eqref{eq:giant-step3} implies that $\P(|S_n^{\gc}(F)-\E^{\infty,\otimes k}F(\mathfrak F_k^\infty)|>2\eta)\to0$. Since $S_n^{\gc}(F)$ does not depend on $K$, this completes the proof. \end{proof} We now state the main result of this part. For an oriented edge $(u,w)$ of $G_n$, let $[G_n,u\to w]\in\mathscr G_2$ denote the component of $u$, rooted at $(u,w)$. Let $\mathfrak E$ denote the following random element of $\mathscr G_2$: take two independent marked $\PGW(p)$ trees $T^-$ and $T^+$ with roots $o^-$ and $o^+$, join $o^-$ and $o^+$ by an edge with mark $h_{o^-o^+}=w/\sqrt p$, where $w$ is an independent standard complex Gaussian, and root the resulting tree at $(o^-,o^+)$. Let $\vec E(\gc)$ be the set of oriented edges of $\gc$. \begin{proposition}\label{prop:giant-local} Fix $p>1$ and $k\ge1$. On the event $\{N_n\ge k\}$, conditionally on the marked graph, let $\mathbf V=(V_1,\dots,V_k)$ be a uniformly chosen $k$-tuple of distinct vertices of $\gc$. \begin{enumerate}[label=\textup{(\roman*)}] \item \textup{(Local convergence.)} For every admissible $F\colon\mathscr G_k\to\C$, in particular for every bounded continuous $F$, \[ \E\bigl[F([G_n,\mathbf V])\bigm|G_n,(w_{ij})\bigr]=S_n^{\gc}(F)\longrightarrow\E^{\infty,\otimes k}F(\mathfrak F_k^\infty) \] in probability. In particular, $[G_n,\mathbf V]\to\mathfrak F_k^\infty$ in distribution in $\mathscr G_k$, and $\operatorname{dist}(V_a,V_b)\to\infty$ in probability for $a\ne b$. The same holds if $V_1,\dots,V_k$ are independent and uniform on $\gc$. \item \textup{(Oriented edges.)} We have \[ \frac{|\vec E(\gc)|}{n}\longrightarrow p\bigl(1-(1-\varrho_p)^2\bigr)=p\varrho_p(2-\varrho_p) \] in probability. Further, for every bounded continuous $\Psi\colon\mathscr G_2\to\C$, \[ \frac{1}{|\vec E(\gc)|}\sum_{(u,w)\in\vec E(\gc)}\Psi([G_n,u\to w])\longrightarrow\E\bigl[\Psi(\mathfrak E)\bigm|T^-\text{ or }T^+\text{ is infinite}\bigr] \] in probability. In particular, a uniformly chosen oriented edge of $\gc$ converges in distribution to $\mathfrak E$ conditioned on the event that at least one of $T^-$ and $T^+$ is infinite. \item \textup{(Resolvents.)} For all $z_1,\dots,z_m\in\C\setminus\R$, the array $\bigl(\langle\delta_{V_a},(H_n^{\gc}-z_l)^{-1}\delta_{V_b}\rangle\bigr)_{a,b\le k,\,l\le m}$ converges in distribution to $\bigl(\delta_{ab}\,G^{(a)}_{oo}(z_l)\bigr)_{a,b\le k,\,l\le m}$, where $G^{(1)}_{oo},\dots,G^{(k)}_{oo}$ are the root resolvents of $k$ independent trees with law $\P^\infty$. In particular, the entries with $a\ne b$ converge to zero in probability. Further, for every bounded continuous $\phi\colon\C^m\to\C$, \[ \frac{1}{N_n}\sum_{v\in\gc}\phi\bigl(\langle\delta_v,(H_n^{\gc}-z_l)^{-1}\delta_v\rangle_{l\le m}\bigr)\longrightarrow\E^\infty\phi\bigl((G_{oo}(z_l))_{l\le m}\bigr) \] in probability. \item \textup{(Empirical spectral measure.)} The empirical spectral measure $\nu_n^{\gc}$ converges weakly in probability to $\nu_p^\infty$. \item \textup{(Local spectral masses.)} Let $I_1,\dots,I_m$ be intervals whose endpoints $c$ satisfy $\nu_p(\{c\})=0$ \textup{(}by \Cref{prop:giant-dos}, this holds for every $c\ne0$\textup{)}. Then the array $(\mu_{n,V_a}(I_l))_{a\le k,\,l\le m}$ converges in distribution to $(\mu^{(a)}(I_l))_{a\le k,\,l\le m}$, where $\mu^{(1)},\dots,\mu^{(k)}$ are the root spectral measures of $k$ independent trees with law $\P^\infty$. Further, for every bounded continuous $\phi\colon[0,1]^m\to\C$, \[ \frac{1}{N_n}\sum_{v\in\gc}\phi\bigl((\mu_{n,v}(I_l))_{l\le m}\bigr)\longrightarrow\E^\infty\phi\bigl((\mu_o(I_l))_{l\le m}\bigr) \] in probability, and the analogous statement holds for averages over ordered pairs of distinct vertices of $\gc$, with limit $\E^{\infty,\otimes2}$. \end{enumerate} \end{proposition} \begin{proof}[Proof of \Cref{prop:giant-local}] \proofpart{Proof of (i).} The identity $\E[F([G_n,\mathbf V])\mid G_n,(w_{ij})]=S_n^{\gc}(F)$ holds by the definition of $\mathbf V$, and the convergence is \Cref{lem:giant-quenched}. For bounded continuous $F$, taking expectations and using the bounded convergence theorem gives $\E F([G_n,\mathbf V])\to\E^{\infty,\otimes k}F(\mathfrak F_k^\infty)$, which is the convergence in distribution. For $a\ne b$ and $R\ge0$, the function $x\mapsto\one\{\operatorname{dist}(o_a,o_b)\le R\}$ is continuous and vanishes at $\mathfrak F_k^\infty$, whose roots lie in distinct components; hence $\P(\operatorname{dist}(V_a,V_b)\le R)\to0$. If $V_1,\dots,V_k$ are independent and uniform on $\gc$, then conditionally on the marked graph they are distinct with probability at least $1-k^2/N_n$, and conditionally on being distinct they form a uniform $k$-tuple of distinct vertices of $\gc$. Then their conditional expectation of $F([G_n,\mathbf V])$ differs from $S_n^{\gc}(F)$ by at most $2k^2\|F\|_\infty/N_n$, which tends to zero in probability. \proofpart{Proof of (ii).