\chapter{Introduction}\label{s:intro}

\section{Background}\label{s:intro-background}

Random band matrices are random Hermitian matrices whose entries vanish, or are negligible, outside a band of width $W$ around the diagonal. They interpolate between Wigner matrices, for $W$ comparable to $N$, and random Schr\"odinger operators on a one-dimensional lattice, for $W$ of order one. In physics, they were introduced to describe level statistics intermediate between those of regular and chaotic systems \cite{seligman1985spectral} (see \cite[Section 5.3]{mirlin2000statistics}). For band matrices of size $N$ in one dimension, the eigenvectors are expected to be localized on a scale of order $W^2$, so that one expects a transition at $W \approx N^{1/2}$, with delocalized eigenvectors and the local eigenvalue statistics of the GOE or the GUE, according to the symmetry class, for $W \gg N^{1/2}$, and localized eigenvectors and Poisson statistics for $W \ll N^{1/2}$ \cite[(0.2)]{bourgade2019random}. Casati, Molinari, and Izrailev found numerically that the localization of the eigenvectors and the distribution of the level spacings depend on $N$ and on the bandwidth only through the ratio of the square of the bandwidth to $N$ \cite{casati1990scaling,casati1991scaling}. Fyodorov and Mirlin explained this scaling in \cite{fyodorov1991scaling}, where they mapped band matrices with a general profile onto the one-dimensional supersymmetric sigma model of a thick disordered wire, with a localization length proportional to the square of the bandwidth (see \cite[(5.39)--(5.41)]{mirlin2000statistics}). We refer to \cite{spencer2012susy,spencer2015random,bourgade2019random} for surveys.

On the localized side, Schenker proved in \cite{schenker2009eigenvector} that the eigenvectors are localized when $W^8 \ll N$, and the exponent $8$ was improved to $7$ in \cite[Theorem 4]{peled2019wegner}. Localization for $W \ll N^{1/4}$ was proved in \cite{cipolloni2024dynamical} for the Wegner orbital model and in \cite{chen2022random} for Gaussian band matrices. Recently, Drogin \cite{drogin2025localization} obtained localization at the sharp scale $W^2$ when $W^2 \ll N$, for real symmetric block band matrices with regular entry distributions. Hislop and Krishna showed that, for real symmetric band matrices with Gaussian entries and $W \sim N^\alpha$ with $\alpha < 1/7$, the limit points of the eigenvalue counts in fixed intervals are Poisson distributed, for almost every energy \cite{hislop2022local}. Fan and Zou proved convergence of the local eigenvalue statistics to a Poisson process for real symmetric Gaussian block band matrices with $W^2 \log N/N \to 0$ \cite[Theorem 1.3]{fan2026poisson}.

On the delocalized side, Erd\H{o}s and Knowles proved quantum diffusion and showed that most eigenvectors have localization length larger than $W^{7/6}$ \cite{erdos2011quantum,erdos2011quantumgeneral}, and delocalization was established for $W \gg N^{4/5}$ in \cite{erdos2013delocalization}. Bourgade, Yau, and Yin \cite{bourgade2020delocalizedI} proved quantum unique ergodicity, delocalization, and local GOE statistics for real symmetric band matrices with $W \ge N^{3/4 + \eps}$, using the estimates of Bourgade, Yang, Yau, and Yin \cite{bourgade2019delocalizedII} and of Yang and Yin \cite{yang2021delocalizedIII}, after the case $W \asymp N$ \cite{bourgade2017universality}; see also \cite{bao2017delocalization,dubova2026quantum}.

Delocalization down to the threshold was proved recently. Yau and Yin \cite{yau2025delocalization} proved delocalization, quantum unique ergodicity, and GUE universality for Gaussian periodic block band matrices with $W \ge N^{1/2 + c}$. Erd\H{o}s and Riabov \cite{erdos2025zigzag} extended these results to general variance profiles and entry distributions with $W^2 \ge N^{1 + c}$, and Fan, Yang, and Yin \cite{fan2025block} proved delocalization for general variance profiles with $W \ge N^{1/2 + c}$. With supersymmetric methods, M.~Shcherbina and T.~Shcherbina \cite[Theorem 1.1]{shcherbina2021universality} proved that the two-point correlation function of Gaussian block band matrices converges to that of the GUE when the block size $W$ and the number $n = N/W$ of blocks satisfy $W \ge C n \log^5 n$. As noted in \cite{drogin2025localization,fan2026poisson}, these results establish the localization--delocalization transition of the eigenvectors and the transition between Poisson and random-matrix local statistics, for large classes of band matrices. None of them describes the critical regime $W^2 \asymp N$, although some of them hold there, as we explain below. Other critical scalings occur at the spectral edge, where the transition to Airy statistics happens at $W \asymp N^{5/6}$ \cite{sodin2010spectral} (see \cite{liu2024edge} for higher dimensions), and the transition at $W \asymp N^{1/2}$ is already visible in the fourth moment of the level density on global scales \cite{olver2018evidence}.

In the critical regime $W^2 \asymp N$, the physics prediction is that the statistics of the eigenvalues and eigenvectors are those of the one-dimensional supersymmetric sigma model on a wire of finite length, whose only parameter is the ratio $X$ of the length of the wire to the localization length \cite{efetov1983kinetics,fyodorov1991scaling} (see \cite[Section 3.2.1]{mirlin2000statistics}). For the eigenvectors, Fyodorov and Mirlin solved this sigma model exactly for several statistics, including the inverse participation ratio and the distribution of the components \cite{fyodorov1992analytical,mirlin1993statistics,fyodorov1993level,fyodorov1994statistical}, using the simplification that occurs when the imaginary part of the frequency tends to zero and only the noncompact variable of the sigma model survives \cite[Section 3.2.1]{mirlin2000statistics}. For the eigenvalues, Altland and Fuchs studied the level correlations of a finite wire across the crossover from Wigner--Dyson to Poisson statistics, from numerical solutions of the transfer-matrix equations \cite{altland1995spectral} (see \cite[Section V]{micklitz2016spectral}), and Kettemann studied the analogous autocorrelation of spectral determinants \cite{kettemann1999exploring,kettemann2000symmetry}. Closed formulas are known for long wires at all frequencies \cite{micklitz2016spectral,marinho2018spectral}, and, for every ratio of length to localization length, at frequencies that are large compared with the level spacing of a localization volume \cite[(25)--(26)]{micklitz2016spectral}. We compare them with our results in \zcref{s:intro-physics}.

The rigorous results that describe the critical regime itself concern characteristic polynomials. In \cite[Theorem 1.2]{shcherbina2020characteristic}, T.~Shcherbina considered the band matrix that we study, whose variance profile is the inverse of $\Id - W^2 \Delta$ for the Neumann Laplacian $\Delta$ on $\{1, \dots, N\}$ (see \zcref{d:model} below), with $N$ a fixed multiple of $W^2$. She proved that the normalized second mixed moment of the characteristic polynomials of this matrix, at two energies at distance of order $1/N$ around $E$, converges to a matrix element of the semigroup generated by the Laplacian on the sphere $S^2$ plus a potential. Her parameter coincides with the parameter $X_N(E) = N/((4 - E^2) W^2)$ of our results, since $(2\pi\rho(E))^2 = 4 - E^2$ for the semicircle density $\rho(E) = (4 - E^2)^{1/2}/(2\pi)$. Her Laplacian, written in the variable $F = 1 - 2|U_{12}|^2$ of her unitary matrices $U$, is the operator $\partial_F((1 - F^2) \partial_F)$, which reappears below as the compact part of the operator $\Delta_D$ of \zcref{d:kernels}. This completed the results of \cite{shcherbina2014second,shcherbina2017characteristic} on the two sides of the threshold, and analogous critical results were proved for real symmetric block band matrices in \cite[Theorem 1.1]{shcherbina2022susy} and for a non-Hermitian model in \cite{shcherbina2026characteristicthreshold}. Characteristic polynomials involve only the compact sector of the sigma model, which here is the sphere $S^2$, while the statistics of eigenvalues and eigenvectors require inverse determinants, and therefore its noncompact sector. The noncompact sector, together with the anticommuting variables, appears in the analysis of the two-point function above the threshold by M.~Shcherbina and T.~Shcherbina \cite{shcherbina2018universality,shcherbina2021universality}.

