\chapter{Overview of the proof}\label{s:overview}

In this section, we explain the main steps of our proofs and the ideas behind them. The discussion is heuristic, and the precise statements and proofs are in Parts II--IV; \zcref{s:ov-organization} describes what each section of the paper does. We use the notation of \zcref{s:intro} and the conventions of \zcref{s:intro-notation}. This section can be read without any background in supersymmetry.

\section{Strategy and main difficulty}\label{s:ov-strategy}

All our results are derived from expectations of ratios of characteristic polynomials of $H$. Recall from \eqref{e:intro-ratio} that the source ratio $T_N^{(k)}(\mathbf u, \mathbf b)$ is the expectation of a ratio of $2k$ characteristic polynomials of $H$ to $2k$ others, evaluated at distance of order $1/N$ from the energy $E$, above and below the real axis. We also use its derivatives in perturbations of the spectral parameters at finitely many sites, which are expectations of products of resolvent entries at these sites.

The proof has two halves, which are joined by the interface theorem, \zcref{t:interface}. In its part (a), it states that $T_N^{(k)}(\mathbf u, \mathbf b)$ converges to the partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ of the supersymmetric sigma model on $[0, 1]$ of \zcref{d:partition}, as $N, W \to \infty$ with $X_N(E) \to X$, for sources near the base point $(\mathbf u, \mathbf b) = (0, \eps \1)$. Its part (b) states that the derivatives in perturbations at sites $i_1 < \dots < i_m$ with $i_a/N \to x_a \in (0, 1)$ converge to the same sigma-model expression with an explicit function of the field $Q$ inserted at each point $x_a$, also when neighboring sites are perturbed. We let $\kappa_E = E/2$. Then $-\kappa_E + \iu s_E$ is the value at $E + \iu 0$ of the Stieltjes transform of the semicircle law. Parts (c) and (d) state that the moments of the diagonal resolvent entries $G_{ii}(z)$, at distance of order $1/N$ from $E$, are bounded uniformly over all sites, and that $\E G_{ii}(z) \to -\kappa_E + \iu s_E$ at sites $i$ with $i/N$ away from $0$ and $1$. We prove the interface theorem in Part IV, and from it and from the sigma model we derive the eigenvalue and eigenvector statistics of \zcref{s:intro-results} in Part III. \zcref[S]{f:ov-structure} shows how the sections depend on each other.

The theorem holds for every source rank $k$ and every damping $\eps > 0$, with two energies $0 < E_* < E_{\mathrm{out}} \le 1/2$ that do not depend on $k$ or $\eps$. Its limits (a) and (b) hold up to the outer energy bound $E_{\mathrm{out}}$, the energy up to which the estimates of Part IV hold. Its uniform statements (c) and (d) hold up to the inner energy bound $E_*$, and the margin between the two bounds absorbs the shift of the energy by $O(1/N)$ in their proof. Part III also uses this margin to control the eigenvalues far from $E$, and our results are therefore stated for $|E| \le E_*$. \zcref[S]{r:iface-energies} explains the two bounds.

The main difficulty is that three features occur together at $W^2 \asymp N$. First, eigenvalue statistics require inverse determinants at energies in both half planes, and these bring in a noncompact symmetry group, as we explain in \zcref{s:ov-chain}. Characteristic polynomials, which are the subject of the earlier critical results recalled in \zcref{s:intro-background}, involve only the compact part of the sigma model. Second, the supersymmetric representation of $T_N^{(k)}$ is a one-dimensional chain of $N$ matrix-valued variables, and at $W^2 \asymp N$ it does not reduce to an integral over finitely many variables. Its slow variables diffuse over a time of order one along the chain, and we must control the chain over all $N \asymp W^2$ steps, with errors that remain summable over that many steps. Third, eigenvector statistics require all derivatives in the local perturbations, including every coefficient in the anticommuting variables, and they appear only in the limit $\eps \downarrow 0$ of the damping $\eps$ of \eqref{e:intro-ratio}, which comes after the limit $N \to \infty$.

Three steps of our argument are new, and we point out each where it is explained. The first is the control of the whole chain at $W^2 \asymp N$, including its noncompact part, at every source rank $k$ and with all anticommuting coefficients, by averaging over its fast variables, in \zcref{s:ov-averaging}. It replaces the spectral analysis of the transfer operator in the earlier work recalled in \zcref{s:intro-sources}, which concerns the regime above the threshold or, at the threshold, the compact part alone. The second is the attachment of the two ends of the chain with errors of order $e^{-\gamma W}$ for some $\gamma > 0$, in \zcref{s:ov-ends}. It replaces arguments by weak convergence or compactness, which give no rate, and a rate is needed, since the comparison along the chain multiplies these errors by quantities that we control only up to $e^{o(W)}$. The third is the extraction theorem for eigenvector moments, in \zcref{s:ov-eigenvectors}, which replaces the formal residue identities through which eigenvector statistics are computed in physics \cite[(3.4) and (3.69)--(3.70)]{mirlin2000statistics}, and which has, to our knowledge, no rigorous antecedent.