} For $x\in\mathscr G_1$ with root $o$ and a neighbor $w$ of $o$, let $x^{o\to w}\in\mathscr G_2$ denote $x$ rooted at $(o,w)$. Fix a bounded continuous $\Psi\colon\mathscr G_2\to\C$, and define \[ \mathsf F(x)=\sum_{w\sim o}\Psi(x^{o\to w}), \qquad \mathsf F_L(x)=\one\{\deg(o)\le L\}\,\mathsf F(x),\qquad L\ge1. \] Since every neighbor of a vertex of $\gc$ lies in $\gc$, we have \begin{equation}\label{eq:giant-edge-sum} \sum_{(u,w)\in\vec E(\gc)}\Psi([G_n,u\to w])=\sum_{u\in\gc}\mathsf F([G_n,u]). \end{equation} We first show that $\mathsf F_L$ is continuous. Let $x_m\to x$ in $\mathscr G_1$, and let $R\ge1$. For $m$ large, there exists an isomorphism $\varphi_m$ of $B_{R+1}(x)$ onto $B_{R+1}(x_m)$ with mark differences at most $(R+1)^{-1}$. In particular, the roots of $x$ and $x_m$ have the same degree, and $\varphi_m$ maps the neighbors of $o$ bijectively onto those of the root $o_m$ of $x_m$. For a neighbor $w$ of $o$, the vertices within distance $R$ of $\{o,w\}$ lie within distance $R+1$ of $o$. Then $\varphi_m$ restricts to an isomorphism of $B_R(x^{o\to w})$ onto $B_R(x_m^{o_m\to\varphi_m(w)})$ with mark differences at most $R^{-1}$. This gives $d_{\mathrm{loc}}(x_m^{o_m\to\varphi_m(w)},x^{o\to w})\le(1+R)^{-1}$ and \[ |\mathsf F_L(x_m)-\mathsf F_L(x)|\le\sum_{w\sim o}\sup\Bigl\{|\Psi(y)-\Psi(x^{o\to w})|:\ d_{\mathrm{loc}}(y,x^{o\to w})\le(1+R)^{-1}\Bigr\} \] for $m$ large. Since $\Psi$ is continuous at the finitely many points $x^{o\to w}$, the right side tends to zero as $R\to\infty$. We conclude that $\mathsf F_L$ is continuous; it is bounded by $L\|\Psi\|_\infty$. By part (i) with $k=1$ and \eqref{eq:giant-size}, we have \[ \frac1n\sum_{u\in\gc}\mathsf F_L([G_n,u])=\frac{N_n}{n}\,S_n^{\gc}(\mathsf F_L)\longrightarrow\varrho_p\,\E^\infty\mathsf F_L(\mathcal T)=\E\bigl[\mathsf F_L(\mathcal T)\one_{\mathcal S}\bigr] \] in probability, where we regard the rooted tree $\mathcal T$ as an element of $\mathscr G_1$. Since $|\mathsf F-\mathsf F_L|(x)\le\|\Psi\|_\infty\deg(o)\one\{\deg(o)>L\}\le\|\Psi\|_\infty\deg(o)^2/L$, by \Cref{lem:giant-explore} we have \[ \E\Bigl[\frac1n\sum_{u\in[n]}\bigl|\mathsf F-\mathsf F_L\bigr|([G_n,u])\Bigr]\le\frac{\|\Psi\|_\infty}{L}\E[\deg(V_1)^2]\le\frac{(p+p^2)\|\Psi\|_\infty}{L}, \] where $V_1$ is uniform on $[n]$. In the same way, since the number $N$ of children of $o$ satisfies $\E N^2=p+p^2$, we have $\E|\mathsf F-\mathsf F_L|(\mathcal T)\le(p+p^2)\|\Psi\|_\infty/L$. Letting $L\to\infty$ and using Markov's inequality for the first error term, we conclude from \eqref{eq:giant-edge-sum} that \begin{equation}\label{eq:giant-edge-limit} \frac1n\sum_{(u,w)\in\vec E(\gc)}\Psi([G_n,u\to w])\longrightarrow\E\bigl[\mathsf F(\mathcal T)\one_{\mathcal S}\bigr] \end{equation} in probability. We now compute the right side of \eqref{eq:giant-edge-limit} using the Mecke formula. The children $j$ of $o$, together with the marks $h_{oj}$ and the marked forward subtrees $\mathcal T_j$, form a Poisson process $\Pi=\sum_j\delta_{M_j}$, $M_j=(h_{oj},\mathcal T_j)$, whose intensity is $p$ times the product of the law of $w/\sqrt p$ and the law of the marked $\PGW(p)$ tree. For each $j$, the rooted tree $\mathcal T^{o\to j}$ and the indicator $\one_{\mathcal S}$ are measurable functions of $(M_j,\Pi-\delta_{M_j})$: the tree $\mathcal T^{o\to j}$ is obtained by joining the root of the tree with root $o$ and child process $\Pi-\delta_{M_j}$ to the root of $\mathcal T_j$ by an edge with mark $h_{oj}$, and $\mathcal T$ is infinite if and only if one of these two trees is infinite. By the Mecke formula of \Cref{lem:palm-mecke}, applied to the positive and negative parts of the real and imaginary parts of $\Psi$ (each side is finite, since $|\mathsf F(\mathcal T)|\le\|\Psi\|_\infty N$ and $\E N=p$), we have \[ \E\bigl[\mathsf F(\mathcal T)\one_{\mathcal S}\bigr]=\E\sum_j\Psi(\mathcal T^{o\to j})\one_{\mathcal S}=p\,\E\Bigl[\Psi(\mathfrak E)\one\{T^-\text{ or }T^+\text{ is infinite}\}\Bigr]. \] Indeed, on the right side of the Mecke formula the residual process is an independent copy of $\Pi$, which means that the tree with root $o^-$ built from it is a marked $\PGW(p)$ tree $T^-$, and the added point $(h,T^+)$ is independent of it with the law of $(w/\sqrt p,\text{marked }\PGW(p)\text{ tree})$. Taking $\Psi\equiv1$ in \eqref{eq:giant-edge-limit}, we obtain $|\vec E(\gc)|/n\to p\,\P(T^-\text{ or }T^+\text{ is infinite})=p(1-(1-\varrho_p)^2)$, which is positive. Dividing \eqref{eq:giant-edge-limit} by this limit proves the second claim of (ii). The last claim follows by taking expectations, since the left side of the second display in (ii) is the conditional expectation of $\Psi$ at a uniformly chosen oriented edge of $\gc$, given the marked graph. \proofpart{Proof of (iii).} By \Cref{lem:giant-admissible}(ii), the function $x\mapsto\phi((G_{ab}(x,z_l))_{a,b,l})$ is admissible for every bounded continuous $\phi$. By the paragraph after \Cref{def:giant-functions}, its value at $[G_n,\mathbf V]$ is $\phi$ applied to the array in (iii). Since $\P^{\infty,\otimes k}$ is absolutely continuous with respect to the law of $k$ independent unconditioned trees, $\mathfrak F_k^\infty\in\mathscr G_k^{\mathrm{esa}}$ almost surely, and \Cref{lem:giant-continuity}(ii) identifies $G_{ab}(\mathfrak F_k^\infty,z_l)$ with $\delta_{ab}G^{(a)}_{oo}(z_l)$. The convergence in distribution then follows from part (i) and the bounded convergence theorem, and the convergence of the average over $\gc$ follows from part (i) with $k=1$. \proofpart{Proof of (iv).