Several estimates are valid at $W^2 \asymp N$ but do not detect the transition: the density of states of Disertori, Lohmann, and Sodin \cite[Theorem 1]{disertori2021density}, the Wegner and Minami estimates of \cite{peled2019wegner}, the lower bounds of Erd\H{o}s and Knowles on the localization length \cite{erdos2011quantum,erdos2011quantumgeneral}, which are uniform in $N$, their Altshuler--Shklovskii formulas for the density--density correlations on mesoscopic scales $\eta \ge W^{-1/3 + c}$ \cite[Theorems 2.2--2.3]{erdos2015altshulerI}, and, for periodic profiles with sufficiently fast polynomial decay, the delocalization bound $\|\psi\|_\infty^2 \le N^{\tau}(W^{-2} + N^{-1})$ of Fan, Yang, and Yin \cite[Theorem 2.6]{fan2026localization}, which holds for all bulk eigenvectors $\psi$ and all $W \ge N^c$, for every $\tau > 0$ with probability at least $1 - N^{-D}$ for every $D > 0$ and all large $N$. To our knowledge, no rigorous result on the local eigenvalue process, on the eigenvalue correlation functions on the scale of the level spacing, or on the distribution of the eigenvector mass was previously available at $W^2 \asymp N$.

A rigorous analogue of the critical regime is known for one-dimensional random Schr\"odinger operators on a box of $n$ sites whose potential has variance of order $1/n$. Kritchevski, Valk\'o, and Vir\'ag \cite{kritchevski2012scaling} showed that their eigenvalues near a bulk energy converge to a point process $\mathrm{Sch}_\tau$, where $\tau$ depends on the variance of the potential and on the energy. This regime lies between a nearly periodic spectrum, for a potential of smaller variance, and Poisson statistics, for a potential of larger variance \cite{kritchevski2012scaling} (see also \cite{dumaz2023localization}). (For CMV matrices with decaying coefficients, Killip and Stoiciu found a transition from Poisson statistics to the clock process \cite{killip2009eigenvalue}.) Rifkind and Vir\'ag \cite[Theorem 1.1]{rifkind2018eigenvectors} proved that the mass of a uniformly chosen eigenvector of the critical operator, viewed as a measure on $[0, 1]$, converges in law to the probability measure with density proportional to $S(\tau(t - U))$, where
\[
S(t) = \exp\Big( \frac{\beta'_t}{\sqrt 2} - \frac{|t|}{4} \Big), \qquad t \in \R,
\]
$\beta'$ is a two-sided Brownian motion, and $U$ is uniform on $[0, 1]$ and independent of $\beta'$. They expect this behavior to be universal for eigenvectors of critical random operators in one dimension, and they state conjectures for several such models \cite[Conjectures 1.2--1.3]{rifkind2018eigenvectors}. Dumaz and Labb\'e proved one of them, for the continuous Anderson Hamiltonian on a long interval at energies of the order of its length \cite[Theorems 3 and 4]{dumaz2023delocalized}, and found the same shape, on an infinite interval, in the localized crossover regime of that operator \cite[Theorem 3]{dumaz2023localization}; see also \cite{nakano2022shape} for decaying potentials.

In this article, we determine the limit of the eigenvalue process of the band matrix of \zcref{d:model} at the critical bandwidth, for energies near zero, together with the macroscopic and local statistics of the eigenvectors. Several of our results confirm predictions obtained by physicists from the sigma model, and our law of the eigenvector mass coincides with the law of Rifkind and Vir\'ag; we discuss these relations in \zcref{s:intro-physics}.

\section{Main results}\label{s:intro-results}

We now state our results. We begin with the definition of the model and the scaling under which we study it, and we then state the results, grouped by the observables they concern.

\subsection{The model}

\begin{definition}\label{d:model}
Let $N \ge 2$ be an integer and let $W > 0$. Let $\Delta$ denote the Neumann Laplacian on $\{1, \dots, N\}$, which acts on functions $f \colon \{1, \dots, N\} \to \C$ by
\[
(\Delta f)(i) = \sum_{j \colon |i - j| = 1} \big( f(j) - f(i) \big).
\]
The matrix $\Id - W^2 \Delta$ is positive definite, since $-\Delta$ is nonnegative. We set $J = (\Id - W^2 \Delta)^{-1}$. We let $H = H_{N, W}$ denote the $N \times N$ complex Hermitian random matrix whose entries $H_{ij}$ with $i \le j$ are independent centered Gaussian random variables, such that $H_{ii}$ is real with variance $J_{ii}$ and, for $i < j$, the real and imaginary parts of $H_{ij}$ are independent, each with variance $J_{ij}/2$. Equivalently,
\begin{equation}\label{e:intro-covariance}
\E H_{ij} H_{kl} = \delta_{il} \delta_{jk} J_{ij}, \qquad i, j, k, l \in \{1, \dots, N\}.
\end{equation}
We call $H$ the \emph{Gaussian band matrix} with bandwidth $W$, and we call $J$ its \emph{variance profile}.
\end{definition}

We let $\iu$ denote the imaginary unit, since the letter $i$ denotes a site. We use the conventions of \zcref{s:intro-notation} for constants, asymptotic notation, and covariances.

The matrix $H$ belongs to the unitary symmetry class, for which the target space of the sigma model below is simplest. For real symmetric matrices one expects the orthogonal sigma model, with a localization length half as large \cite[(52)--(53)]{efetov1983kinetics} and with the Gaussian factor $\E \gamma^4/(\E \gamma^2)^2 = 3$ of a real Gaussian variable $\gamma$ in place of the factor $2$ in \zcref{t:intensity,t:profile} below; the orthogonal analogue of the critical result for characteristic polynomials is in \cite{shcherbina2022susy}.

The profile $J$ is a smoothed version of the usual band structure. We set
\[
\theta_W = 2 \operatorname{arsinh}\Big( \frac{1}{2W} \Big), \qquad J_{\Z}(m) = \frac{e^{-\theta_W |m|}}{\sqrt{1 + 4W^2}}, \qquad m \in \Z.
\]
Then $J_{\Z}(i - j)$ is the kernel of the inverse of $\Id - W^2 \Delta_{\Z}$, where $\Delta_{\Z}$ is the Laplacian on $\Z$, and the method of images gives
\begin{equation}\label{e:intro-images}
J_{ij} = \sum_{m \in \Z} \big( J_{\Z}(i - j + 2mN) + J_{\Z}(i + j - 1 + 2mN) \big), \qquad i, j \in \{1, \dots, N\}.
\end{equation}
All terms in \eqref{e:intro-images} are positive, and $J$ decays exponentially in $|i - j|/W$, since $W \theta_W \to 1$ as $W \to \infty$. Since $\Delta$ annihilates constants, every row of $J$ sums to one, and $\E \sum_j |H_{ij}|^2 = 1$ for all $i$, as for a Wigner matrix. When $N/W$ is large, the second moment $\sum_j J_{ij} (i - j)^2$ is of order $W^2$, uniformly in $i$, and for the kernel on $\Z$ we have $\sum_{m \in \Z} J_{\Z}(m) m^2 = 2W^2$.

The reason for this choice of profile is that its inverse $\Id - W^2 \Delta$ is a nearest-neighbor operator. Our analysis starts from averages of ratios of characteristic polynomials of $H$, such as \eqref{e:intro-ratio} below. To compute them, one writes each determinant as a Gaussian integral over commuting or anticommuting variables, averages over $H$, and decouples the resulting quartic interaction by a Hubbard--Stratonovich transformation, which introduces one variable for each site. This variable is a matrix whose entries are commuting or anticommuting variables. We call such a matrix a supermatrix (see \zcref{s:grass-supermatrix}). The Gaussian weight of these variables is given by the quadratic form of $J^{-1}$. Since $J^{-1}$ couples only nearest neighbors, the integrand is a product of factors attached to single sites and to pairs of neighboring sites. The integral is therefore a one-dimensional chain, whose value is a matrix element of a power of a transfer operator; we derive this representation in \zcref{p:rep-exact}. (This structure underlies earlier supersymmetric work on band matrices with the profile $J$, such as \cite{disertori2002density,shcherbina2016transfer,disertori2021density}; see \zcref{s:intro-sources}.)