\begin{figure}[t]
\centering
\begin{tikzpicture}[x=1cm, y=1cm, font=\footnotesize, >={Stealth[length=4pt]},
  box/.style={draw, rounded corners=2pt, align=center, inner sep=3pt, minimum height=3.1em, text width=#1},
  part/.style={font=\scriptsize\scshape, text=black!60, rotate=90}]
 \node[part] at (-0.3, 0.4) {Part II};
 \node[box=3.3cm, minimum height=4.3em] (rep) at (2.025, 0.4) {exact transfer chain\\ \zcref{s:rep}};
 \node[box=3.3cm, minimum height=4.3em] (gr) at (6.045, 0.4) {Grassmann calculus\\ and localization\\ \zcref{s:grassmann}};
 \node[box=3.3cm, minimum height=4.3em] (sig) at (10.065, 0.4) {sigma model on $[0, 1]$\\ \zcref{s:sigma}};
 \node[box=3.3cm, minimum height=4.3em] (rad) at (14.085, 0.4) {sigma model\\ of rank one\\ \zcref{s:sigma-radial}};
 \node[part] at (-0.3, -3.0) {Part IV};
 \node[box=2.75cm, minimum height=4.3em] (hist) at (1.75, -2.1) {a priori bounds $e^{o(W)}$\\ \zcref{s:hist}};
 \node[box=2.75cm, minimum height=4.3em] (tube) at (4.9, -2.1) {fast modes, Ward identity\\ \zcref{s:tube}};
 \node[box=2.75cm, minimum height=4.3em] (stab) at (8.05, -2.1) {stability of powers\\ \zcref{s:stab}};
 \node[box=2.75cm, minimum height=4.3em] (cons) at (11.2, -2.1) {averaging\\ \zcref{s:cons}};
 \node[box=2.75cm, minimum height=4.3em] (end) at (14.35, -2.1) {ends of the chain\\ errors $e^{-\gamma W}$\\ \zcref{s:end}};
 \node[box=6.8cm] (asm) at (8.05, -3.95) {comparison and assembly, \zcref{s:assembly}\\ interface theorem, \zcref{t:interface}};
 \node[part] at (-0.3, -6.7) {Part III};
 \node[box=2.7cm] (proc) at (1.75, -5.8) {point process\\ \zcref{s:process}};
 \node[box=2.7cm] (ext) at (6.6, -5.8) {extraction\\ \zcref{s:extract}};
 \node[box=2.7cm] (res) at (9.7, -5.8) {noncompact boundary\\ \zcref{s:residue}};
 \node[box=2.7cm] (sca) at (1.75, -7.65) {the function $I_X$\\ \zcref{s:metallic,s:scales}};
 \node[box=2.7cm] (int) at (6.6, -7.65) {intensities, IPR\\ \zcref{s:intensity}};
 \node[box=2.7cm] (coa) at (9.7, -7.65) {coarse law\\ \zcref{s:coarse}};
 \node[box=6.0cm] (out) at (8.05, -9.5) {main results, \zcref{s:intro-results}};
 \draw[->] (gr) -- (rep);
 \draw[->] (gr) -- (sig);
 \draw[->] (sig) -- (rad);
 \draw[->] (rep.south) -- (hist.north);
 \draw[->] (rep.south) -- (tube.north);
 \draw[->] (sig.south) -- (cons.north);
 \draw[->] (tube) -- (stab);
 \draw[->] (stab) -- (cons);
 \draw[->] (hist.south) -- (asm.west);
 \draw[->] (tube.south) -- (asm.north west);
 \draw[->] (stab.south) -- (asm.north);
 \draw[->] (cons.south) -- (asm.north east);
 \draw[->] (end.south) -- (asm.east);
 \draw[->] (asm.south west) -- (proc.north);
 \draw[->] (asm.south) -- (ext.north);
 \draw[->] (rad.south) -- (12.8, -1.2) -- (12.8, -4.75) -- (res.north east);
 \draw[->] (proc) -- (sca);
 \draw[->] (ext) -- (int);
 \draw[->] (res) -- (coa);
 \draw[->] (ext.south east) -- (coa.north west);
 \draw[->] (res.south west) -- (int.north east);
 \draw[->] (proc.south east) -- (out.north west);
 \draw[->] (sca.south east) -- (out.west);
 \draw[->] (int.south) -- (out.north);
 \draw[->] (coa.south) -- (out.north east);
\end{tikzpicture}
\caption{The logical structure of the proof. An arrow from one box to another means that the second uses results of the first, and only the main dependencies are drawn (for instance, \zcref{s:end} also uses results of \zcref{s:tube,s:cons}). \zcref[S]{s:rep} is used throughout Part IV, and \zcref{s:sigma,s:sigma-radial} throughout Part III, which uses from Part IV only the interface theorem. The a priori bounds $e^{o(W)}$ on partial chains (integrals over consecutive sites of the chain) make the errors $e^{-\gamma W}$, for some $\gamma > 0$, of the attachment of the ends negligible.}
\label{f:ov-structure}
\end{figure}

\section{The transfer chain}\label{s:ov-chain}

We begin by explaining why a ratio of determinants becomes a one-dimensional chain. A determinant is a Gaussian integral over anticommuting, or Grassmann, variables, and an inverse determinant is a Gaussian integral over commuting complex variables, provided that the Hermitian part of the quadratic form is positive definite. We write each of the $4k$ determinants of $T_N^{(k)}$ in this way, as in \zcref{s:grass-motivation}. In the resulting integral, $H$ appears linearly in the exponent, and the normalizations of the commuting and the anticommuting Gaussians cancel. We may then take the average over $H$ inside the integral, and since $H$ is Gaussian, this average is the exponential of a quartic form in the integration variables, with the coefficients $J_{ij}$. We decouple this quartic form by a Hubbard--Stratonovich transformation (a Gaussian integral that turns the exponential of a quadratic form into an average of exponentials of linear forms), which introduces a supermatrix $S_i$ of size $(2k|2k)$, in the sense of \zcref{s:intro-results}, for each site $i \in [N]$. The BB and FF blocks of $S_i$ have commuting entries, while the BF and FB blocks have anticommuting entries.

The Gaussian field of this transformation has covariance $J$, so that its precision $J^{-1} = \Id - W^2 \Delta$ appears in the integrand. This is where the variance profile matters. For all functions $f$ on $[N]$, we have
\begin{equation}\label{e:ov-precision}
\sum_{i, j = 1}^N (J^{-1})_{ij} f(i) f(j) = \sum_{i = 1}^N f(i)^2 + W^2 \sum_{i = 1}^{N - 1} \big( f(i + 1) - f(i) \big)^2 ,
\end{equation}
and the same identity holds for supermatrices, with products replaced by supertraces of products. The precision thus couples only neighboring sites, although $J$ itself couples all pairs of sites. After the original variables are integrated out site by site, we obtain the exact representation of \zcref{p:rep-exact}. To state it, we define the chain integral $\mathcal Z_N$ and the onsite weight $w_\Gamma$ by
\begin{equation}\label{e:ov-chain}
\begin{gathered}
\mathcal Z_N(\Gamma) = \int \prod_{i = 1}^N w_\Gamma(S_i) \prod_{i = 1}^{N - 1} e^{-W^2 \Str(S_{i + 1} - S_i)^2/2} \dd S , \\
w_\Gamma(S) = e^{-\Str(S^2)/2 - \iu a_N^{-1} \Str(S \Gamma)} \, \Sdet(E - S)^{-1} ,
\end{gathered}
\end{equation}
where $\Gamma$ is the source supermatrix of \zcref{s:intro-results}, and $\Sdet$ is the superdeterminant of \zcref{s:grass-supermatrix}, which plays the role of the determinant for supermatrices. We let $\Gamma_0$ denote the value of $\Gamma$ at the base point of the sources, at which $T_N^{(k)} = 1$. Up to an explicit factor that tends to one, $T_N^{(k)}$ is the ratio $\mathcal Z_N(\Gamma)/\mathcal Z_N(\Gamma_0)$, so that the normalization of the integral cancels, and by \zcref{t:interface}(a) this ratio converges to the partition function $\mathcal Z^{(k)}_{X, \eps}$ of the sigma model. We have $\Str \Gamma = 0$, since $\sum_j u_j = 0$.

The integrand of \eqref{e:ov-chain} factors into terms that involve one site or two neighboring sites. The integral is a matrix element of $T_\Gamma^{N - 1}$, where the transfer operator $T_\Gamma$ of \zcref{d:rep-chain} acts on functions of one supermatrix by applying the Gaussian link and multiplying the result by $w_\Gamma$. The link is the heat kernel on supermatrices at time $(2W^2)^{-1}$, and \eqref{e:ov-chain} resembles the Feynman--Kac representation of a walk whose Gaussian steps have size $W^{-1}$, in a potential. The analogy is only formal. The weights are complex and contain anticommuting variables, so that probabilistic arguments do not apply to the chain, and every estimate of Part IV is an estimate for operators.