} Fix $z\in\Hup$. By \eqref{eq:giant-esd}, the Stieltjes transform of $\nu_n^{\gc}$ is $N_n^{-1}\sum_{v\in\gc}\langle\delta_v,(H_n^{\gc}-z)^{-1}\delta_v\rangle$. We apply part (iii) with $m=1$ and $\phi(\zeta)=\zeta\min\{1,(|\Im z|\,|\zeta|)^{-1}\}$, which is bounded, continuous, and equal to $\zeta$ for $|\zeta|\le|\Im z|^{-1}$. Since all resolvent entries involved are bounded by $|\Im z|^{-1}$, we find that the Stieltjes transform of $\nu_n^{\gc}$ at $z$ converges in probability to $\E^\infty G_{oo}(z)$, the Stieltjes transform of $\nu_p^\infty$. Applying \Cref{lem:giant-stieltjes}(ii) with $D=\{z\in\Hup:\Re z,\Im z\in\mathbb Q\}$ completes the proof. \proofpart{Proof of (v).} By \Cref{lem:giant-admissible}(iii), the function $x\mapsto\phi((\mu_{x,a}(I_l))_{a,l})$ is admissible for every bounded continuous $\phi$, and by the paragraph after \Cref{def:giant-functions} its value at $[G_n,\mathbf V]$ is $\phi((\mu_{n,V_a}(I_l))_{a,l})$. As in the proof of (iii), \Cref{lem:giant-continuity}(ii) identifies $\mu_{\mathfrak F_k^\infty,a}$ with the root spectral measure of the $a$-th tree of $\mathfrak F_k^\infty$, almost surely. The claims follow from part (i) with $k$, $1$ and $2$ roots. \end{proof} \section{The density of states}\label{sec:giant-dos} In this section, we prove the following two-sided bound for the averaged root spectral measure $\nu_p^\infty$ of \eqref{eq:giant-nup}. \begin{proposition}\label{prop:giant-dos} Fix $p>1$ and a compact set $J\subset\R\setminus\{0\}$. There exist constants $00$, and the resolvent identity gives the Aronszajn--Krein formula $F_u=F/(1+uF)$ \cite{Simon1995spectral}. Writing $F=\alpha+\iu\beta$ with $\beta>0$, we have $\Im F_u=\beta/|1+uF|^2$ and \[ \int_\R\Im F_u(\zeta)\dd u=\int_\R\frac{\beta\dd u}{(\alpha^2+\beta^2)u^2+2\alpha u+1}=\frac{2\pi\beta}{\sqrt{4(\alpha^2+\beta^2)-4\alpha^2}}=\pi. \] Let $B$ be an open interval and let $f$ be continuous with compact support in $B$ and $0\le f\le1$. Since $\pi^{-1}\Im F_u(E+\iu\eps)$ is the convolution of $\mu_u$ with the Poisson kernel $P_\eps(x)=\eps/(\pi(x^2+\eps^2))$, and $P_\eps*f\to f$ uniformly, we have $\int f\dd\mu_u=\lim_{\eps\downarrow0}\int f(E)\pi^{-1}\Im F_u(E+\iu\eps)\dd E$ for every $u$. By Fatou's lemma and Tonelli's theorem, \begin{align*} \int g(u)\int f\dd\mu_u\dd u &\le\liminf_{\eps\downarrow0}\int f(E)\,\frac1\pi\int g(u)\Im F_u(E+\iu\eps)\dd u\dd E\\ &\le\|g\|_\infty\int f(E)\dd E\le\|g\|_\infty|B|. \end{align*} Letting $f$ increase to $\one_B$ and using monotone convergence proves the claim for open intervals. A general interval $B$ is contained in open intervals of length $|B|+\eta$ for every $\eta>0$. \end{proof} Let $H_n^{\mathrm{bip}}$ be the $2n\times2n$ matrix of model (b), with the vertices $[n]\times\{0\}$ listed first. Then \[ H_n^{\mathrm{bip}}=\begin{pmatrix}0&A_n^*\\A_n&0\end{pmatrix}, \] where $A_n$ is the $n\times n$ matrix with $A_n(j,i)=h_{(j,1),(i,0)}$ if $\{(i,0),(j,1)\}$ is an edge and $A_n(j,i)=0$ otherwise. For $y\in[n]$, let $a_y=A_n^*\delta_y\in\C^n$ be the conjugate of the $y$-th row of $A_n$, let $d_y$ be the degree of $(y,1)$, and let $U_y=\|a_y\|^2$. If $d_y\ge1$, set $v_y=a_y/\sqrt{U_y}$. Then \begin{equation}\label{eq:giant-rowsum} K_n=A_n^*A_n=\sum_{y\in[n]:\,d_y\ge1}U_y|v_y\rangle\langle v_y|. \end{equation} \begin{lemma}\label{lem:giant-wegner} Let $0<\alpha_-<\alpha_+$. For every integer $n>p$ and every interval $I\subset[\alpha_-,\alpha_+]$, \[ \E\Tr\one_I(H_n^{\mathrm{bip}})\le\frac{2\sqrt p\,\alpha_+}{\sqrt{2\pi}\,\alpha_-^2}\,n|I|. \] \end{lemma} \begin{proof} We first record the conditional law of the rows. Conditionally on the graph, the rows of $A_n$ are independent, since they involve disjoint sets of marks. If $d_y=d\ge1$, the nonzero entries of $a_y$ are $d$ independent copies of $w/\sqrt p$ (up to conjugation, which does not change their law). The law of this vector in $\C^d$ has density $(p/\pi)^de^{-p\|\zeta\|^2}$, which is invariant under unitary maps. Then its direction $v_y$ is uniform on the unit sphere of the coordinate subspace, independent of $U_y$, and $U_y=p^{-1}\sum_{j=1}^d|w_j|^2$ is a sum of $d$ independent exponential variables with rate $p$, with density \[ f_{d,p}(u)=\frac{p^d}{\Gamma(d)}u^{d-1}e^{-pu}\one_{\{u>0\}}. \] We conclude that, conditionally on the graph and on all directions $(v_y)_{y:d_y\ge1}$, the squared row norms $(U_y)_{y:d_y\ge1}$ are independent, and $U_y$ has density $f_{d_y,p}$. This is the finite-volume analogue of \Cref{prop:tree-bipartite}. Next, we reduce to the square block. Let $E>0$. If $H_n^{\mathrm{bip}}(\varphi,\chi)=E(\varphi,\chi)$, then $A_n^*\chi=E\varphi$ and $A_n\varphi=E\chi$, which gives $K_n\varphi=E^2\varphi$; further, $\varphi\ne0$ unless $\chi=0$. Conversely, if $K_n\varphi=E^2\varphi$, then $(\varphi,E^{-1}A_n\varphi)$ is an eigenvector of $H_n^{\mathrm{bip}}$ with eigenvalue $E$. This shows that $(\varphi,\chi)\mapsto\varphi$ is a linear bijection from $\ker(H_n^{\mathrm{bip}}-E)$ onto $\ker(K_n-E^2)$. Summing over eigenvalues, we obtain $\Tr\one_I(H_n^{\mathrm{bip}})=\Tr\one_{I^2}(K_n)$ for every interval $I\subset(0,\infty)$, where $I^2=\{\lambda^2:\lambda\in I\}$. Set $B=I^2\subset[\alpha_-^2,\alpha_+^2]$. Since $K_n\one_B(K_n)\ge\alpha_-^2\one_B(K_n)$, using \eqref{eq:giant-rowsum} we obtain \[ \Tr\one_B(K_n)\le\alpha_-^{-2}\Tr\bigl(K_n\one_B(K_n)\bigr)=\alpha_-^{-2}\sum_{y:\,d_y\ge1}U_y\langle v_y,\one_B(K_n)v_y\rangle. \] Fix $y$ with $d_y=d\ge1$, and let $L_y=K_n-U_y|v_y\rangle\langle v_y|$. Let $\mathcal F_y$ be the $\sigma$-algebra generated by the graph, all directions, and $(U_{y'})_{y'\ne y}$. Conditionally on $\mathcal F_y$, the matrix $L_y$ and the vector $v_y$ are fixed, and $U_y$ has density $f_{d,p}$. By \Cref{lem:giant-averaging} with $g(u)=uf_{d,p}(u)$, we have \[ \E\bigl[U_y\langle v_y,\one_B(K_n)v_y\rangle\bigm|\mathcal F_y\bigr] =\int_0^\infty uf_{d,p}(u)\,\langle v_y,\one_B(L_y+u|v_y\rangle\langle v_y|)v_y\rangle\dd u\le c_d|B|, \] where \[ c_d=\sup_{u>0}uf_{d,p}(u)=\frac{d^de^{-d}}{\Gamma(d)}\le\sqrt{\frac{d}{2\pi}},\qquad d\ge1. \] The supremum is attained at $u=d/p$, and the inequality follows from Stirling's lower bound $\Gamma(d+1)\ge\sqrt{2\pi d}\,(d/e)^d$. We set $c_0=0$; with this convention, the rows with $d_y=0$, which have $U_y=0$ and do not appear in \eqref{eq:giant-rowsum}, contribute $c_{d_y}=0$. Taking expectations and using that $d_y$ is binomial with parameters $n$ and $p/n$, we obtain \[ \E\Tr\one_B(K_n)\le\alpha_-^{-2}|B|\sum_{y=1}^n\E c_{d_y}\le\frac{|B|}{\sqrt{2\pi}\,\alpha_-^2}\sum_{y=1}^n\E\sqrt{d_y}\le\frac{\sqrt p\,|B|}{\sqrt{2\pi}\,\alpha_-^2}\,n, \] where in the last inequality we used Jensen's inequality and $\E d_y=p$. Finally, if $I$ has endpoints $\alpha\le\beta$, then $|B|=(\beta+\alpha)(\beta-\alpha)\le2\alpha_+|I|$. \end{proof} \begin{lemma}\label{lem:giant-upper} Let $J=[E_-,E_+]\subset(0,\infty)$ with $E_->0$, and set $C_J=4(2\pi)^{-1/2}\sqrt p\,(E_++1)E_-^{-2}$. For every interval $I\subset J$, we have $\nu_p(I)\le C_J|I|$ and $\nu_p^\infty(I)\le\varrho_p^{-1}C_J|I|$. \end{lemma} \begin{proof} Let $V$ be uniform on the $2n$ vertices of model (b), independent of the marked graph, and let $m_n=\E\bigl[(2n)^{-1}\sum_{v}\mu^{\mathrm{bip}}_{n,v}\bigr]$, where $\mu^{\mathrm{bip}}_{n,v}$ is the spectral measure of $\delta_v$ for $H_n^{\mathrm{bip}}$. Then $m_n$ is a deterministic probability measure with $m_n(I)=(2n)^{-1}\E\Tr\one_I(H_n^{\mathrm{bip}})$, and its Stieltjes transform at $z\in\Hup$ is $\E G_{11}([G_n,V],z)$, in the notation of \Cref{def:giant-functions}. By \Cref{lem:giant-admissible}(ii) (applied with a bounded continuous $\phi$ that equals the identity on the disc of radius $(\Im z)^{-1}$) and \Cref{lem:giant-annealed} for model (b), this converges to $\E G_{11}(\mathfrak F_1,z)=\E G_{oo}(z)$, which is the Stieltjes transform of $\nu_p$. By \Cref{lem:giant-stieltjes}(i), $m_n\to\nu_p$ weakly. Let $J'=[E_-/2,E_++1]$. For an open interval $O\subset J'$, \Cref{lem:giant-wegner} with $\alpha_-=E_-/2$ and $\alpha_+=E_++1$ implies that $m_n(O)\le C_J|O|$, and by the portmanteau theorem, $\nu_p(O)\le\liminf_nm_n(O)\le C_J|O|$. If $I\subset J$ has endpoints $\alpha\le\beta$ and $0<\eta\le\min\{1,E_-/2\}$, then $I\subset(\alpha-\eta,\beta+\eta)\subset J'$ and $\nu_p(I)\le C_J(|I|+2\eta)$. Letting $\eta\downarrow0$ proves the first bound. By \Cref{lem:palm-survival}, $\E^\infty Z=\varrho_p^{-1}\E[Z\one_{\mathcal S}]\le\varrho_p^{-1}\E Z$ for every nonnegative random variable $Z$; with $Z=\mu_o(I)$, this gives the second bound. \end{proof} \subsection{The lower bound} For the lower bound, we work directly on the tree. We use the following consequence of the parity symmetry. By \Cref{prop:tree-symmetry}(i), almost surely, the spectral measure $\mu_o$ of $\delta_o$ for $\mathcal H$ is symmetric under $\lambda\mapsto-\lambda$. The same holds for the spectral measure of $\delta_v$ for the forward subtree operator $\mathcal H_v$, for every vertex $v$, since $V(\mathcal T_v)$ is a component of the finite removal that deletes the edge $\{v^-,v\}$. Let $\kappa_o$ be the image of $\mu_o$ under $\lambda\mapsto\lambda^2$, and let $F_o=s_{\kappa_o}$ be its Stieltjes transform. Then for every interval $I\subset(0,\infty)$, since $I\cup(-I)$ is the preimage of $I^2=\{\lambda^2:\lambda\in I\}$ under $\lambda\mapsto\lambda^2$, \begin{equation}\label{eq:giant-square-measure} \mu_o(I)=\tfrac12\bigl(\mu_o(I)+\mu_o(-I)\bigr)=\tfrac12\,\kappa_o(I^2). \end{equation} Let $Q=\{z\in\C:\Re z>0,\ \Im z>0\}$ be the open first quadrant, so that $z^2\in\Hup$ for $z\in Q$. For $z\in Q$, we have \begin{equation}\label{eq:giant-square-transform} G_{oo}(z)=\int\frac{(\lambda+z)\,\mu_o(\dd\lambda)}{\lambda^2-z^2}=z\int\frac{\mu_o(\dd\lambda)}{\lambda^2-z^2}=zF_o(z^2), \end{equation} where the term $\int\lambda\,\mu_o(\dd\lambda)/(\lambda^2-z^2)$ vanishes by symmetry (the integrand is bounded, since $|\lambda^2-z^2|\ge\Im z^2>0$ and $|\lambda|/|\lambda^2-z^2|\to0$ as $|\lambda|\to\infty$). By \Cref{prop:tree-bipartite}(v), $\kappa_o$ is the spectral measure of $\delta_o$ for $A^*A$, and the formula \eqref{eq:giant-pendant} below describes the effect of a pendant leaf as a rank-one perturbation of $A^*A$. We do not need this interpretation: the proof below only uses the Schur formula of \Cref{prop:tree-schur} and \eqref{eq:giant-square-transform}. \begin{lemma}\label{lem:giant-lower} Let $J=[E_-,E_+]\subset(0,\infty)$ with $E_->0$. There exists $c_J>0$, depending only on $p$ and $J$, such that $\nu_p^\infty(I)\ge c_J|I|$ for every interval $I\subset J$. \end{lemma} \begin{proof} We find a configuration at the root, of positive probability, on which $F_o(\lambda+\iu\eps)$ is an explicit function of the weight $t$ of a leaf edge. Averaging over $t$ produces a Poisson kernel, which gives a lower bound on $\E^\infty\Im F_o(\lambda+\iu\eps)$ uniformly in $\lambda\in[E_-^2,E_+^2]$ and small $\eps>0$. We then let $\eps\downarrow0$ and use \eqref{eq:giant-square-measure}. \proofstep{1}{The configuration.