However, the inverse of a sharp profile such as $J_{ij} = (2r_0 + 1)^{-1} \1_{|i - j| \le r_0}$, when it exists, is not a band matrix, and the same procedure produces an interaction between all pairs of sites. In the physics literature, band matrices with a general normalized profile are mapped onto the same one-dimensional sigma model, in which the profile appears only through its second moment \cite{fyodorov1991scaling} (see \cite[(5.39)--(5.41)]{mirlin2000statistics} and \cite[(7)]{fyodorov1995statistical}). More precisely, for a profile whose rows sum to one, we obtain from the lattice sigma model of \cite[(7)]{fyodorov1995statistical}, as in \zcref{s:intro-physics}, the parameter $2N/((4 - E^2) \sum_r J(r) r^2)$, where the second moment is taken away from the ends of the chain. Our profile has $\sum_r J(r) r^2 = 2W^2$ there, and this parameter becomes $N/((4 - E^2) W^2) = X_N(E)$, with $X_N(E)$ as in \eqref{e:scaling} below. The sharp profile of half-width $r_0$ has second moment $r_0(r_0 + 1)/3$, and \zcref{f:intro-profile} compares the two profiles when their second moments agree. Our results should therefore describe general profiles, and we conjecture that they hold for the sharp profile of half-width $r_0$, with $W^2$ replaced by $r_0(r_0 + 1)/6$ in $X_N(E)$. Our proof, however, uses the nearest-neighbor structure of $J^{-1}$ in an essential way. The Neumann boundary condition corresponds to a wire with two ends \cite[Section 3.2.1]{mirlin2000statistics}.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\begin{axis}[
  width=0.78\linewidth, height=0.4\linewidth,
  xmin=-60, xmax=60, ymin=0, ymax=0.055,
  xlabel={$i - j$}, ylabel={$J_{ij}$},
  scaled y ticks=false, yticklabel style={/pgf/number format/fixed, /pgf/number format/precision=2},
  tick label style={font=\small}, label style={font=\small},
  axis lines=left, clip=true]
\addplot[thick, domain=-60:60, samples=241] {exp(-0.0999584*abs(x))/sqrt(401)};
\addplot[dashed, gray] coordinates {(-60, 0) (-24.5, 0) (-24.5, 0.0204082) (24.5, 0.0204082) (24.5, 0) (60, 0)};
\end{axis}
\end{tikzpicture}
\caption{The kernel $J_{\Z}(i - j)$ of \eqref{e:intro-images} for $W = 10$ (solid line), which is a row of the variance profile $J$ far from the ends of the chain, up to exponentially small terms, and the sharp profile $(2r_0 + 1)^{-1} \1_{|i - j| \le r_0}$ with $r_0 = 24$ (dashed gray line). Both have total mass one and the second moment $2W^2 = r_0(r_0 + 1)/3 = 200$.}
\label{f:intro-profile}
\end{figure}

The law of $H$ has a positive density on the space of Hermitian matrices, since the entries of $J$ are positive. In particular, the eigenvalues of $H$ are simple almost surely. We let $\lambda_1 < \dots < \lambda_N$ denote them, and we let $u_1, \dots, u_N$ denote corresponding normalized eigenvectors, with coordinates $u_\alpha(i)$. We set $w_\alpha(i) = |u_\alpha(i)|^2$. These weights do not depend on the choice of phases, and $\sum_i w_\alpha(i) = 1$ and $\sum_\alpha w_\alpha(i) = 1$. Throughout, $\alpha$ denotes an eigenvalue index and $i, j$ denote sites. For $E \in (-2, 2)$, we set
\begin{equation}\label{e:scaling}
\begin{gathered}
s_E = \sqrt{1 - E^2/4}, \qquad \rho(E) = \frac{s_E}{\pi}, \qquad a_N = N \rho(E), \\
X_N(E) = \frac{N}{(4 - E^2) W^2}, \qquad x_\alpha = a_N (\lambda_\alpha - E).
\end{gathered}
\end{equation}
By \zcref{l:iface-density}, the expected density of states of $H$ converges to the semicircle density $\rho(E) = (4 - E^2)^{1/2}/(2\pi)$, and the unfolded eigenvalues $x_\alpha$ have mean spacing one near zero. The parameter $X_N(E)$ compares $N$ with $W^2$. In the physics literature, $W^2$ is the order of the localization length of the eigenvectors when $N = \infty$ \cite{casati1990scaling,fyodorov1991scaling}, and $X_N(E) = N/\xi$ is the length of the chain in units of the localization length $\xi = (4 - E^2) W^2$ (see \zcref{s:intro-physics}).
For $N \ll W^2$ one expects the local statistics of the GUE and delocalized eigenvectors, and for $N \gg W^2$ one expects Poisson statistics and localized eigenvectors (see \zcref{s:intro-background}). This paper concerns the critical regime $N \asymp W^2$, in which $X_N(E)$ stays bounded and bounded away from zero, and in which features of the two regimes appear on different scales.

The paper has four parts, which are described in \zcref{s:intro-organization}. Part I consists of this section and the overview of the proof, Part II introduces supersymmetry and the sigma model, Part III derives the results below from the sigma model and from the interface theorem, \zcref{t:interface}, and Part IV proves the interface theorem. This theorem states that averages of ratios of characteristic polynomials of $H$, and their derivatives in perturbations at single sites, converge to quantities of the sigma model, and that the diagonal entries of the resolvent of $H$ have bounded moments and, at sites away from $1$ and $N$, the mean that the semicircle law predicts; we describe it in \zcref{s:ov-strategy}. Every result of \zcref{s:intro-results} uses it.

We now fix the energies and the limits that our results concern. \zcref[S]{t:interface} involves two universal energies $0 < E_* < E_{\mathrm{out}} \le 1/2$, the inner and the outer energy bound of \zcref{r:iface-energies}. Our analysis of the transfer operator in Part IV applies for $|E| \le E_{\mathrm{out}}$, and this bound will be fixed in \eqref{e:asm-Eout}. The inner bound $E_*$ will be fixed in \zcref{s:asm-choices}. The margin $E_{\mathrm{out}} - E_*$ absorbs the shifts of the energy by $O(1/N)$ in the proof of \zcref{t:interface}, and Part III uses it to control the eigenvalues far from $E$. Both bounds are the same for all our results, and in particular they do not depend on the number of characteristic polynomials in the averages below or on the order of any moment. Their values are not explicit, since they depend on universal constants in Part IV that we do not compute. All our results are stated for $|E| \le E_*$. These restrictions come from the estimates of Part IV, and we expect the results to hold for all $|E| < 2$.

\begin{definition}\label{d:intro-critical}
Let $X \in (0, \infty)$. We say that $(N, W, E)$ tends to the \emph{critical limit at $X$} if $N, W \to \infty$ and $E = E_{N, W}$ satisfies $|E| \le E_*$ and $X_N(E) \to X$. We say that a statement holds \emph{uniformly in the critical regime} if for all $0 < X_- \le X_+ < \infty$ it holds uniformly over all $(N, W, E)$ with $|E| \le E_*$ and $X_- \le X_N(E) \le X_+$, as $N \to \infty$.
\end{definition}

In these limits the energy $E$ may depend on $N$ and $W$. Several statements below concern limits $X \downarrow 0$, $X \to \infty$, or $L \to \infty$ of objects defined below for fixed $X \in (0, \infty)$, such as the limiting point process $\Xi_X$ and the kernels $K_X$, which arise in the critical limit at $X$. In all such statements the critical limit is taken first, and the second limit is a statement about the limiting objects. We do not consider joint limits in which $X_N(E) \to 0$ or $X_N(E) \to \infty$ as $N \to \infty$.

\subsection{The eigenvalue process}

We identify a locally finite point configuration on $\R$ with its counting measure. We equip the set of locally finite counting measures on $\R$ with the vague topology, the coarsest topology for which the maps $\zeta \mapsto \int f \dd \zeta$ are continuous for all continuous functions $f$ with compact support. We consider the unfolded eigenvalue process
\[
\Xi_N = \sum_{\alpha = 1}^N \delta_{x_\alpha}.
\]

Our description of the limit of $\Xi_N$ is based on averages of ratios of characteristic polynomials. Fix $\eps > 0$ and an integer $k \ge 1$. For $\mathbf u = (u_1, \dots, u_k) \in \R^k$ with $\sum_j u_j = 0$ and $\mathbf b = (b_1, \dots, b_k) \in (0, \infty)^k$, we set
\begin{equation}\label{e:intro-ratio}
T_N^{(k)}(\mathbf u, \mathbf b) = \E \prod_{\alpha = 1}^N \prod_{j = 1}^k \frac{(x_\alpha - u_j)^2 + b_j^2}{x_\alpha^2 + \eps^2}.
\end{equation}
We call $T_N^{(k)}$ the \emph{source ratio}, $(\mathbf u, \mathbf b)$ the \emph{sources}, $k$ the \emph{source rank}, and $\eps$ the \emph{damping}. The dependence of $T_N^{(k)}$ on $\eps$ is not displayed. Writing $z_j = E + (u_j + \iu b_j)/a_N$ and $z_0 = E + \iu \eps/a_N$, we have
\[
T_N^{(k)}(\mathbf u, \mathbf b) = \E \frac{\prod_{j = 1}^k \det(H - z_j) \det(H - \bar z_j)}{\big( \det(H - z_0) \det(H - \bar z_0) \big)^k}.
\]
Each determinant in the numerator is a Gaussian integral over anticommuting (Grassmann) variables, and each determinant in the denominator is a Gaussian integral over complex variables (see \zcref{s:grass-motivation}). Averages of this kind are the quantities that the supersymmetric method computes. Since $\sum_j u_j = 0$, we have
\[
\prod_{j = 1}^k \frac{(x - u_j)^2 + b_j^2}{x^2 + \eps^2} = 1 + O(x^{-2}), \qquad |x| \to \infty,
\]
and the corresponding product over the points of a process of intensity one converges absolutely.