The commuting variables need care, and this is where inverse determinants make a difference. We call the $k$ pairs of determinants and inverse determinants with $\Im z > 0$ the retarded flavors, and the $k$ pairs with $\Im z < 0$ the advanced flavors. The Gaussian integral that represents $\det(H - z)^{-1}$ has an exponent of the form $\iu \zeta^* (z - H) \zeta$, with a complex vector $\zeta$, when $\Im z > 0$, and the opposite sign is needed when $\Im z < 0$. The commuting variables of the $2k$ inverse determinants carry an indefinite quadratic form, with $k$ positive and $k$ negative directions, whose symmetry group is the noncompact group $U(k, k)$. In the Hubbard--Stratonovich transformation, this has the following consequence. Integrating the commuting block of each $S_i$ over Hermitian matrices, as for the Gaussian unitary ensemble in \zcref{s:grass-example}, would produce an inverse superdeterminant with poles on the domain of integration. Following Sch\"afer and Wegner \cite{schafer1980disordered} (see \zcref{s:intro-sources}), we integrate this block instead over a cycle in complex space, on which the commuting block of $E - S$, with its advanced rows multiplied by $-1$, has a positive definite imaginary part. We construct the product of these cycles for the whole chain, with all anticommuting coefficients, in \zcref{s:contour}. The cycle is noncompact, and along its noncompact directions the integrand decays only through the damping $\eps/a_N$, which is of order $W^{-2}$. We will see in \zcref{s:ov-eigenvectors} that this slow decay is the origin of the Brownian motion in our eigenvector laws.

Local sources are treated in the same way. We call a supermatrix $Y = \diag(Y^{\mathrm B} \,|\, Y^{\mathrm F})$ that is added to $E + \iu a_N^{-1} \Gamma$ at one site $i$ a local source at $i$. It perturbs the spectral parameters at $i$ and multiplies the onsite weight at $i$ by $\Sdet(\Id + (E - S_i)^{-1} Y)^{-1}$. The derivatives of this factor at $Y = 0$ are polynomials in the entries of $(E - S_i)^{-1}$, and the dictionary of \zcref{l:rep-dictionary} identifies their integrals with expectations of polynomials in the resolvent entries of $H$ (for instance, a first derivative gives $-\E G_{ii}(z)$). We call the collection of all these derivatives at one site its complete local jet. The eigenvector observables need it in full, including its mixed and off-diagonal terms, since two derivatives at one site and two derivatives at neighboring sites have different limits.

\section{The saddle manifold and the two scales}\label{s:ov-scales}

We now explain why the chain is close to a sigma model for large $W$. The source-free onsite weight $w_0$, obtained by setting $\Gamma = 0$ in $w_\Gamma$, is stationary at the supermatrices $S$ with $S = (E - S)^{-1}$. For $S = \tau \Id$ with $\tau \in \C$, this equation reads $\tau (E - \tau) = 1$. Its solutions $\kappa_E \mp \iu s_E$ are minus the semicircle values $-\kappa_E \pm \iu s_E$ at $E \pm \iu 0$ recalled in \zcref{s:ov-strategy}, and the saddle-point equation is the self-consistent equation of the semicircle law. Since the equation is invariant under conjugation, its supermatrix solutions $\kappa_E \Id - \iu s_E Q$, with $Q^2 = \Id$, form orbits. Recall the supermatrix $\Lambda = \diag(\Lambda_{\mathrm B} \,|\, \Lambda_{\mathrm B})$ of \zcref{s:intro-results}, whose diagonal entries are $1$ on the retarded and $-1$ on the advanced flavors. The damping selects the orbit of the point $\kappa_E \Id - \iu s_E \Lambda$, at which the retarded flavors take the value $\kappa_E - \iu s_E$ and the advanced flavors the value $\kappa_E + \iu s_E$. This orbit is a copy of the target space $\mathcal M_k$ of the sigma model, on which $Q$ is the coordinate, and the onsite weight $w_0$ is identically one on it. In the FF block, the two values can also be distributed differently over the flavors. We call the union of all these orbits the saddle manifold, and we call the number $p \in \{0, \dots, 2k\}$ of diagonal entries of the FF block at which the value $\kappa_E - \iu s_E$ occurs the signature of an orbit. The saddle manifold has $2k + 1$ components, one for each signature. The copy of $\mathcal M_k$ is the component $p = k$, the balanced component of \zcref{s:rep-heuristics}, and we explain in \zcref{s:ov-averaging} why the other components disappear in the limit.

The cycle of \zcref{s:ov-chain} passes through the saddle manifold, and by \zcref{l:end-onsite}, the onsite weight on the cycle decays away from the saddle manifold. This is why the chain lives near the saddle manifold. Along the saddle manifold, the onsite weight is constant, while in the normal directions it has a nondegenerate critical point, with second derivatives of order one. The links $e^{-W^2 \Str(S_{i + 1} - S_i)^2/2}$ make the increments of the chain of size $W^{-1}$, and the two kinds of directions respond to them differently. We set $h = W^{-1/2}$, so that the increment of one link is of size $h^2$. In the normal directions, the chain is a discretized Ornstein--Uhlenbeck process (a Gaussian autoregressive chain) with steps of size $h^2$ and a confinement of order one. Its normal coordinate relaxes by a factor $1 - c h^2$ per step, over about $h^{-2} = W$ steps. Its stationary size is of order $h$, since a chain with steps of size $h^2$ and relaxation rate $h^2$ has a stationary variance of order $h^4/h^2 = h^2$. In the tangent directions, the chain performs a random walk along the saddle manifold, each step of which has variance $\delta_W = h^4 = W^{-2}$ (the square of the increment of one link). Over the $N \asymp h^{-4}$ steps of the chain, it moves by a distance of order one, while over the $h^{-2}$ steps that the normal coordinate needs to relax, it moves only by $h$. (In the language of physics, the normal and the tangent variables are the massive and the Goldstone modes.) \zcref[S]{f:tube-scales} shows the two scales.

The total time of the tangent walk is the parameter $X_N(E)$ of our results. Indeed, to first order, one link acts on functions of the tangent variables as the heat semigroup $e^{-t \mathsf C_k}$ at time $t = (4 s_E^2 W^2)^{-1}$, where $\mathsf C_k$ is the Casimir operator of \zcref{s:intro-results}, minus the invariant Laplace operator of $\mathcal M_k$. The reason is that the link factor, a Gaussian in the distance between neighbors, is the heat kernel of the ambient space at the time $(2 W^2)^{-1}$, and \zcref{l:tube-casimir} identifies the restriction of the ambient heat generator to the saddle manifold as $-(2 s_E^2)^{-1} \mathsf C_k$. Since $4 s_E^2 = 4 - E^2$, the $N$ links of the chain run this semigroup for the time
\begin{equation}\label{e:ov-time}
\frac{N}{4 s_E^2 W^2} = \frac{N}{(4 - E^2) W^2} = X_N(E) .
\end{equation}
The source factor $e^{-\iu a_N^{-1} \Str(S \Gamma)}$ of an onsite weight becomes $e^{-s_E a_N^{-1} \Str(Q \Gamma)}$ on the saddle manifold, since $\Str \Gamma = 0$. Since $N s_E/a_N = \pi$, the $N$ source factors combine to $\exp(-\pi \int_0^1 \Str(Q(t) \Gamma) \dd t)$, where $Q(t)$ is the tangent variable at the site $tN$.

Heuristically, the chain should therefore converge to the sigma model of \zcref{s:intro-results}. We let $\mathsf G_{X, \Gamma} = X \mathsf C_k + \pi \Str(Q \Gamma)$ denote its generator. In this notation, $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) = \int e^{-\mathsf G_{X, \Gamma}} 1$, with the invariant Berezin integral over $\mathcal M_k$ of \zcref{s:intro-results}. In probabilistic terms, this would be the expectation of $\exp(-\pi \int_0^1 \Str(Q(t) \Gamma) \dd t)$ for a Brownian motion $Q(t)$ on $\mathcal M_k$ with generator $-X \mathsf C_k$, integrated over its starting point, where the time $t$ is the position along the chain. The noncompact factor $U(k, k)/(U(k) \times U(k))$ of the underlying manifold of $\mathcal M_k$ comes from the inverse determinants, and the compact factor from the determinants. A Brownian motion on this superspace is only a heuristic notion, and \zcref{s:sigma} defines the semigroup $e^{-t \mathsf G_{X, \Gamma}}$ analytically, on weighted $L^2$ spaces. The weights are needed since the constant function $1$, to which the semigroup is applied, is not square integrable over the noncompact factor.