} Let $\mathcal E$ be the event that the root $o$ has exactly two children, one of which has no children while the forward tree of the other is infinite. On $\mathcal E$, we call these children $\ell$ and $y$, respectively, and set $t=|h_{o\ell}|^2$ and $t_0=|h_{oy}|^2$. Conditionally on $\{N=2\}$, the pairs $(h_{oj},\mathcal T_j)$, $j=1,2$, consisting of the edge mark and the marked forward tree of the $j$-th child, are independent, with $h_{oj}$ distributed as $w/\sqrt p$ and $\mathcal T_j$ a marked $\PGW(p)$ tree. The event $\mathcal E\cap\{N=2\}=\mathcal E$ is the disjoint union of $\{|\mathcal T_1|=1,\ |\mathcal T_2|=\infty\}$ and $\{|\mathcal T_2|=1,\ |\mathcal T_1|=\infty\}$, so \[ \P(\mathcal E)=\P(N=2)\cdot2e^{-p}\varrho_p=p^2e^{-2p}\varrho_p. \] Further, conditionally on $\mathcal E$, the variables $t$, $t_0$ and the marked forward tree $\mathcal T_y$ are independent; $t$ and $t_0$ have density $pe^{-ps}\one_{\{s>0\}}$, since $|w|^2$ is exponential with mean one; and $\mathcal T_y$ has law $\P^\infty$. Since $\mathcal E\subset\mathcal S$, by \Cref{lem:palm-survival} we have $\E^\infty[Z\one_{\mathcal E}]=\varrho_p^{-1}\E[Z\one_{\mathcal E}]$ for nonnegative $Z$. \proofstep{2}{The Schur formula on $\mathcal E$.} On $\mathcal E$, the forward tree of $\ell$ is a single vertex, whose operator is zero, and we have $\Gamma_\ell(z)=-1/z$. By \Cref{prop:tree-schur}, for $z\in Q$, \[ G_{oo}(z)=-\Bigl(z+t_0\Gamma_y(z)-\frac tz\Bigr)^{-1}=\frac{z}{t-q(z)}, \qquad q(z)=z^2+zt_0\Gamma_y(z). \] Combining this with \eqref{eq:giant-square-transform}, we obtain \begin{equation}\label{eq:giant-pendant} F_o(z^2)=\frac{1}{t-q(z)}\qquad\text{on $\mathcal E$, for $z\in Q$}. \end{equation} We claim that $\Im q(z)\ge\Im z^2>0$ for $z\in Q$. Let $m_y$ be the spectral measure of $\delta_y$ for $\mathcal H_y$, which is symmetric. As in \eqref{eq:giant-square-transform}, we have \[ z\Gamma_y(z)=\int\frac{z^2\,m_y(\dd\lambda)}{\lambda^2-z^2}=-1+\int\frac{\lambda^2\,m_y(\dd\lambda)}{\lambda^2-z^2}, \] and $\Im[\lambda^2/(\lambda^2-z^2)]=\lambda^2\Im z^2/|\lambda^2-z^2|^2\ge0$. This implies that $\Im(zt_0\Gamma_y(z))\ge0$, which proves the claim. Note that $q(z)$ is a function of $t_0$ and of the marked tree $\mathcal T_y$ only. \proofstep{3}{A Poisson kernel bound.} Let $\Lambda=[E_-^2,E_+^2]$ and $\delta=E_-^2/4$. For $q\in\Hup$, let $P_q(s)=\pi^{-1}\Im q/((s-\Re q)^2+(\Im q)^2)$ be the Poisson kernel. We claim that if $\dist(\Re q,\Lambda)\le\delta$ and $0<\Im q\le\delta$, then \begin{equation}\label{eq:giant-poisson} \int_0^\infty pe^{-ps}P_q(s)\dd s\ge c_*, \qquad c_*=\frac p2\,e^{-p(E_+^2+2\delta)}. \end{equation} Indeed, the interval $[\Re q-\Im q,\Re q+\Im q]$ is contained in $[E_-^2-2\delta,E_+^2+2\delta]\subset(0,E_+^2+2\delta]$, where $pe^{-ps}\ge pe^{-p(E_+^2+2\delta)}$, and $\int_{\Re q-\Im q}^{\Re q+\Im q}P_q(s)\dd s=\pi^{-1}(\arctan1-\arctan(-1))=1/2$. \proofstep{4}{Choice of parameters.} We choose, in this order, $\eps_0$, $M$ and $\tau$. For $\lambda\in\Lambda$ and $\eps>0$, let $z(\lambda,\eps)\in Q$ be the square root of $\lambda+\iu\eps$ in the first quadrant. Then \[ E_-\le\sqrt\lambda\le\Re z\le|z|=(\lambda^2+\eps^2)^{1/4}\le\sqrt{\lambda+\eps}\le\sqrt\lambda+\frac{\eps}{2E_-}, \qquad \Im z=\frac{\eps}{2\Re z}\le\frac{\eps}{2E_-}, \] where the first inequality for $\Re z$ uses $\Re z=((|\lambda+\iu\eps|+\lambda)/2)^{1/2}$. Set $\eps_0=\min\{1,2E_-,\delta/2\}$ and $J_+=[E_-,E_++1]$. Then for all $\lambda\in\Lambda$ and $0<\eps\le\eps_0$, the point $z=z(\lambda,\eps)$ satisfies \begin{equation}\label{eq:giant-chart} \Re z\in J_+,\qquad 0<\Im z\le1,\qquad |z|\le E_++1. \end{equation} By \Cref{lem:rde-moments} with $r=1/2$ and the compact interval $J_+$, there exists $C_1=C(p,J_+,1/2)$ with $\E^\infty|\Gamma_o(z)|^{1/2}\le C_1$ for all $z$ with $\Re z\in J_+$ and $0<\Im z\le1$. (This bound follows from the cavity equation of \Cref{prop:tree-schur} and \Cref{lem:palm-affine}, since the constant term $z$ has modulus at least $E_-$.) Set $M=16C_1^2$. By Markov's inequality, $\P^\infty(|\Gamma_o(z)|>M)\le C_1M^{-1/2}=1/4$ for all such $z$. Finally, set $\tau=\delta/(2(E_++1)M)$. \proofstep{5}{Lower bound for the averaged transform.