The limits of $T_N^{(k)}$ are partition functions of the supersymmetric sigma model of a disordered wire \cite{efetov1983kinetics,efetov1997supersymmetry}, which Fyodorov and Mirlin derived for band matrices \cite{fyodorov1991scaling}.

\begin{definition}\label{d:partition}
Let $X > 0$, $\eps > 0$, $k \ge 1$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$, and $\mathbf b \in (0, \infty)^k$. We let $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ denote the partition function of the supersymmetric sigma model on $[0, 1]$ with coupling $X$ and sources $(\mathbf u, \mathbf b)$, which is defined in \zcref{d:sigma-partition}.
\end{definition}

The reader may treat $\mathcal Z^{(k)}_{X, \eps}$ as an explicit superintegral, defined in \zcref{s:sigma}, and \zcref{t:process} below can be read without its definition.

We describe it briefly. We call a $4k \times 4k$ matrix a supermatrix of size $(2k | 2k)$ if its two diagonal blocks of size $2k$, the BB and the FF block, have commuting entries and its two off-diagonal blocks have anticommuting entries (see \zcref{s:grass-supermatrix}). We let $\diag(A \,|\, D)$ denote the supermatrix with BB block $A$, FF block $D$, and vanishing off-diagonal blocks, and we let $\Str$ denote the supertrace, given by $\Str M = \Tr A - \Tr D$ for a supermatrix $M$ with BB block $A$ and FF block $D$. Let $\mathcal M_k$ denote the symmetric superspace $U(k, k | 2k)/(U(k | k) \times U(k | k))$, a quotient of Lie supergroups that \zcref{s:sigma} realizes as a set of supermatrices $Q$ of size $(2k | 2k)$ with $Q^2 = \Id$. Its underlying manifold is the product of the noncompact symmetric space $U(k, k)/(U(k) \times U(k))$ and the compact symmetric space $U(2k)/(U(k) \times U(k))$. We call the functions on $\mathcal M_k$ superfunctions. Besides a point of the underlying manifold, they depend on $4k^2$ anticommuting variables (see \zcref{s:grassmann}). The partition function $\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ is the Berezin integral over $\mathcal M_k$, an integral over the commuting and the anticommuting variables, of the heat semigroup at time one, applied to the constant function one, of the operator $X \mathsf C_k$ plus a potential. Here the Casimir operator $\mathsf C_k$ is minus the invariant Laplace--Beltrami operator of $\mathcal M_k$, and the potential is linear in $Q$ and depends on $\eps$, $\mathbf u$, and $\mathbf b$.

Precisely, let $\Lambda_{\mathrm B} = \diag(\Id_k, -\Id_k)$, and let $\Gamma = \diag(\eps \Lambda_{\mathrm B} \,|\, \Gamma_{\mathrm F})$ denote the diagonal supermatrix with
\[
\Gamma_{\mathrm F} = \diag(b_1 - \iu u_1, \dots, b_k - \iu u_k, -b_1 - \iu u_1, \dots, -b_k - \iu u_k).
\]
The diagonal entries of $E + \iu \Gamma/a_N$ are the spectral parameters $z_0, \bar z_0$ and $z_j, \bar z_j$ of $T_N^{(k)}$. Then
\[
\mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b) = \int e^{-(X \mathsf C_k + \pi \Str(Q \Gamma))} 1,
\]
where $\int$ is the invariant Berezin integral over $\mathcal M_k$ of \zcref{d:sigma-integral}, normalized so that $\int e^{-d_k} = 1$ with $d_k(Q) = \frac12 \Str(\Lambda Q)$ and $\Lambda = \diag(\Lambda_{\mathrm B} \,|\, \Lambda_{\mathrm B})$. The time variable of the semigroup is the position along the chain, rescaled to $[0, 1]$. We will use two properties of $\mathcal Z^{(k)}_{X, \eps}$ in this section. By \zcref{c:sigma-normalization}, we have $\mathcal Z^{(k)}_{X, \eps}(0, \eps \1) = 1$, in agreement with $T_N^{(k)}(0, \eps \1) = 1$. At rank one, the second derivative of $\mathcal Z^{(1)}_{X, \eps}(0, b)$ in $b$ at $b = \eps$ is given by the function $I_X$ of \zcref{d:kernels} below.

The underlying manifold of $\mathcal M_1$ is the product of the hyperbolic plane and the sphere $S^2$. The rank-one computations of \zcref{s:sigma-radial} reduce to functions of two coordinates, the hyperbolic cosine $B \ge 1$ of the distance to a base point of the hyperbolic plane and the cosine $F \in [-1, 1]$ of the distance to a base point of $S^2$. The operator $\Delta_D$ in the next definition is the Laplace--Beltrami operator of the product acting on functions of $(B, F)$.

\begin{definition}\label{d:kernels}
Let $D = [1, \infty) \times [-1, 1]$, with coordinates $(B, F)$, and set $d = B - F$. We define the differential operator
\[
\Delta_D = \partial_B \big( (B^2 - 1) \partial_B \big) + \partial_F \big( (1 - F^2) \partial_F \big).
\]
Let $X > 0$. We let $(\tilde B_t, \tilde F_t)_{t \ge 0}$ denote the diffusion on $D$ with generator $X \Delta_D$, and we let $\E_p$ denote the expectation when $(\tilde B_0, \tilde F_0) = p \in D$. For $z \in \C$ with $\Re z > 0$, $t \ge 0$, and bounded measurable functions $f$ on $D$, we set
\[
\big( e^{t(X \Delta_D - z d)} f \big)(p) = \E_p \Big[ f(\tilde B_t, \tilde F_t) \exp\Big( -z \int_0^t (\tilde B_s - \tilde F_s) \dd s \Big) \Big],
\]
and we define the \emph{singlet state} $\Psi^{X, z}_t \colon D \to \C$ by
\begin{equation}\label{e:intro-singlet}
\Psi^{X, z}_t = 1 - z d \int_0^t e^{v (X \Delta_D - z d)} 1 \dd v.
\end{equation}
For $0 < x \le y < 1$ and $\Re z > 0$, we define
\begin{align}
K_X(x, y; z) &= \int_D \Psi^{X, z}_x \, \big( e^{(y - x)(X \Delta_D - z d)} \Psi^{X, z}_{1 - y} \big) \dd B \dd F, \label{e:intro-K} \\
\widetilde K_X(x; z) &= \int_D \frac{B}{d} \, \Psi^{X, z}_x \Psi^{X, z}_{1 - x} \dd B \dd F, \label{e:intro-Kc} \\
I_X(z) &= \int_0^1 \int_x^1 K_X(x, y; z) \dd y \dd x, \label{e:intro-I}
\end{align}
and we set $K_X(x, y; z) = K_X(y, x; z)$ for $0 < y < x < 1$. We call $\widetilde K_X$ the \emph{contact kernel}.
\end{definition}

The coordinates $\tilde B_t$ and $\tilde F_t$ are independent one-dimensional diffusions, with generators $X \partial_B ((B^2 - 1) \partial_B)$ on $[1, \infty)$ and $X \partial_F ((1 - F^2) \partial_F)$ on $[-1, 1]$. A computation of their scale functions and speed measures shows that their finite endpoints are entrance boundaries in Feller's classification. This implies that the diffusion is determined by its generator and does not reach the boundary of $D$ at positive times. The integrals in \eqref{e:intro-K}--\eqref{e:intro-I} converge absolutely, and $K_X$, $\widetilde K_X$, and $I_X$ are holomorphic in the half-plane $\Re z > 0$ (see \zcref{s:sigma-radial,s:residue}). The singularity of $B/d$ in \eqref{e:intro-Kc} is at the corner $(1, 1)$, where it is integrable.