The sigma model inherits the normalization of the source ratios, $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$. This is a consequence of supersymmetry, and since the same mechanism appears twice more below, we describe it. In flat space, there exists an odd vector field $\mathfrak q$, the supercharge of \zcref{s:grass-susy}, which turns commuting variables into anticommuting ones and has zero divergence. Consider a superfunction that $\mathfrak q$ annihilates. An integration by parts shows that inserting a factor $e^{-c \mathfrak q \sigma}$, where $\sigma$ is an odd function for which $\mathfrak q \sigma$ is a Gaussian exponent in all variables, does not change its integral, for all $c > 0$. As $c \to \infty$, the inserted Gaussian concentrates the integral at the origin, where the Gaussian factors of the commuting and the anticommuting variables cancel. Therefore, the integral of the superfunction is its value at the origin, as in the dimensional reduction of Parisi and Sourlas \cite{parisi1979random}. We call a superfunction closed if a supercharge annihilates it. By the same argument, which is \zcref{l:sigma-localization}, the integral over $\mathcal M_k$ of a closed superfunction is its value at the base point $Q = \Lambda$. At the base point of the sources, the derivative in $t$ of the integral of $e^{-t \mathsf G_{X, \Gamma_0}} 1$ reduces, after an integration by parts, to the integral of a closed superfunction that vanishes at $\Lambda$. By localization, this derivative is zero, and we obtain the normalization in \zcref{c:sigma-normalization}.

\section{Averaging over the fast modes}\label{s:ov-averaging}

We now describe how Part IV proves the convergence of the chain to the sigma model, which is the main part of the proof of \zcref{t:interface}(a). We write a supermatrix near the saddle manifold in terms of a point of the saddle manifold and a normal displacement $U$, in a tube of fixed small radius around the saddle manifold. We conjugate the transfer operator by the square root of the source-free onsite weight and by a Gaussian in $U$, so that the normal variables live in the Gaussian Hilbert space of a harmonic oscillator, with all their anticommuting coefficients. We let $A_h$ denote the resulting operator, which we call the adapted transfer operator, as in \zcref{s:tube-gauge}.

We call the operator obtained from $A_h$ by removing the tangent motion and the source the frozen normal transfer. By \zcref{l:tube-quadratic}, it is to leading order the transfer operator of a supersymmetric harmonic oscillator, namely the Ornstein--Uhlenbeck semigroup at a complex time of order $h^2$, tensored with its finite-dimensional analogue for the anticommuting normal variables. We let $\Omega_0$ denote the vacuum of the oscillator, which is the constant function in its Gaussian Hilbert space, and we call the number by which one step of the oscillator multiplies $\Omega_0$ its zero-point factor. The commuting and the anticommuting normal coordinates contribute reciprocal factors to it, and in the balanced component, where the two kinds of coordinates have equal dimensions, these factors cancel. There the oscillator fixes $\Omega_0$ and contracts the orthogonal complement of the vacuum by a factor $1 - ch^2$ per step.

Since the normal variables equilibrate after about $h^{-2}$ steps, long before the tangent variables move, the evolution of the tangent variables should involve the normal ones only through their equilibrium. In the language of semigroup theory, this is averaging to first order over the fast variables, as in Kurtz's theorem on perturbed semigroups \cite{kurtz1973limit}. Rather than an averaging theorem, our argument uses the Trotter--Kurtz approximation theorem \cite{trotter1958approximation,kurtz1969extensions}, and we check its two conditions, consistency on a suitable set of vectors and stability of the powers of $A_h$. The difficulty in both is the length of the chain. An error of size $h^3$ per step, which looks small, adds up to $h^{-1}$ over the $h^{-4}$ steps, and every error must be compared either with the gap $h^2$ of the oscillator or with the variance $\delta_W$ of the tangent step.

The first obstacle is the normalization of the fast chain. The frozen normal transfer differs from the oscillator by the anharmonic terms of the onsite weight, which connect the vacuum with excited states at order $h^3$ and change the vacuum itself at order $h^4$. A perturbative argument places its top eigenvalue at $1 + O(h^3 \cdot h^3/h^2 + h^4) = 1 + O(h^4)$. However, $(1 + C h^4)^{h^{-4}}$ tends to $e^C$, so that such an argument would have to determine the coefficient of $h^4$ exactly, including all couplings of the anticommuting variables, to keep the normalization of the chain. To overcome this obstacle, we use supersymmetry, which shows that the relevant mean is preserved exactly. The normal oscillator has its own supercharge $\mathfrak q_{\rm nor}$, an odd operator with $\mathfrak q_{\rm nor}^2 = 0$. As in \zcref{s:ov-scales}, we call the vectors that it annihilates closed. All factors of the frozen normal transfer are invariant, so that it commutes with $\mathfrak q_{\rm nor}$ and maps the closed vectors into closed vectors. For a closed vector, the normalized Gaussian integral over the normal variables is its value at the origin, by the localization argument of \zcref{s:ov-scales}, and the invariant factors of the step equal one at the origin. It follows that the frozen normal transfer preserves the Gaussian mean of closed vectors exactly, whatever its anharmonic terms are. This is \zcref{p:tube-gap}, and we call it the Ward identity, a name that the physics literature gives to all identities obtained from supersymmetry in this way.

Since the frozen normal transfer also contracts the closed vectors of mean zero by the factor $e^{-ch^2}$ per step, Banach's fixed point theorem implies that there exists a unique closed vector $\Omega_h$ of mean one that it leaves invariant. By \zcref{l:cons-fixed-state}, it differs from the vacuum by $O(h)$, and every closed vector of mean one relaxes to it at the rate $h^2$ per step. Once the ends of the chain are attached to the tube, as in \zcref{s:ov-ends}, they relax to $\Omega_h$ in this way. (A supersymmetry of the transfer operator was used by Disertori, Lohmann, and Sodin \cite{disertori2021density} to fix its top eigenvalue.)

We call the set of vectors $\Omega_h \otimes f$, where $f$ is a function on the saddle manifold, the invariant normal line. The adapted step acts on it, to first order, as one step of the semigroup of the sigma model. Using the time normalization \eqref{e:ov-time} in \zcref{p:cons-consistency}, we find that
\begin{equation}\label{e:ov-consistency}
A_h (\Omega_h \otimes f) = \Omega_h \otimes \big( f - N^{-1} \mathsf G_{X_N(E), \Gamma} f \big) + O(N^{-1} h)
\end{equation}
for smooth functions $f$ with compact support, with an error measured in norms of finitely many derivatives of $f$. The tangent heat of one link contributes the Casimir operator, and the source factor contributes the potential. This averaging on the invariant normal line, with its normalization made exact by the Ward identity, is the first new step of our argument. Over the $N$ steps of a macroscopic time, the errors add up to $O(h)$, which tends to zero. Heuristically, first order suffices because the tangent generator moves vectors off the invariant normal line only at order $h$, since its normal derivatives annihilate $\Omega_h$, so that the correction of second order in the averaging vanishes in the limit.