} Let $\lambda\in\Lambda$, $0<\eps\le\eps_0$, $\zeta=\lambda+\iu\eps$ and $z=z(\lambda,\eps)$, so that $z^2=\zeta$. On the event $\mathcal E\cap\{t_0\le\tau\}\cap\{|\Gamma_y(z)|\le M\}$, \eqref{eq:giant-chart} gives \[ |q(z)-\lambda|\le|z^2-\lambda|+|z|\,t_0\,|\Gamma_y(z)|\le\eps+(E_++1)\tau M\le\frac\delta2+\frac\delta2=\delta, \] so that $\dist(\Re q(z),\Lambda)\le\delta$ and, by Step~2, $0<\Im q(z)\le\delta$. By \eqref{eq:giant-pendant}, on $\mathcal E$ we have $\pi^{-1}\Im F_o(\zeta)=\pi^{-1}\Im q/|t-q|^2=P_{q(z)}(t)$. Since $\Im F_o\ge0$ everywhere, Step~1 gives \[ \frac1\pi\E^\infty\Im F_o(\zeta)\ge\varrho_p^{-1}\E\Bigl[\one_{\mathcal E}\one\{t_0\le\tau\}\one\{|\Gamma_y(z)|\le M\}P_{q(z)}(t)\Bigr]. \] Conditionally on $\mathcal E$, the variable $t$ is independent of $(t_0,\mathcal T_y)$ and has density $pe^{-ps}$, while $q(z)$ is a function of $(t_0,\mathcal T_y)$. Integrating first over $t$ and using \eqref{eq:giant-poisson}, then using the independence of $t_0$ and $\mathcal T_y$ given $\mathcal E$, we obtain \begin{equation}\label{eq:giant-lower-transform} \begin{split} \frac1\pi\E^\infty\Im F_o(\lambda+\iu\eps) &\ge\varrho_p^{-1}c_*\,\P(\mathcal E)\,\P(t_0\le\tau\mid\mathcal E)\,\P^\infty\bigl(|\Gamma_o(z)|\le M\bigr)\\ &\ge\frac34\,c_*\,p^2e^{-2p}(1-e^{-p\tau})=c_1. \end{split} \end{equation} In the second inequality, we used that $\Gamma_y(z)$ is the root resolvent of $\mathcal T_y$, which has law $\P^\infty$ given $\mathcal E$, and Step~4. The constant $c_1>0$ depends only on $p$ and $J$, and \eqref{eq:giant-lower-transform} holds for all $\lambda\in\Lambda$ and $0<\eps\le\eps_0$. \proofstep{6}{From the transform to the measure.} Let $\bar\kappa=\E^\infty\kappa_o$, and let $f\ge0$ be continuous with compact support in $(E_-^2,E_+^2)$. For every sample, $\int f(\lambda)\pi^{-1}\Im F_o(\lambda+\iu\eps)\dd\lambda=\int(P_{\iu\eps}*f)\dd\kappa_o$, which converges to $\int f\dd\kappa_o$ as $\eps\downarrow0$ and is bounded by $\|f\|_\infty$. By dominated convergence, Tonelli's theorem and \eqref{eq:giant-lower-transform}, we have \[ \int f\dd\bar\kappa=\lim_{\eps\downarrow0}\int f(\lambda)\,\frac1\pi\E^\infty\Im F_o(\lambda+\iu\eps)\dd\lambda\ge c_1\int f(\lambda)\dd\lambda. \] Approximating the indicator of an open interval $O\subset(E_-^2,E_+^2)$ from below by such functions, we obtain $\bar\kappa(O)\ge c_1|O|$. \proofstep{7}{Conclusion.} Let $I\subset J$ be an interval with endpoints $\alpha<\beta$. Then $I^2\supset(\alpha^2,\beta^2)$, and \eqref{eq:giant-square-measure} and Step~6 give \[ \nu_p^\infty(I)=\tfrac12\,\E^\infty\kappa_o(I^2)\ge\tfrac12\,\bar\kappa\bigl((\alpha^2,\beta^2)\bigr)\ge\tfrac{c_1}2(\beta+\alpha)(\beta-\alpha)\ge c_1E_-|I|. \] This proves the lemma with $c_J=c_1E_-$; the case of a degenerate interval is trivial. \end{proof} \begin{proof}[Proof of \Cref{prop:giant-dos}] Since $J$ is compact and does not contain $0$, there exist $00$, the set $B$ is covered by countably many intervals of total length at most $|B|+\eta$, and their intersections with $[E_-,E_+]$ are intervals. It follows that the restriction of $\nu_p^\infty$ to $[E_-,E_+]$ is absolutely continuous with a density at most $C_{p,J}$ almost everywhere. By the Lebesgue differentiation theorem, for almost every $E\in(E_-,E_+)$ the density at $E$ is the limit of $\nu_p^\infty((E-h,E+h))/(2h)$ as $h\downarrow0$, which is at least $c_{p,J}$ by the lower bound. The same holds on $[-E_+,-E_-]$. Finally, let $E\ne0$, and apply \eqref{eq:giant-dos} to a compact interval around $E$ not containing $0$ and to $I=\{E\}$. This gives $\nu_p(\{E\})=\nu_p^\infty(\{E\})=0$, that is, $\E\mu_o(\{E\})=\E^\infty\mu_o(\{E\})=0$. \end{proof} \section{Eigenvalue counts and the transfer of phases}\label{sec:giant-transfer} \subsection{Eigenvalue counts} \begin{lemma}\label{lem:giant-polya} Let $m_n$ be random probability measures on $\R$ that converge weakly in probability to a deterministic probability measure $m$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every interval $I$ whose endpoints are not atoms of $m$, we have $m_n(I)\to m(I)$ in probability. \item If $m$ has no atoms in a compact interval $[a,b]$, then $\sup_I|m_n(I)-m(I)|\to0$ in probability, where the supremum is over all intervals $I\subset[a,b]$. \end{enumerate} \end{lemma} \begin{proof} For (i), first suppose that $I$ is bounded, with endpoints $\alpha\le\beta$, and let $r>0$. Let $f_+$ be continuous with $\one_{[\alpha,\beta]}\le f_+\le\one_{(\alpha-r,\beta+r)}$, and let $f_-$ be continuous with $\one_{[\alpha+r,\beta-r]}\le f_-\le\one_{(\alpha,\beta)}$ (with $f_-=0$ if $\beta-\alpha\le2r$). Then $f_-\le\one_I\le f_+$, and $\int(f_+-f_-)\dd m\le m((\alpha-r,\alpha+r))+m((\beta-r,\beta+r))$, which tends to $m(\{\alpha\})+m(\{\beta\})=0$ as $r\downarrow0$. Given $\eta>0$, we choose $r$ such that $\int(f_+-f_-)\dd m\le\eta$. Since $\int f_-\dd m_n\le m_n(I)\le\int f_+\dd m_n$ and $\int f_-\dd m\le m(I)\le\int f_+\dd m$, we obtain \[ \P\bigl(|m_n(I)-m(I)|>2\eta\bigr)\le\sum_{\pm}\P\Bigl(\Bigl|\int f_\pm\dd m_n-\int f_\pm\dd m\Bigr|>\eta\Bigr)\longrightarrow0. \] For a half-line, choose continuous upper and lower cutoffs only at its finite endpoint; their difference is supported in its $r$-neighborhood, so the same argument applies. The cases $I=\R$ and $I=\varnothing$ are immediate. For (ii), if $a=b$, every nonempty interval $I\subset[a,b]$ is $\{a\}$, and part (i) gives $m_n(\{a\})\to m(\{a\})=0$ in probability. Hence