\begin{remark}\label{r:intro-singlet}
The time variables in \zcref{d:kernels} are positions along the chain. In \eqref{e:intro-K}, the diffusion runs for time $x$ from the left end of $[0, 1]$, for time $y - x$ between the two points $x$ and $y$, and for time $1 - y$ from the right end. The singlet state has the following probabilistic meaning. A direct computation shows that $1/d$ is harmonic for $\Delta_D$ away from the corner $(1, 1)$. Let $(B_t, F_t)_{t \ge 0}$ denote the Doob transform of $(\tilde B_t, \tilde F_t)$ by $1/d$, which is the diffusion with generator
\[
X d \, \Delta_D \, d^{-1} = X \Big( \Delta_D - \frac{2(B^2 - 1)}{d} \partial_B + \frac{2(1 - F^2)}{d} \partial_F \Big),
\]
stopped when it reaches the corner. We call it the singlet diffusion. For $\Re z > 0$, the defining property of the Doob transform and \eqref{e:intro-singlet} give
\[
\Psi^{X, z}_t(p) = \E^{\mathrm s}_p \Big[ \exp\Big( -z \int_0^t (B_v - F_v) \dd v \Big) \Big],
\]
where $\E^{\mathrm s}_p$ denotes the expectation when $(B_0, F_0) = p$ (see \zcref{l:res-singlet}). Consequently $|\Psi^{X, z}_t| \le \Psi^{X, \Re z}_t \le 1$ and $\Psi^{X, z}_t(1, 1) = 1$, since $d$ vanishes at the corner. By \zcref{l:sigma-localization}, the Berezin integral over $\mathcal M_1$ of a superfunction that is annihilated by the odd vector field (supercharge) of that lemma equals the value of the superfunction at the point $\Lambda$ of $\mathcal M_1$, and this point corresponds to the corner. We explain in \zcref{s:sigma-radial} how $\Psi^{X, z}_t$ and the kernels arise from the sigma model. In the notation of the physics literature, we have $X \Delta_D - zd = -2X d^{-1} \mathcal H d$, where $\mathcal H$ is the transfer-matrix Hamiltonian of the unitary wire \cite[(12)]{ivanov2009correlations}, $d^{-1} \mathcal H d$ is the operator of \cite[(18)--(20)]{ivanov2009correlations}, and their dimensionless frequency is set to $z/(2X)$; time there is position in units of the localization length.
\end{remark}

Our first main result is the following theorem, which identifies the limit of $\Xi_N$.

\begin{theorem}\label{t:process}
Let $X \in (0, \infty)$. In the critical limit at $X$, the following statements hold.
\begin{enumerate}[label=(\alph*)]
\item The point processes $\Xi_N$ converge in law, in the vague topology, to a point process $\Xi_X$ on $\R$ that is simple, stationary, invariant under the reflection $x \mapsto -x$, and of intensity one. The law of $\Xi_X$ depends only on $X$.
\item For all $\eps > 0$, $k \ge 1$, $\mathbf u \in \R^k$ with $\sum_j u_j = 0$, and $\mathbf b \in (0, \infty)^k$, we have $T_N^{(k)}(\mathbf u, \mathbf b) \to \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b)$ and
\begin{equation}\label{e:intro-source-law}
\E \prod_{x \in \Xi_X} \prod_{j = 1}^k \frac{(x - u_j)^2 + b_j^2}{x^2 + \eps^2} = \mathcal Z^{(k)}_{X, \eps}(\mathbf u, \mathbf b).
\end{equation}
For every fixed $\eps > 0$, the values of the left side of \eqref{e:intro-source-law} for all $k$, $\mathbf u$, and $\mathbf b$ as above determine the law of $\Xi_X$.
\item For all $\eps > 0$,
\begin{equation}\label{e:intro-jets}
\partial_t T_N^{(1)}(0, \eps + t) \big|_{t = 0} \longrightarrow 2\pi, \qquad \partial_t^2 T_N^{(1)}(0, \eps + t) \big|_{t = 0} \longrightarrow 4\pi^2 \big( 1 + I_X(2\pi\eps) \big).
\end{equation}
\end{enumerate}
\end{theorem}

The proof of \zcref{t:process} is given in \zcref{s:process}. The matrices appear in the proof of part (b) only through \zcref{t:interface}, which is the main result of Part IV. We derive parts (a) and (c) from part (b), from the counting estimates of \zcref{l:iface-counts}, which follow from known results, and, for part (c), from a computation in the sigma model of rank one. By part (b), a single damping $\eps$ suffices to determine the law. The reason is that the logarithms of the rational functions in \eqref{e:intro-source-law}, which all have the same poles $\pm \iu \eps$, approximate every nonnegative continuous function $f$ with compact support in the norm $\sup_x (x^2 + \eps^2) |f(x)|$. After the change of variables $x = \eps \tan(\vartheta/2)$, this follows from Fej\'er's theorem.

The limits in part (c) concern sources of rank one. Let $P_\eps(v) = \eps/(\pi(v^2 + \eps^2))$ denote the Poisson kernel. Differentiating the product in \eqref{e:intro-ratio}, we find
\[
\partial_t T_N^{(1)}(0, \eps + t) \big|_{t = 0} = 2\pi \E \sum_{\alpha = 1}^N P_\eps(x_\alpha),
\]
and the first limit in \eqref{e:intro-jets} expresses that $\Xi_N$ has asymptotic intensity one. The second derivative is a two-point function of $\Xi_N$ smoothed at scale $\eps$. Together with stationarity, it identifies the pair correlation of $\Xi_X$, as follows. \zcref[S]{t:process} confirms the prediction, recalled in \zcref{s:intro-background}, that the local eigenvalue statistics at the critical bandwidth form a family of laws indexed by $X$. We call a positive Radon measure $\nu$ on $\R$ translation-bounded if $\sup_{s \in \R} \nu([s, s + 1]) < \infty$. We call the measure $\nu_X$ of the next corollary the \emph{pair measure} of $\Xi_X$. It is the reduced second factorial moment measure of $\Xi_X$ in the sense of \cite[Section 8.1]{daley2003introduction}.

\begin{corollary}\label{c:intro-pair}
Let $X \in (0, \infty)$. There exists an even, positive, and translation-bounded Radon measure $\nu_X$ on $\R$ such that
\begin{equation}\label{e:intro-pair}
\E \sum_{\substack{x, y \in \Xi_X \\ x \neq y}} f(x, y) = \int_\R \int_\R f(v, v + s) \, \nu_X(\dd s) \dd v
\end{equation}
for all nonnegative measurable functions $f$ on $\R^2$. It satisfies
\begin{equation}\label{e:intro-pair-transform}
(P_\eps * \nu_X)(v) = 1 + \Re I_X(\pi \eps - \iu \pi v) - P_\eps(v), \qquad \eps > 0, \ v \in \R,
\end{equation}
and it is the unique translation-bounded measure with this property. Further, for all bounded intervals $A_1, A_2 \subset \R$, in the critical limit at $X$,
\begin{equation}\label{e:intro-count-cov}
\Cov\big( \Xi_N(A_1), \Xi_N(A_2) \big) \longrightarrow |A_1 \cap A_2| + \int_\R |A_1 \cap (A_2 - s)| \, \nu_X(\dd s) - |A_1| |A_2|,
\end{equation}
and the right side equals $\Cov(\Xi_X(A_1), \Xi_X(A_2))$.
\end{corollary}

\zcref[S]{c:intro-pair} is proved in \zcref{s:scales}. By \eqref{e:intro-pair-transform}, the function $1 + \Re I_X(\pi \eps - \iu \pi v)$ is the Poisson smoothing at scale $\eps$ of the two-point function $\delta_0 + \nu_X$ of $\Xi_X$, including its diagonal term. For $L \ge 0$, we set
\begin{equation}\label{e:intro-variance}
\Sigma_X^2(L) = L + \int_{-L}^L (L - |s|) \, \nu_X(\dd s) - L^2.
\end{equation}
Applying \zcref{c:intro-pair} with $A_1 = A_2 = [c, c + L]$, we see that
\[
\Sigma_X^2(L) = \Var \Xi_X([c, c + L]) = \lim \Var \Xi_N([c, c + L]), \qquad c \in \R,
\]
where the limit is the critical limit at $X$. We call $\Sigma_X^2(L)$ the \emph{number variance} of $\Xi_X$.