The second obstacle is stability. The Trotter--Kurtz theorem also requires that the powers $A_h^n$ with $n \le T h^{-4}$ be bounded, uniformly in $h$, on all vectors and not only on the invariant normal line. It is natural to try to prove this by an energy estimate, showing that the generator of the adapted step is dissipative in the Gaussian Hilbert space of the normal oscillator. However, this approach fails for the anticommuting normal variables. Their part of the step has an eigenvalue of modulus at least $1 + ch^2$, which is compensated by the zero-point factor of the commuting variables alone, and \zcref{r:stab-fock} explains why every metric of product form loses this compensation. To avoid this loss, we integrate the anticommuting normal variables against the complete Gaussian factor of the step before we take any norm.

We then compare $A_h$ with a reference contraction $C_h$, the product of the oscillator and of the heat flow on the saddle manifold during one link. We bound $A_h - C_h$ relative to the defects $\Id - C_h^* C_h$ and $\Id - C_h C_h^*$, which measure how much $C_h$ dissipates. A perturbation that is small relative to these defects keeps the powers bounded, by \zcref{l:tools-defect}. In this way, we obtain $\|A_h^n\| \le e^{CT}$ for $n \delta_W \le T$, which is \zcref{t:stab-main} and the most technical estimate of Part IV. The comparison rests on the kernel of the step in coordinates centered at the midpoint of the link. In these coordinates, the Gaussian factor of a link is an even function of the tangent increment, and the terms of the kernel that are odd in the increment gain a factor $h^2$ when they are integrated.

With stability and consistency, the Trotter--Kurtz theorem implies that $A_h^n (\Omega_h \otimes f) \to \Omega_h \otimes e^{-t \mathsf G_{X, \Gamma}} f$ as $n/N \to t$, which is \zcref{t:cons-limit}. The other components of the saddle manifold disappear in the same framework. In a component of signature $p \ne k$, the normal variables have $2(p - k)^2$ more commuting than anticommuting dimensions, with an excess of $(p - k)^2$ in each of the two root blocks. The zero-point factors of the oscillator then do not cancel, and they leave a factor of modulus $e^{-c (p - k)^2 h^2}$ per step. Over a stretch of $\theta N$ steps, this gives the factor $e^{-c \theta W}$.

\section{The ends of the chain}\label{s:ov-ends}

We now turn to the ends of the chain. The chain begins and ends with the constant function $1$ on the cycle of \zcref{s:ov-chain}, and this function is far from the invariant normal line in three ways. It does not decay in the normal variables. In the Gaussian space of the normal oscillator, it behaves like the inverse of the vacuum, and a decaying Gaussian factor is needed to give it a finite norm. It is spread over all $2k + 1$ components of the saddle manifold. Finally, it lives on the cycle and not on the tube. The second new step of our argument is the attachment of the ends to the tube with exponentially small errors.

The errors must be exponentially small for the following reason. We call the chain on the cycle the physical chain, to distinguish it from the adapted chain on the tube, and we call $T_\Gamma^j 1$, the integral over $j$ consecutive sites as a function of the variable at its free end, a partial chain. In the final comparison, every change from the cycle to the tube near an end is paired with a physical partial chain, and the size of these partial chains is not obvious. One physical step has norm at most $e^{C/W}$ in a suitable weighted space, which gives only $e^{CW}$ over $N \asymp W^2$ steps. Observe that this is the same exponential order as the errors that Gaussian concentration produces, which are of size $e^{-cWr^2}$ at normal distance $r$ from the saddle manifold, since the normal fluctuations are of size $h = W^{-1/2}$. We need two estimates, an a priori bound $e^{o(W)}$ on the partial chains and attachment errors $e^{-\gamma W}$ for some $\gamma > 0$.

The a priori bound of \zcref{t:hist-bound}, which is of order $e^{C \sqrt W (\log W)^2}$, combines two estimates of a different nature. Away from cores of radius $h \log W$ around the saddle manifold, one step contracts by the factor $e^{-c (\log W)^2/W}$, since the onsite weight decays there. Near the saddle manifold, we return to random matrices. The Gaussian part of a partial chain whose first site is linked to a given point $S$ of the cycle can be integrated back, by reading the Hubbard--Stratonovich transformation backward. The resulting anchored identity, \zcref{p:rep-anchored}, writes the partial chain as the expectation, over an $n \times n$ Gaussian band matrix, of a quotient of two determinants, one from commuting and one from anticommuting variables, in which $S$ appears linearly. Using Hua's determinant inequality \cite{hua1955inequalities}, we compare the two determinants, and we bound the expectation by integrating over the diagonal entries of the band matrix, with the version of Minami's estimate \cite{minami1996local} proved by Graf and Vaghi \cite{graf2007remark}.

Next, we explain the attachment, which takes place in a block at one end of the chain, whose length is a universal constant times $W$. Since each link is a heat kernel at time $(2W^2)^{-1}$, the links of this block amount to a heat flow of total time of order $1/W$, which is short, so that a growth rate of order $W$ per unit of heat time adds up to a constant factor over the block. This is why plain weighted energy estimates suffice at the ends, while the long stretches of the chain need the stability analysis of \zcref{s:ov-averaging}. Three such estimates are combined in \zcref{s:end}. The first concerns the region far from the saddle manifold, where the constant function is not small but the onsite weight decays. An estimate of Agmon type \cite{agmon1982lectures} (an energy estimate with an exponential weight) shows that the partial chain over the block stays bounded against a weight that grows with the distance from the saddle manifold and is raised gradually during the block, and hence that it is exponentially small far from the saddle manifold.

The second estimate compares the block near the saddle manifold with reference vectors on the tube, which are built from adapted steps. The two satisfy the same local equation between steps and agree initially near the saddle manifold, so that their difference is created only at a normal distance of order one from it. A Davies--Gaffney estimate \cite{davies1989heat} shows that a heat flow of duration of order $1/W$ carries a disturbance over a distance of order one only with a factor $e^{-cW}$, which yields the error $e^{-\gamma W}$. The third estimate gives the references a Gaussian decay in the normal variables, which the comparison along the rest of the chain needs, although at the start of the block they have finite norm only with a decaying Gaussian weight. Heuristically, since each step relaxes the normal coordinate by a factor $1 - ch^2$, the exponent of a Gaussian weight in the normal variables can be raised by an amount of order $h^2$ per step at a bounded cost, and over a stretch of order $W$ steps it can move from negative to positive values.

One point requires care. In \zcref{t:interface}, the energy bounds $E_*$ and $E_{\mathrm{out}}$ are fixed first, and the rank $k$, the damping $\eps$, and the local sources of part (b) only afterward. The estimates that restrict the energy, at the ends of the chain and in the consistency analysis of \zcref{s:cons}, must therefore hold with constants that depend on none of these choices. These universal constants are not computed. They determine $E_{\mathrm{out}}$, together with the restriction $|E| \le 1/2$ of the a priori bound of \zcref{s:hist}, and this is why $E_{\mathrm{out}}$, and with it the energy $E_* < E_{\mathrm{out}}$ of our results, is not explicit.