assume $a0$. Since $m$ has no atoms in $[a,b]$ and has countably many atoms, there exist points $x_01$ and a compact set $J\subset\R\setminus\{0\}$, and let $c_{p,J},C_{p,J}$ and $\rho_p^\infty$ be as in \Cref{prop:giant-dos}. \begin{enumerate}[label=\textup{(\roman*)}] \item For every interval $I\subset J$, we have $N_n^{-1}\Tr\one_I(H_n^{\gc})\to\nu_p^\infty(I)$ in probability. If $J$ is an interval, the convergence is uniform over intervals $I\subset J$, in probability. \item Let $E\ne0$ be a Lebesgue point of $\rho_p^\infty$. Then for every $\delta>0$, \[ \lim_{h\downarrow0}\limsup_{n\to\infty}\P\Bigl(\Bigl|\frac{1}{2hN_n}\Tr\one_{I_h(E)}(H_n^{\gc})-\rho_p^\infty(E)\Bigr|>\delta\Bigr)=0. \] \item For every nondegenerate interval $I\subset J$, \[ \P\Bigl(\tfrac12c_{p,J}N_n|I|\le\Tr\one_I(H_n^{\gc})\le2C_{p,J}N_n|I|\Bigr)\longrightarrow1. \] \end{enumerate} \end{proposition} \begin{proof} By \eqref{eq:giant-esd}, $N_n^{-1}\Tr\one_I(H_n^{\gc})=\nu_n^{\gc}(I)$. By \Cref{prop:giant-local}(iv), $\nu_n^{\gc}\to\nu_p^\infty$ weakly in probability, and by \Cref{prop:giant-dos}, $\nu_p^\infty$ has no atoms in $\R\setminus\{0\}$. Part (i) now follows from \Cref{lem:giant-polya}. For (ii), let $h_0>0$ be such that $[E-h_0,E+h_0]\subset\R\setminus\{0\}$, and let $00$, and (i) implies that $\nu_n^{\gc}(I)$ lies in $[\nu_p^\infty(I)/2,2\nu_p^\infty(I)]$ with probability tending to one. \end{proof} \subsection{Deficits} Recall the normalized local masses $P_{I,n}(v)$ and the deficit $\mathcal D_{I,n}(R)$ of \Cref{def:intro-phases}, and that $I_h(E)=(E-h,E+h]$. We define their limiting counterparts as follows. \begin{definition}\label{def:giant-deficit} For an interval $I$ with $\nu_p^\infty(I)>0$ and $R\in[0,\infty)$, the \emph{tree deficit} is \[ \mathcal D_I^\infty(R)=1-\E^\infty\min\Bigl\{\frac{\mu_o(I)}{\nu_p^\infty(I)},R\Bigr\}. \] \end{definition} By \Cref{prop:giant-dos}, $\nu_p^\infty(I)>0$ for every nondegenerate interval $I$ whose closure does not contain $0$. \begin{proposition}\label{prop:giant-transfer} Fix $p>1$. \begin{enumerate}[label=\textup{(\roman*)}] \item For every nondegenerate interval $I$ whose closure is contained in $\R\setminus\{0\}$ and every $R\in[0,\infty)$, we have $\mathcal D_{I,n}(R)\to\mathcal D_I^\infty(R)$ in probability. \item Let $U\subset\R\setminus\{0\}$ be open, and suppose that, $\P^\infty$-almost surely, the restriction of $\mu_o$ to $U$ is absolutely continuous. Then for Lebesgue-almost every $E\in U$, the limit $d_E(R)=\lim_{h\downarrow0}\mathcal D_{I_h(E)}^\infty(R)$ exists for every $R$, $\lim_{R\to\infty}d_E(R)=0$, and the giant component is uniformly integrable delocalized at $E$. \item Let $U\subset\R\setminus\{0\}$ be open, and suppose that, $\P^\infty$-almost surely, the restriction of $\mu_o$ to $U$ is singular with respect to Lebesgue measure (for instance, pure point). Then for Lebesgue-almost every $E\in U$ and every $R\in[0,\infty)$, we have $\lim_{h\downarrow0}\mathcal D_{I_h(E)}^\infty(R)=1$, and the giant component is spectrally condensed at $E$. \end{enumerate} \end{proposition} \begin{proof} \proofpart{Proof of (i).} Set $\bar m=\nu_p^\infty(I)$, which is positive by \Cref{prop:giant-dos}. For $v\in\gc$, write $X_{n,v}=\mu_{n,v}(I)\in[0,1]$, and let $\bar X_n=N_n^{-1}\sum_{v\in\gc}X_{n,v}=\bar\mu_n(I)$. Since the endpoints of $I$ are nonzero, \Cref{prop:giant-dos} implies that they are not atoms of $\nu_p$, and \Cref{prop:giant-local}(v) applies to $I$: for every bounded continuous $\psi\colon[0,1]\to\R$, \begin{equation}\label{eq:giant-psi} \frac1{N_n}\sum_{v\in\gc}\psi(X_{n,v})\longrightarrow\E^\infty\psi\bigl(\mu_o(I)\bigr) \end{equation} in probability. With $\psi(x)=x$, we find $\bar X_n\to\bar m$ in probability. On the event $\{|\bar X_n-\bar m|\le\bar m/2\}$, whose probability tends to one, we have $\bar X_n>0$ and $\mathcal D_{I,n}(R)=1-N_n^{-1}\sum_{v\in\gc}\min\{X_{n,v}/\bar X_n,R\}$, and for $x\in[0,1]$, \[ \Bigl|\min\Bigl\{\frac{x}{\bar X_n},R\Bigr\}-\min\Bigl\{\frac{x}{\bar m},R\Bigr\}\Bigr|\le x\Bigl|\frac1{\bar X_n}-\frac1{\bar m}\Bigr|\le\frac{2|\bar X_n-\bar m|}{\bar m^2}. \] Averaging over $v\in\gc$, we find that $|\mathcal D_{I,n}(R)-1+N_n^{-1}\sum_{v\in\gc}\min\{X_{n,v}/\bar m,R\}|\le2|\bar X_n-\bar m|/\bar m^2$ on this event. Applying \eqref{eq:giant-psi} with $\psi(x)=\min\{x/\bar m,R\}$, we conclude that $\mathcal D_{I,n}(R)\to1-\E^\infty\min\{\mu_o(I)/\bar m,R\}=\mathcal D_I^\infty(R)$ in probability. \proofpart{Differentiation of the root measure.