\subsection{Eigenvectors}

We select the eigenvectors of $H$ through their eigenvalues, with a smooth window. We fix a nonnegative function $\phi \in C_c^\infty((-1, 1))$ with $\int \phi = 1$ for the rest of the paper. For $E \in \R$ and $r > 0$, we set $\phi_{E, r}(\lambda) = r^{-1} \phi((\lambda - E)/r)$.

\begin{definition}\label{d:selection}
Let $E \in (-2, 2)$ and $r > 0$, and let $w_\alpha = (w_\alpha(1), \dots, w_\alpha(N))$. For each $\alpha$, let $\Phi_\alpha = \Phi(\lambda_\alpha, w_\alpha)$, where $\Phi \colon \R \times [0, 1]^N \to \R$ is a measurable function that is bounded or nonnegative. We define the \emph{annealed selection} of $\Phi_\alpha$ at energy $E$ and scale $r$ by
\[
\langle \Phi_\alpha \rangle_{E, r} = \frac{\E \sum_\alpha \phi_{E, r}(\lambda_\alpha) \Phi_\alpha}{\E \sum_\alpha \phi_{E, r}(\lambda_\alpha)},
\]
and, for $i \in \{1, \dots, N\}$, its \emph{site-rooted} version by
\[
\langle \Phi_\alpha \rangle^i_{E, r} = \frac{\E \sum_\alpha \phi_{E, r}(\lambda_\alpha) w_\alpha(i) \Phi_\alpha}{\E \sum_\alpha \phi_{E, r}(\lambda_\alpha) w_\alpha(i)}.
\]
\end{definition}

The denominators are positive, since the law of $H$ has a positive density. The two selections are expectations under probability measures on pairs $(H, \alpha)$. The first tilts the law of $H$ by $\sum_\alpha \phi_{E, r}(\lambda_\alpha)$, and given $H$ it chooses $\alpha$ with probability proportional to $\phi_{E, r}(\lambda_\alpha)$. The second tilts further by $w_\alpha(i)$. They are ratios of expectations, not expectations of random ratios (this is the sense of the word annealed), and they are well defined when $r$ is much smaller than the spacing of the eigenvalues near $E$, in which case a typical sample of $H$ has no eigenvalue in the support of $\phi_{E, r}$. Our results hold uniformly for $0 < r \le \bar r_N$, for every deterministic sequence $\bar r_N \to 0$. This covers all windows that are small on the macroscopic scale, including windows much smaller than the spacing.

We let $\mathcal P([0, 1])$ denote the space of Borel probability measures on $[0, 1]$, with the topology of weak convergence. We set $\mu(f) = \int f \dd \mu$ for $\mu \in \mathcal P([0, 1])$ and bounded measurable functions $f$ on $[0, 1]$. We define the \emph{mass measure} of $u_\alpha$ by
\[
\mu_\alpha^{(N)} = \sum_{i = 1}^N w_\alpha(i) \delta_{i/N} \in \mathcal P([0, 1]).
\]
Its limit is described by the following Brownian objects.

\begin{definition}\label{d:brownian}
Let $X > 0$, and let $\beta = (\beta_t)_{t \in [0, 1]}$ be a standard Brownian motion, with law $\Prob$ and expectation $\E$. We set
\[
g_X(t) = \sqrt{2X} \beta_t - X t, \qquad Z_X = \int_0^1 e^{g_X(t)} \dd t, \qquad \mu_X(\dd t) = \frac{e^{g_X(t)}}{Z_X} \dd t.
\]
Since $\E e^{g_X(t)} = 1$ for all $t$, we have $\E Z_X = 1$. We let $\mathbb Q_X$ denote the probability measure defined by $\dd \mathbb Q_X = Z_X \dd \Prob$. For $k \ge 1$ and $x_1, \dots, x_k \in [0, 1]$, we set
\[
M_k^X(x_1, \dots, x_k) = \E \bigg[ \frac{\exp\big( g_X(x_1) + \dots + g_X(x_k) \big)}{Z_X^{k - 1}} \bigg], \qquad \mathcal C_X(x_1, x_2) = M_2^X(x_1, x_2).
\]
\end{definition}

The random measure $\mu_X$ is the normalized exponential of a Brownian motion with drift, and $\mathbb Q_X$ tilts its law by the total mass $Z_X$. Using the definition of $\mathbb Q_X$, we see that $M_k^X$ is the density of the $k$-th moment measure of $\mu_X$ under $\mathbb Q_X$,
\[
\E_{\mathbb Q_X} \int_{[0, 1]^k} f \dd \mu_X^{\otimes k} = \int_{[0, 1]^k} f(x) M_k^X(x) \dd x
\]
for all nonnegative measurable functions $f$ on $[0, 1]^k$. In particular $M_1^X = 1$. We call $\mathcal C_X$ the \emph{eigenvector correlation}.

\begin{theorem}\label{t:coarse}
Let $\bar r_N \to 0$ be a deterministic sequence, and let $\Phi \colon \mathcal P([0, 1]) \to \R$ be continuous. Then, uniformly in the critical regime and in $0 < r \le \bar r_N$,
\[
\big\langle \Phi(\mu_\alpha^{(N)}) \big\rangle_{E, r} - \E_{\mathbb Q_{X_N(E)}} \big[ \Phi(\mu_{X_N(E)}) \big] \longrightarrow 0.
\]
In particular, in the critical limit at $X$, the law of $\mu_\alpha^{(N)}$ under $\langle \cdot \rangle_{E, r}$ converges weakly to the law of $\mu_X$ under $\mathbb Q_X$, for every choice of $r = r_N \in (0, \bar r_N]$.
\end{theorem}

\zcref[S]{t:coarse} is proved in \zcref{s:coarse}. It describes the distribution of the mass of an eigenvector, selected by its eigenvalue, on the macroscopic scale. In the critical regime the mass is spread over the whole chain, but it is not flat. Its density is the exponential of a Brownian motion with drift, normalized to have total mass one, under the law tilted by the total mass. This confirms the law of the envelope of an eigenfunction derived by Fyodorov and Mirlin \cite{fyodorov1994statistical}, in the form of \cite[(3.32)--(3.34)]{mirlin2000statistics}. The same law, as a law of exponential functionals of Brownian motion, was proved by Rifkind and Vir\'ag for the critical one-dimensional random Schr\"odinger operator recalled in \zcref{s:intro-background}, with their parameter $\tau$ equal to $4X$ \cite[Theorem 1.1]{rifkind2018eigenvectors}; we check this identification in \zcref{r:coarse-rv}. \zcref[S]{t:coarse} shows that band matrices at the critical bandwidth have the same coarse eigenvector law. The Brownian motion comes from the noncompact direction $B$ of the sigma model. In \zcref{s:residue}, we show that the diffusion of \zcref{d:kernels} becomes a geometric Brownian motion on the scale where $B$ is large, with logarithm given by the process $g_X$. As $X \downarrow 0$, the measure $\mu_X$ tends to the Lebesgue measure, which is the macroscopic profile of the eigenvectors of the GUE. As $X \to \infty$, the mass concentrates near a uniformly distributed point, as expected for localized eigenvectors (see \zcref{p:endpoints}). The coarse law does not control the mass at the scale of single sites, since functions such as $\sum_i w_\alpha(i)^2$ are not continuous functions of $\mu_\alpha^{(N)}$. We treat these in \zcref{t:profile}.

The next corollary concerns the mean and the covariance of the mass. We let $\Cov_{\mathbb Q_X}$ denote the covariance under $\mathbb Q_X$, and for observables $\Phi_\alpha$ and $\Phi'_\alpha$ as in \zcref{d:selection}, we define their covariance under the annealed selection by
\[
\Cov_{E, r}(\Phi_\alpha, \Phi'_\alpha) = \langle \Phi_\alpha \Phi'_\alpha \rangle_{E, r} - \langle \Phi_\alpha \rangle_{E, r} \langle \Phi'_\alpha \rangle_{E, r}.
\]

\begin{corollary}\label{c:coarse-cov}
Let $X > 0$, and let $f, g \colon [0, 1] \to \R$ be bounded and Riemann integrable. Then
\begin{equation}\label{e:intro-coarse-cov}
\begin{gathered}
\E_{\mathbb Q_X} \mu_X(f) = \int_0^1 f(t) \dd t, \\
\Cov_{\mathbb Q_X}\big( \mu_X(f), \mu_X(g) \big) = \int_0^1 \int_0^1 f(x) g(y) \big( \mathcal C_X(x, y) - 1 \big) \dd x \dd y.
\end{gathered}
\end{equation}
Further, let $\bar r_N \to 0$ be a deterministic sequence. In the critical limit at $X$, uniformly in $0 < r \le \bar r_N$, the quantities $\langle \mu_\alpha^{(N)}(f) \rangle_{E, r}$ and $\Cov_{E, r}(\mu_\alpha^{(N)}(f), \mu_\alpha^{(N)}(g))$ converge to the corresponding quantities in \eqref{e:intro-coarse-cov}.
\end{corollary}

The exact identities in \eqref{e:intro-coarse-cov} follow from \zcref{d:brownian} and Tonelli's theorem, and the matrix limits are proved in \zcref{s:coarse}. In particular $\E_{\mathbb Q_X} \mu_X$ is the Lebesgue measure on $[0, 1]$, and a selected eigenvector is flat on average. The function $\mathcal C_X - 1$ is the covariance density of the mass.