We compare the physical chain with the adapted one through a telescoping identity in one-sided form, \zcref{l:asm-telescope}. We attach the chain to the tube only at its right end, and we write $\mathcal Z_N(\Gamma)$ as the pairing of the constant function $1$ with the chain applied to the attached block. In this form, each local error is paired with a physical partial chain started from $1$, which the a priori bound controls, and the total error is of order $N e^{-\gamma W + o(W)}$. Finally, \zcref{l:asm-return} moves the pairing with the constant function from the cycle back to the tube by Stokes' theorem along an explicit homotopy, with an error of the same exponential order. On the tube, an initial layer of about $h^{-2} \log(1/h)$ adapted steps, whose tangent time tends to zero, relaxes the factors at both ends to $\Omega_h \otimes 1$, by the relaxation of closed vectors described in \zcref{s:ov-averaging}, and the unbalanced components are suppressed on stretches of length of order $N$. By the limit of the adapted chain on the invariant normal line, the balanced term converges to the integral over $\mathcal M_k$ of the product of $e^{-\theta \mathsf G_{X, \Gamma}} 1$ and $e^{-(1 - \theta) \mathsf G_{X, \Gamma}} 1$, for every splitting point $\theta \in (0, 1)$, and this integral is $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$. The normal factors contribute the normalized Berezin integral of $\Omega_h^2$ over the normal variables, which \zcref{s:tube-oscillator} calls the analytic Berezin pairing of $\Omega_h$ with itself, and which is one by localization. This proves \zcref{t:interface}(a).

Local sources need one further idea. The physical comparison keeps exactly the complete local jet at each site that carries a local source. A complete jet changes the bounds only by a constant factor and a bounded number of polynomial weights, and it leaves the Gaussian decay unchanged, so that the estimates at the ends hold for blocks that contain such sites, with the same universal constants, also when such sites are adjacent. This is \zcref{l:asm-jet-block}.

Only afterward, inside the adapted chain, whose powers are bounded, do we replace the jet by its value on the saddle manifold. We call $\mathcal G(Q) = -\kappa_E \Id + \iu s_E Q$ the orbit resolvent. On the tube, we have $(E - S)^{-1} = -\mathcal G(Q) + O(h)$. The retarded and advanced diagonal entries of $\mathcal G(\Lambda)$ are the semicircle limits $-\kappa_E \pm \iu s_E$ of $G_{ii}(E \pm \iu 0)$. In this order of the two replacements, the algebraic error $O(h)$ is paired with the stable adapted chain and not with a physical partial chain, whose size is known only to be $e^{o(W)}$. We call the function obtained from the jet by replacing $(E - S)^{-1}$ with $-\mathcal G(Q)$ the orbit multiplier of the site, as in \eqref{e:iface-multiplier}. In the limit, the jet acts as the multiplication by its orbit multiplier. This multiplier is a polynomial in the entries of $\mathcal G(Q)$. It acts trivially on the normal factor and therefore preserves $\Omega_h$, and we obtain \zcref{t:interface}(b). Derivatives at distinct sites, even adjacent ones, become separate multipliers, while all derivatives at one site become a single multiplier, which contains the off-diagonal entries of $\mathcal G(Q)$.

Parts (c) and (d) of \zcref{t:interface} follow from the same comparison. For the moments of $G_{ii}$ at every site, also close to the ends, we attach the chain at the end farther from $i$, so that the stretch of the chain that arrives at $i$ has length of order $N$, and we use the boundedness of the pairing with the adapted chain rather than its limit. The mean of $G_{ii}$ at sites away from the ends is the case of one first derivative, which localization evaluates as $-\kappa_E + \iu s_E$.

\section{From the sigma model to the point process}\label{s:ov-process}

We now turn to Part III, which uses \zcref{t:interface} and the sigma model, and nothing else from Part IV. The convergence of $\Xi_N$ in \zcref{t:process} rests on three properties. The bounds of \cite{peled2019wegner} on factorial moments of eigenvalue counts, which are estimates of Minami type, give tightness, and together with the asymptotics of the expected density of eigenvalues from \cite{disertori2021density} they show that every limit is simple and has intensity one. Next, $H$ has a Gaussian component proportional to the identity, independent of the rest of $H$, whose standard deviation after unfolding is at least of order $W^{1/2}$ by \zcref{l:iface-gaussian}. A translation of the unfolded spectrum by a bounded amount therefore changes its law by $O(W^{-1/2})$, and every limit is stationary. Finally, the sources at one damping determine the law, by Fej\'er's theorem as explained after \zcref{t:process}, since they determine the Laplace functionals of the process. Since \zcref{t:interface}(a) identifies $T_N^{(k)}$ only near the base point, we also use that $T_N^{(k)}$ is entire in the parameters $\mathbf u$ and $\mathbf b$, which appear only in the numerator of \eqref{e:intro-ratio}, with bounds that are uniform in $N$, and a normal-families argument (Montel's theorem) extends the convergence to all parameters.

The rank-one derivatives in \zcref{t:process}(c) are where the singlet state and the kernels of \zcref{d:kernels} come from. Superfunctions on $\mathcal M_1$ that are invariant under the stabilizer of $\Lambda$ are functions of the two coordinates $(B, F) \in D$ of \zcref{s:intro-results}. By \zcref{l:sigma-radial-part}, the operator $\mathsf C_1$ acts on them as $-d \Delta_D d^{-1}$, where $d = B - F$ and the conjugation by $d$ comes from the four anticommuting variables. The generator reduces to $\mathsf G_{X, z} = X \mathsf C_1 + z d$, with $z = 2\pi\eps$, at the base point of the sources. Thus $e^{-t \mathsf G_{X, z}} 1$ is the probability that the singlet diffusion of \zcref{r:intro-singlet} survives killing at rate $z d$ up to time $t$, which is the singlet state $\Psi^{X, z}_t$. In the coordinates $(B, F)$, the invariant integral over $\mathcal M_1$ consists of an integral over $D$ and a point mass at the corner $(1, 1)$, which corresponds to $\Lambda$. We call this point mass the corner term. It is the Efetov--Wegner term of polar coordinates described in \zcref{s:grass-boundary}, and localization computes it. The second derivative in \zcref{t:process}(c) is a sum of pairings with two insertions along the wire. Each insertion splits into a value at the corner, which the semigroup preserves by localization, and a part that evolves by the diffusion $(\tilde B_t, \tilde F_t)$ of \zcref{d:kernels}, and these parts produce the kernel $K_X$ and the function $I_X$.

\section{Eigenvector moments}\label{s:ov-eigenvectors}

Eigenvector moments are extracted from local densities at positive damping. Recall that the local density $\ell^\eps_{i, N}$ of \zcref{s:intro-results} is the Poisson smoothing of the measure $N \sum_\alpha w_\alpha(i) \delta_{x_\alpha}$. Let $\Pi_N$ denote the expectation of the measure $N^2 \sum_{\alpha, \beta} w_\alpha(i) w_\beta(j) \delta_{(x_\alpha, x_\beta)}$ on $\R^2$. The part of $\Pi_N$ with $\alpha = \beta$ lies on the diagonal, with a density that is the eigenvector moment we want. We have
\begin{equation}\label{e:ov-poisson}
\E \ell^\eps_{i, N}(0) \ell^\eps_{j, N}(0) = \int_{\R^2} P_\eps(t) P_\eps(v) \, \Pi_N(\dd t, \dd v) ,
\qquad
\int_\R P_\eps(t)^2 \dd t = \frac{1}{2 \pi \eps} .
\end{equation}
A diagonal part with constant density $c$ therefore contributes $c/(2\pi\eps)$. For the stationary limiting measures considered below, whose integrals against products of Poisson kernels are finite, the part without mass on the diagonal contributes $o(\eps^{-1})$ as $\eps \downarrow 0$. The density $c$ is the residue at $z = 0$ of the limit at the damping $\eps = z/(2\pi)$, and \zcref{t:interface}(b) expresses this limit through the sigma model.