} For the remaining parts, we use the Lebesgue differentiation theorem in the following form \cite[Chapter~7]{Rudin1987real}: if $m$ is a locally finite Borel measure on an open set $U\subset\R$, with Lebesgue decomposition $m=f\dd\lambda+m_{\mathrm s}$, then $m((E-h,E+h))/(2h)\to f(E)$ as $h\downarrow0$ for Lebesgue-almost every $E\in U$. The same holds with $I_h(E)$ in place of $(E-h,E+h)$, since $(E-h,E+h)\subset I_h(E)\subset(E-h',E+h')$ for $h'>h$; taking $h'=(1+\theta)h$ and then $\theta\downarrow0$ gives the claim. For $E\in\R$ and $h>0$, set \[ A_h(E)=\frac{\mu_o(I_h(E))}{2h}, \qquad b_h(E)=\E^\infty A_h(E)=\frac{\nu_p^\infty(I_h(E))}{2h}. \] For fixed $h$, the map $(\omega,E)\mapsto A_h(E)$ is jointly measurable, since $c\mapsto\mu_o((-\infty,c])$ is right-continuous and measurable in $\omega$ for each $c$. Since $h\mapsto\mu_o(I_h(E))$ is nondecreasing for fixed $E$, we have $(h_1/h)A_{h_1}(E)\le A_h(E)\le(h_2/h)A_{h_2}(E)$ for $h_10$, and choose $R_0$ with $d_E(R_0)<\delta$. Let $R\ge R_0$, so that $d_E(R)<\delta$. For all sufficiently small $h>0$, we have $\mathcal D_{I_h(E)}^\infty(R)<\delta$ and the closure of $I_h(E)$ is contained in $\R\setminus\{0\}$. For such $h$, part (i) implies that $\P(\mathcal D_{I_h(E),n}(R)>\delta)\to0$ as $n\to\infty$. We conclude that $\limsup_{h\downarrow0}\limsup_{n\to\infty}\P(\mathcal D_{I_h(E),n}(R)>\delta)=0$ for every $R\ge R_0$, and in particular the limit as $R\to\infty$ is zero. This is uniformly integrable delocalization at $E$ in the sense of \Cref{def:intro-phases}. \proofpart{Proof of (iii).} Since, for $\P^\infty$-almost every sample, $\mu_o$ restricted to $U$ is singular, the differentiation theorem implies that $A^+(E)=0$ for Lebesgue-almost every $E\in U$. As in the proof of (ii), Tonelli's theorem gives a Lebesgue-null set $\mathcal N_1'$ such that for $E\in U\setminus\mathcal N_1'$, we have $A_h(E)\to0$ as $h\downarrow0$, $\P^\infty$-almost surely. Fix $E\in U\setminus(\mathcal N_0\cup\mathcal N_1')$ and $R\in[0,\infty)$. By \eqref{eq:giant-bh}, $Z_h=A_h(E)/b_h(E)\to0$ almost surely, and $0\le\min\{Z_h,R\}\le R$. By dominated convergence, $\E^\infty\min\{Z_h,R\}\to0$, that is, $\mathcal D_{I_h(E)}^\infty(R)\to1$. Now fix $\delta>0$. For all sufficiently small $h>0$, we have $\mathcal D_{I_h(E)}^\infty(R)>1-\delta$ and the closure of $I_h(E)$ is contained in $\R\setminus\{0\}$, and part (i) implies that $\P(\mathcal D_{I_h(E),n}(R)>1-\delta)\to1$ as $n\to\infty$. This shows that $\lim_{h\downarrow0}\liminf_{n\to\infty}\P(\mathcal D_{I_h(E),n}(R)>1-\delta)=1$. Since $R$ and $\delta$ were arbitrary, the giant component is spectrally condensed at $E$. \end{proof} In both (ii) and (iii), the limit $n\to\infty$ is taken first, at fixed $h$ and $R$, then $h\downarrow0$, and in (ii) finally $R\to\infty$; this is the order in \Cref{def:intro-phases}. \subsection{Proof of \texorpdfstring{\Cref{thm:giant}}{the theorem on the giant component}} \begin{proof}[Proof of \Cref{thm:giant}] For part (i), let $p>1$ and let $J\subset\R\setminus\{0\}$ be compact. By \Cref{prop:giant-counts}(iii), the conclusion holds with the constants $c_{p,J}/2$ and $2C_{p,J}$ from \Cref{prop:giant-dos}. For part (ii), let $p\ge p_0$, and set $U_{\mathrm{ac}}=(-E_c(p),0)\cup(0,E_c(p))$ and $U_{\mathrm{pp}}=(-\infty,-E_c(p))\cup(E_c(p),\infty)$, which are open subsets of $\R\setminus\{0\}$. By \Cref{thm:main}, $\P^\infty$-almost surely, $\mathcal H$ is purely absolutely continuous on $U_{\mathrm{ac}}$. This means that the spectral measure of every vector, restricted to $U_{\mathrm{ac}}$, is absolutely continuous; in particular, so is the restriction of $\mu_o$. By \Cref{prop:giant-transfer}(ii), the giant component is uniformly integrable delocalized at Lebesgue-almost every $E\in U_{\mathrm{ac}}$. Again by \Cref{thm:main}, $\P^\infty$-almost surely, $\mathcal H$ has pure point spectrum on $U_{\mathrm{pp}}$. In particular, the restriction of $\mu_o$ to $U_{\mathrm{pp}}$ is a countable sum of point masses, which is singular with respect to Lebesgue measure. By \Cref{prop:giant-transfer}(iii), the giant component is spectrally condensed at Lebesgue-almost every $E\in U_{\mathrm{pp}}$. \end{proof} \begin{remark}\label{rem:giant-full-graph} The restriction to the giant component in \Cref{thm:giant} cannot be removed. For every $p>1$, the graph $G_n$ has linearly many finite components, and these support localized eigenvectors in every interval of nonzero energies. For instance, fix a compact interval $I\subset(0,\infty)$ with nonempty interior, and let $X_n(I)$ be the number of components of $G_n$ that consist of a single edge $e$ with $|h_e|\in I$. The function on $\mathscr G_1$ equal to $1$ if the component of the root is a single edge $e$ with $|h_e|\in I$, and $0$ otherwise, is continuous at every point except those whose root component is a single edge $e$ with $|h_e|$ an endpoint of $I$. Since the latter have probability zero under the law of $\mathfrak F_1$, this function is admissible, and \Cref{lem:giant-lln} implies that \[ \frac{2X_n(I)}{n}\longrightarrow\P\bigl(\mathcal T\text{ is a single edge $e$ with }|h_e|\in I\bigr)=p\,e^{-2p}\,\P\bigl(|w|\in\sqrt p\,I\bigr)>0 \] in probability. Each such component carries an eigenvector of $H_n$ with eigenvalue $|h_e|\in I$, supported on the two vertices of $e$, and these eigenvectors are orthogonal. Since $\Tr\one_I(H_n)\le n$, a fraction at least $X_n(I)/n$ of the eigenvalues of $H_n$ in $I$ belongs to eigenvectors supported on two vertices, and this fraction is bounded below in probability by a positive constant. This holds for every $p>1$, independently of the transition at $E_c(p)$; by \eqref{eq:giant-block}, these eigenvectors do not affect $H_n^{\gc}$. \end{remark} \backmatter \setlength{\bibsep}{3pt plus 1pt minus 1pt} \bibliographystyle{plainnat} \bibliography{refs} \end{document}