We now describe the mass as seen from a point of the chain, which we call the root. There are two natural choices of the root, at random according to the mass of the eigenvector, or at a given site under the site-rooted selection. For the first, we let $\overline{\mathbb Q}_X$ denote the probability measure on pairs $(\beta, U)$, where $\beta$ is a path on $[0, 1]$ and $U \in [0, 1]$, defined by
\[
\overline{\mathbb Q}_X(\dd \beta, \dd U) = e^{g_X(U)} \, \Prob(\dd \beta) \dd U.
\]
Since $\int_0^1 e^{g_X(U)} \dd U = Z_X$ and $\E e^{g_X(U)} = 1$, the measure $\overline{\mathbb Q}_X$ has first marginal $\mathbb Q_X$ and a uniform second marginal. For the second, we let $x \in (0, 1)$ and set
\begin{equation}\label{e:intro-rooted-density}
p_{X, x}(t) = \frac{\exp\big( \sqrt{2X} (\beta_t - \beta_x) - X |t - x| \big)}{\int_0^1 \exp\big( \sqrt{2X} (\beta_s - \beta_x) - X |s - x| \big) \dd s}, \qquad t \in [0, 1].
\end{equation}
For the large-$X$ limit, we let $(\beta_s)_{s \in \R}$ denote a two-sided Brownian motion, so that $(\beta_s)_{s \ge 0}$ and $(\beta_{-s})_{s \ge 0}$ are independent standard Brownian motions, and we set
\begin{equation}\label{e:intro-shape}
p_\infty(s) = \frac{e^{\sqrt 2 \beta_s - |s|}}{\int_\R e^{\sqrt 2 \beta_{s'} - |s'|} \dd s'}, \qquad s \in \R.
\end{equation}
The integral in the denominator is finite almost surely, since $\beta_s/s \to 0$ as $|s| \to \infty$. For $X > 0$ and $t_0 \in [0, 1]$, we let $\sigma_{X, t_0}(t) = X(t - t_0)$. We let $\mathcal P(\R)$ denote the space of Borel probability measures on $\R$, with the topology of weak convergence.

\begin{theorem}\label{t:shape}
Let $X \in (0, \infty)$, and let $\bar r_N \to 0$ be a deterministic sequence. The following statements hold.
\begin{enumerate}[label=(\alph*)]
\item In the critical limit at $X$, uniformly in $0 < r \le \bar r_N$, for every continuous function $\Phi \colon \mathcal P([0, 1]) \times [0, 1] \to \R$,
\[
\Big\langle \int_0^1 \Phi\big( \mu_\alpha^{(N)}, t \big) \, \mu_\alpha^{(N)}(\dd t) \Big\rangle_{E, r} \longrightarrow \E_{\overline{\mathbb Q}_X} \big[ \Phi(\mu_X, U) \big].
\]
\item Let $i = i_N$ satisfy $i/N \to x \in (0, 1)$. In the critical limit at $X$, uniformly in $0 < r \le \bar r_N$, for every continuous function $\Phi \colon \mathcal P([0, 1]) \to \R$,
\[
\big\langle \Phi(\mu_\alpha^{(N)}) \big\rangle^i_{E, r} \longrightarrow \E \big[ \Phi\big( p_{X, x}(t) \dd t \big) \big].
\]
\item As $X \to \infty$, the law of $\big( (\sigma_{X, U})_* \mu_X, U \big)$ under $\overline{\mathbb Q}_X$ converges weakly to the law of $(p_\infty(s) \dd s, U')$, where $U'$ is uniform on $[0, 1]$ and independent of $\beta$. For every $x \in (0, 1)$, the law of $(\sigma_{X, x})_* (p_{X, x}(t) \dd t)$ under $\Prob$ converges weakly to the law of $p_\infty(s) \dd s$, as random elements of $\mathcal P(\R)$.
\end{enumerate}
\end{theorem}

\zcref[S]{t:shape} is proved in \zcref{s:coarse}. In part (a), the integral against $\mu_\alpha^{(N)}(\dd t)$ samples a root $t$ from the mass of the eigenvector, and the theorem states that the pair formed by the mass measure and the root converges in law to $(\mu_X, U)$ under $\overline{\mathbb Q}_X$. By the Cameron--Martin theorem, the random measure $p_{X, x}(t) \dd t$ under $\Prob$ has the law of $\mu_X$ under $e^{g_X(x)} \dd \Prob$. Indeed, tilting by $e^{g_X(x)}$ adds the drift $\sqrt{2X}$ to $\beta$ on $[0, x]$, which reverses the drift of $g_X$ to the left of $x$. The mass of an eigenvector rooted at $x$ decays exponentially away from $x$ in each direction, at rate $X$ per unit length, up to the fluctuations of two independent Brownian motions. At large $X$, part (c) describes this shape on the scale $1/X$. (In the coupling given by Brownian scaling, the densities converge in $L^1$ almost surely.) Undoing the scaling, a distance $s$ in \eqref{e:intro-shape} corresponds to $(4 - E^2) W^2 s$ sites. Heuristically, at large fixed $X$, the mass of a selected eigenvector at distance $m$ from its root is of order $\exp(\sqrt 2 \beta_s - |s|)$ with $s = m/((4 - E^2) W^2)$.

These shapes coincide with those of the critical random Schr\"odinger operator. Under the identification of parameters stated after \zcref{t:coarse}, and with the function $S$ of \zcref{s:intro-background}, the pair formed by the density of Rifkind and Vir\'ag \cite[Theorem 1.1]{rifkind2018eigenvectors} and its root $U$ has the law of $(\mu_X, U)$ under $\overline{\mathbb Q}_X$, which is the limit in \zcref{t:shape}(a), and by Brownian scaling the density on $\R$ proportional to $S(4s)$ has the law of $p_\infty$; we show this in \zcref{r:coarse-rv} and at the start of \zcref{s:coarse-wire}. Recentered at its center of mass, the same shape is the limit of the eigenfunctions of the continuous Anderson Hamiltonian in its crossover regime \cite[Theorem 3]{dumaz2023localization}. We also determine the mean of the random shape.

\begin{corollary}\label{c:mean-shape}
Let $p_\infty$ be as in \eqref{e:intro-shape}, and set $\bar p_\infty(s) = \E p_\infty(s)$ for $s \in \R$. Then $\bar p_\infty$ is even, $\bar p_\infty(0) = 1/3$, $\int_\R \bar p_\infty(s) \dd s = 1$, and
\begin{equation}\label{e:intro-tail}
\bar p_\infty(s) = \frac{\pi^{7/2}}{32 \, s^{3/2}} e^{-s/4} \big( 1 + O(s^{-1}) \big), \qquad s \to \infty.
\end{equation}
\end{corollary}

\zcref[S]{c:mean-shape} is proved in \zcref{s:coarse}. Since $\beta_s/s \to 0$ almost surely, a typical sample of $p_\infty$ satisfies $\log p_\infty(s) = -|s|(1 + o(1))$ as $|s| \to \infty$, while by \eqref{e:intro-tail} the mean decays at rate $1/4$. The mean profile is therefore governed by rare samples, in which the Brownian motion is unusually large far from the root. The tail \eqref{e:intro-tail}, with its constant, confirms a prediction of Efetov and Larkin \cite[(56)]{efetov1983kinetics} (see \zcref{s:intro-physics}).
The value $\bar p_\infty(0) = 1/3$ is half the slope in $X$ of the limiting normalized mean inverse participation ratio in \zcref{t:profile} below. Indeed, we show in \zcref{s:coarse} that $X^{-1} \mathcal C_X(x, x) \to \bar p_\infty(0)$ as $X \to \infty$, for all $x \in (0, 1)$.