To make this rigorous, we must deal with the order of limits, which is the main difficulty, since \zcref{t:interface} identifies the limit $N \to \infty$ only at fixed $\eps > 0$. To handle this, we consider vague limits of the expected measures $\Pi_N$ themselves, and \zcref{l:extract-limits} establishes three properties of all such limits. It is stationary in the energy variable, by the Gaussian component of $H$. Its part from pairs of distinct eigenvalues puts no mass on the diagonal. This follows from the bound on second factorial moments of eigenvalue counts, which shows that the number of pairs of distinct unfolded eigenvalues in an interval of length $r$ has expectation $O(r^2)$. One might expect that estimates on $k$ eigenvalues are needed for the products over $k$ sites that the coarse law involves, but the pair bound suffices at every order. Finally, Poisson kernels may be integrated against the limit. The Gaussian component of $H$ also makes the expected diagonal measures Gaussian mixtures (convolutions with a centered Gaussian density) whose width tends to infinity, and for such mixtures vague convergence implies that the densities converge locally uniformly. Densities give the annealed selections over windows of every width $r \le \bar r_N$, including windows much smaller than the spacing of the eigenvalues. This extraction theorem, \zcref{t:extract-main}, is the third new step of our argument.

It remains to compute residues in the sigma model, and this is where the noncompact sector appears. Consider the kernel $K_X(x, y; z)$ of \zcref{d:kernels}, whose residue at $z = 0$ is twice the eigenvector correlation $\mathcal C_X(x, y)$. It is built from diffusions on $D$ killed at rate $z (B - F)$. For small $z$, the killing is negligible unless $B$ is of order $1/z$, so that the relevant part of $D$ lies far out in the noncompact direction, near the boundary at infinity of the hyperbolic plane. On this scale, the noncompact coordinate $B_t$ of the singlet diffusion becomes a geometric Brownian motion. For the starting point $B_0 = Xq/z$, we have, approximately and in law as processes,
\begin{equation}\label{e:ov-gbm}
\frac{z}{X} B_t \approx q \, e^{g_X(t)} ,
\qquad
z \int_0^1 B_t \dd t \approx X q Z_X ,
\end{equation}
where $t$ is the position along the wire, and $g_X(t) = \sqrt{2X} \beta_t - X t$ and $Z_X = \int_0^1 e^{g_X(t)} \dd t$ are the Brownian motion with drift and its exponential functional of \zcref{d:brownian}. The compact coordinate $F$, the angles and the anticommuting variables decouple on this scale, since their coupling to $B$ is of order $1/B$ by \zcref{l:res-freezing}.

The singlet state at time one therefore tends to $\E e^{-X q Z_X}$, the Laplace transform of the total mass. On this scale, an insertion at the position $x$ multiplies by a factor proportional to $B_x$, and hence to $e^{g_X(x)}$. In these coordinates, the invariant measure of the hyperbolic plane becomes $\dd B = X z^{-1} \dd q$, and the factor $z^{-1}$ is the pole. Integrating over the starting point $q$ and using
\begin{equation}\label{e:ov-laplace}
A^{1 - k} = \frac{1}{(k - 2)!} \int_0^\infty q^{k - 2} e^{-q A} \dd q , \qquad A > 0 , \ k \ge 2 ,
\end{equation}
we obtain the moments $M_k^X = \E[e^{g_X(x_1) + \dots + g_X(x_k)}/Z_X^{k - 1}]$ of \zcref{d:brownian}. Heuristically, the mass profile $e^{g_X(t)}/Z_X$ of a selected eigenvector is the profile of the noncompact coordinate $B$ of the sigma model along the wire, on the scale $1/z$ on which the damping acts, divided by its integral. For $k = 2$, we couple the scaled diffusion with the geometric Brownian motion driven by the same Brownian motion, and this proves $z K_X(x, y; z)/2 \to \mathcal C_X(x, y)$ in \zcref{t:res-scalar}.

For moments of order $k \ge 3$, the extraction needs a nonnegative observable of the resolvent whose part from single eigenvalues is proportional to the product $\prod_a w_\alpha(i_a)$ over $k$ sites. We use permanents of $\Im G$ at these sites. Inverse determinants are Gaussian integrals over commuting variables, so that derivatives at $k$ sites of a local source in the commuting block produce permanents, by Wick's theorem for complex Gaussians, just as derivatives of a determinant produce determinants. We can therefore stay at rank one for all orders. The expected permanent is proportional to the Poisson smoothing of a positive measure whose diagonal part is proportional to $\sum_\alpha \prod_a w_\alpha(i_a) \delta_{x_\alpha}$. The limit of the expected permanent at fixed damping is a pairing with $k$ source insertions in the rank-one sigma model. The source is not invariant, so that this limit needs the complete rank-one model, with its angular and anticommuting variables. The generator acts on the components of a superfunction, on the scale $B \asymp 1/z$, as the scalar singlet generator plus a matrix potential of order $1/B$. The source becomes $B$ times a function of the angle of the hyperbolic plane and of the anticommuting variables, and these variables stay fixed in the limit. We then compute the residue, an explicit multiple of $M_k^X$, by a coupling argument in \zcref{l:res-permanent}, which is the most technical step of \zcref{s:residue}.

The remaining steps of the coarse law are soft. By Scheff\'e's lemma, the moments at sites that are separated from each other and from the ends determine the step densities of all tensor moments of the mass measure in $L^1$, and by the Stone--Weierstrass theorem, these moments determine the law of the random probability measure on $[0, 1]$, and \zcref{t:coarse} follows. The rooted laws of \zcref{t:shape} follow from the Cameron--Martin theorem, as explained after that theorem, and the infinite-wire shape follows by Brownian scaling as $X \to \infty$. The mean shape of \zcref{c:mean-shape} is computed from Dufresne's identity for the exponential functional of a Brownian motion with drift \cite{dufresne1990distribution}.

\section{Local intensities and the participation ratio}\label{s:ov-intensities}

The local intensities of \zcref{t:intensity} involve two sites, and here the complete local jet matters. First derivatives at two distinct sites become two separate multipliers, each an entry of the orbit resolvent $\mathcal G(Q)$, and their pairing in the rank-one sigma model produces the kernel $K_X$. Two derivatives at one site become one multiplier, the corresponding term of the complete jet. Besides the product of two diagonal entries of $\mathcal G(Q)$, this multiplier contains the product of the two off-diagonal entries of the commuting block, which couple the retarded and the advanced flavors, as we explain in \zcref{r:iface-onsite}. Its pairing produces the contact kernel $\widetilde K_X$, whose integrand carries the weight $B/d$. By \eqref{e:intro-residue}, we have $z K_X/2 \to \mathcal C_X$ and $z \widetilde K_X \to 2 \mathcal C_X$ as $z \downarrow 0$, and the extraction turns these residues into the profile of \zcref{t:profile}, with the coefficient $1 + \1_{i = j}$. The full two-energy intensity also needs the part of the pair measure away from its atom at zero separation. We recover it from its values at equal energies, since it is an even function of the separation.

The participation ratio $N \langle \IPR(u_\alpha) \rangle_{E, r}$ is the average over all sites $i$ of $N^2 \langle w_\alpha(i)^2 \rangle_{E, r}$, including the sites near the ends of the chain, where \zcref{t:interface}(b) is not available. There we need only a uniform bound on the diagonal densities, which follows from the moment bounds of \zcref{t:interface}(c) at all sites. Strips of width $\delta N$ at the ends then contribute $O(\delta)$, and the interior contributes $2 \int_0^1 \mathcal C_X(x, x) \dd x$. We evaluate this integral exactly, as $1 + X/3$, in \zcref{p:int-exact}, by a Laplace transform in the length of the wire and two integrated identities for the resolvent of the killed geometric Brownian motion.

\section{Second-order statistics and fluctuations across scales}\label{s:ov-scalar}

All second-order statistics of $\Xi_X$ in \zcref{s:intro-results} are read off from the function $I_X$ of \zcref{d:kernels}, through two transforms of positive measures. Recall from \zcref{c:intro-pair} that the pair measure $\nu_X$ is determined by its Poisson transforms \eqref{e:intro-pair-transform}, which are expressed through $I_X$ by \zcref{t:process}(c) and stationarity. Moreover, the reduced covariance measure $\delta_0 + \nu_X - \dd s$ of $\Xi_X$ is positive definite, so that its Fourier transform is a positive measure. Following \cite[Section~8.2]{daley2003introduction}, we call this measure, in the normalization of our Fourier transform, the Bartlett spectrum $m_X$ of $\Xi_X$. In \zcref{l:scales-fourier}, we prove the Laplace identity
\[
I_X(z) = \int_\R e^{-2 z |k|} \, m_X(\dd k) , \qquad \Re z > 0 .
\]
Asymptotics of $I_X$ therefore become statements about $\nu_X$ and $m_X$, through continuity theorems for these transforms.

As $X \downarrow 0$, the diffusion moves only by $O(\sqrt X)$ during the times along the wire, and the semigroups in \eqref{e:intro-K} are close to multiplications. We expand them in powers of $X$ by Duhamel's formula. The difficulty is to bound the remainders uniformly over the closed simplex $0 \le x \le y \le 1$ of positions, since on its faces some of the times vanish and the corresponding factors give no decay in $B$. Recall that $d = B - F$. We overcome this difficulty by controlling the fluctuations of $\log(1 + d)$ along the paths, which shows that the total killing along the wire produces decay faster than every power of $B$. The coefficients of the expansion are then elementary integrals over the simplex of positions and over $D$, and evaluating them, we obtain the expansion of $I_X$ in \zcref{p:metallic}. Afterward, \zcref{r:met-traces} notes the identity $1/45 = 2 \Tr G_{\mathrm N}^2$ of \zcref{s:intro-results}, which expresses the coefficient of $X^2$ through the eigenvalues of the Neumann Laplacian on $[0, 1]$. The expansion of $\mathcal C_X$ is simpler, since it is a Gaussian integral in the parameter $\sqrt{2X}$.

As $X \to \infty$ with $z/X = a$ fixed, the wire is much longer than the time in which the diffusion relaxes, and $X I_X(Xa)$ is close to a stationary pairing $\mathcal I(a)$, which involves only the stationary singlet state of an infinite wire. This state is the zero mode found by Skvortsov and Ostrovsky \cite{skvortsov2007local}, and $\mathcal I(a)$ is an explicit combination of modified Bessel functions. Inverting the Laplace transform gives the function $m_\infty$ and the crossover of \zcref{t:crossover}. These objects were found in physics from the same zero mode, as recalled in \zcref{s:intro-physics}, and \zcref{t:crossover} makes these predictions rigorous, as statements about the limiting processes $\Xi_X$. At fixed $X$, the same estimate for large $z$ determines the behavior of $m_X$ near zero, which is the law $\Sigma_X^2(L) \sim \pi^{-3/2} \sqrt{XL}$ of \zcref{t:largecov}.

\section{Organization of the paper}\label{s:ov-organization}

In Part II, we introduce the supersymmetric tools and the two exact objects of the proof. \zcref[S]{s:grassmann} collects the facts about Grassmann variables, Berezin integrals, supermatrices, and localization in flat space that the paper uses, for readers who are new to them. \zcref[S]{s:rep} writes the source ratios as the open transfer chain \eqref{e:ov-chain} on the product of Sch\"afer--Wegner cycles, together with the anchored identity for partial chains and the dictionary with resolvent entries; it is the starting point of Part IV. \zcref[S]{s:sigma} defines the sigma model on $[0, 1]$, constructs its semigroup, proves the localization identity and the normalization, and is used in Parts III and IV. \zcref[S]{s:sigma-radial} reduces the model of rank one to diffusions on $D$ and to a calculus of rank-one pairings, from which the singlet state and the kernels of \zcref{d:kernels} arise; Part III uses it throughout.

In Part III, we derive our main results from the sigma model and the interface theorem. \zcref[S]{s:interface} collects the inputs about $H$ that do not involve supersymmetry and states the interface theorem, on which Part III relies as its only input from Part IV. \zcref[S]{s:process} identifies the limiting point process and proves \zcref{t:process}. \zcref[S]{s:extract} proves the extraction theorem, which turns limits at positive damping into eigenvector moments over small energy windows, for use in \zcref{s:intensity,s:coarse}. \zcref[S]{s:residue} computes the residues of the sigma-model kernels at zero damping through geometric Brownian motion at the noncompact boundary, and these residues identify the eigenvector moments. \zcref[S]{s:intensity} proves \zcref{t:intensity,t:profile}, and \zcref{s:coarse} proves \zcref{t:coarse,t:shape} and their corollaries. \zcref[S]{s:metallic} proves the metallic expansion of \zcref{p:metallic}, and \zcref{s:scales} proves the statements on the pair measure, the large-scale covariance, and the crossover, with the remaining rows of \zcref{p:endpoints}.

The interface theorem is proved in Part IV. \zcref[S]{s:hist} proves the bounds on one physical step and the a priori bound $e^{o(W)}$ on partial chains, which makes errors of order $e^{-\gamma W}$ negligible in the comparison. \zcref[S]{s:tube} sets up coordinates near the saddle manifold and the fast normal oscillator, identifies in \zcref{l:tube-casimir} the heat generator on the saddle manifold as a multiple of $\mathsf C_k$, and proves the contraction of the oscillator and its Ward identity. \zcref[S]{s:stab} proves the stability of the powers of the adapted transfer operator over $h^{-4}$ steps. \zcref[S]{s:cons} constructs the state $\Omega_h$, deduces from \zcref{l:tube-casimir} the time normalization \eqref{e:ov-time} of one link, proves the consistency \eqref{e:ov-consistency}, and deduces the limit of the adapted chain for vectors of the form $\Omega_h \otimes f$. \zcref[S]{s:end} attaches the ends of the chain to the tube with exponentially small errors. \zcref[S]{s:assembly} combines these results into the proof of \zcref{t:interface}. Finally, \zcref{s:tools} collects four general tools, \zcref{s:contour} constructs the product contour of \zcref{s:rep}, and \zcref{s:regularity} contains the proof of the regular realization of the semigroup of \zcref{s:sigma}.