\subsection{Local intensities and participation}

We now turn to the local density of states at a single site. For $i \in \{1, \dots, N\}$, $\eps > 0$, and $v \in \R$, we set
\[
\ell^\eps_{i, N}(v) = N \sum_{\alpha = 1}^N w_\alpha(i) P_\eps(x_\alpha - v).
\]
We let $G(z) = (H - z)^{-1}$ denote the resolvent of $H$, for $z \in \C$ with $\Im z \neq 0$. Using the spectral decomposition $G_{ii}(z) = \sum_\alpha w_\alpha(i)/(\lambda_\alpha - z)$ and \eqref{e:scaling}, we find
\begin{equation}\label{e:intro-ldos}
\ell^\eps_{i, N}(v) = s_E^{-1} \Im G_{ii}\big( E + (v + \iu \eps)/a_N \big).
\end{equation}
By \zcref{t:interface}(d), the mean of $G_{ii}(E + (v + \iu \eps)/a_N)$ converges to $-E/2 + \iu s_E$, uniformly over sites $i$ with $i/N$ away from $0$ and $1$, and the expectation of $\ell^\eps_{i, N}(v)$ therefore tends to one at such sites. The next theorem identifies the limit of the two-point function of the local density, at two energies and at two sites, for all pairs of interior sites.

\begin{theorem}\label{t:intensity}
Let $\delta \in (0, 1/2)$. For $\eps_1, \eps_2 > 0$ and $u, v \in \R$, set $z = \pi(\eps_1 + \eps_2) - \iu \pi (u - v)$. Uniformly in the critical regime, in $(\eps_1, \eps_2, u, v)$ in compact subsets of $(0, \infty)^2 \times \R^2$, and in sites $i, j$ with $\delta \le i/N, j/N \le 1 - \delta$, we have
\begin{align}
\E \ell^{\eps_1}_{i, N}(u) \ell^{\eps_2}_{j, N}(v) - 1 - \tfrac12 \Re K_{X_N(E)}(i/N, j/N; z) &\longrightarrow 0 \qquad (i \neq j), \label{e:intro-int-distinct} \\
\E \ell^{\eps_1}_{i, N}(u) \ell^{\eps_2}_{i, N}(v) - 1 - \Re \widetilde K_{X_N(E)}(i/N; z) &\longrightarrow 0, \label{e:intro-int-contact}
\end{align}
and
\begin{multline}\label{e:intro-int-same}
N^2 \E \sum_{\alpha = 1}^N w_\alpha(i) w_\alpha(j) P_{\eps_1}(x_\alpha - u) P_{\eps_2}(x_\alpha - v) \\
- (1 + \1_{i = j}) \, \mathcal C_{X_N(E)}(i/N, j/N) \, P_{\eps_1 + \eps_2}(u - v) \longrightarrow 0.
\end{multline}
\end{theorem}

\zcref[S]{t:intensity} is proved in \zcref{s:intensity}. The sum in \eqref{e:intro-int-same} is the part of $\E \ell^{\eps_1}_{i, N}(u) \ell^{\eps_2}_{j, N}(v)$ that comes from single eigenvalues, and subtracting it from \eqref{e:intro-int-distinct} or \eqref{e:intro-int-contact} gives the contribution of pairs of distinct eigenvalues. The theorem includes adjacent sites $j = i + 1$, for which $i/N$ and $j/N$ tend to the same point $x$. The limit $1 + \frac12 \Re K_X(x, x; z)$ of the two-point function at adjacent sites differs from the limit $1 + \Re \widetilde K_X(x; z)$ at a single site. Skvortsov and Ostrovsky found the same distinction for an infinite wire, between nearby and coincident points, with the same weight $B/(B - F)$ at coincident points \cite[(2), (7)--(8)]{skvortsov2007local}.

The two kinds of limits are related through the behavior of the kernels at $z = 0$. In \zcref{t:res-scalar} and \zcref{s:intensity}, we show that for $0 < x \le y < 1$,
\begin{equation}\label{e:intro-residue}
\frac{z}{2} K_X(x, y; z) \longrightarrow \mathcal C_X(x, y), \qquad z \widetilde K_X(x; z) \longrightarrow 2 \mathcal C_X(x, x), \qquad z \downarrow 0,
\end{equation}
where $z$ tends to zero along the positive real axis. In this sense, the eigenvector correlations are the residues of the kernels at $z = 0$. Since $\Re z^{-1} = P_{\eps_1 + \eps_2}(u - v)$, the term $(1 + \1_{i = j}) \, \mathcal C_X(x, y) \Re z^{-1}$ that comes from these residues is exactly the limit in \eqref{e:intro-int-same}. Our proofs use this relation in the reverse direction. In \zcref{s:extract,s:intensity}, we extract the eigenvector correlations from the local densities at positive damping. The factor $1 + \1_{i = j}$ is the ratio $\E |\gamma|^4/(\E |\gamma|^2)^2 = 2$ for a centered complex Gaussian variable $\gamma$ with $\E \gamma^2 = 0$. This suggests that, on the scale of single sites, the coordinates of an eigenvector behave like complex Gaussian variables whose variances are given by the mass measure.

The eigenvector correlations at the scale of single sites are given by the following theorem. For $E \in (-2, 2)$, $r > 0$, and $i, j \in \{1, \dots, N\}$, we define the \emph{rooted mean profile}
\[
P_N(E, r; i, j) = N \langle w_\alpha(j) \rangle^i_{E, r},
\]
so that $N^{-1} P_N(E, r; i, j)$ is the mean mass at $j$ of an eigenvector rooted at $i$. We let $\IPR(u_\alpha) = \sum_i w_\alpha(i)^2$ denote the inverse participation ratio of $u_\alpha$.

\begin{theorem}\label{t:profile}
Let $\delta \in (0, 1/2)$, and let $\bar r_N \to 0$ be a deterministic sequence. Uniformly in the critical regime and in $0 < r \le \bar r_N$,
\begin{equation}\label{e:intro-profile}
\sup_{\delta \le i/N, \, j/N \le 1 - \delta} \big| P_N(E, r; i, j) - (1 + \1_{i = j}) \, \mathcal C_{X_N(E)}(i/N, j/N) \big| \longrightarrow 0,
\end{equation}
and the same holds with $P_N(E, r; i, j)$ replaced by $N^2 \langle w_\alpha(i) w_\alpha(j) \rangle_{E, r}$. Moreover, uniformly in the critical regime and in $0 < r \le \bar r_N$,
\begin{equation}\label{e:intro-ipr}
N \langle \IPR(u_\alpha) \rangle_{E, r} - 2 - \frac{2 X_N(E)}{3} \longrightarrow 0,
\end{equation}
and $\int_0^1 \mathcal C_X(x, x) \dd x = 1 + X/3$ for all $X > 0$.
\end{theorem}

\zcref[S]{t:profile} is proved in \zcref{s:intensity}. The profile \eqref{e:intro-profile} is a statement at the scale of single sites, with coefficient one for distinct sites, even when they are adjacent, and coefficient two for coincident sites. It does not follow from \zcref{t:coarse}, which concerns weak limits of the mass measure. The limit \eqref{e:intro-ipr} requires, in addition, bounds on the moments of the local densities at all sites, including those near the ends of the chain, which are part (c) of \zcref{t:interface}. Its proof applies to every nonnegative integrable window with compact support and positive integral in place of $\phi$. By \zcref{d:brownian}, we have
\[
\int_0^1 \mathcal C_X(x, x) \dd x = \E_{\mathbb Q_X} \int_0^1 \Big( \frac{e^{g_X(t)}}{Z_X} \Big)^2 \dd t.
\]
The limit \eqref{e:intro-ipr} confirms the prediction of Fyodorov and Mirlin \cite{fyodorov1992analytical,fyodorov1994statistical} (see \zcref{s:intro-physics}). The participation ratio of a selected eigenvector is the product of the factor two from the scale of single sites and the participation ratio of the macroscopic profile. As $X \downarrow 0$, the limit $2 + 2X/3$ tends to $2$, which is the limit of $N \IPR$ for the eigenvectors of the GUE (for these, $\E \IPR = 2/(N + 1)$). For large $X$ it grows like $2X/3$. Heuristically, a selected eigenvector then occupies of the order of $N/X$, or $(4 - E^2) W^2$, sites.

\subsection{Endpoints and the metallic regime}

The parameter $X$ interpolates between the metallic regime $X \to 0$, where one expects the statistics of the GUE, and the localized regime $X \to \infty$, where one expects Poisson statistics and localized eigenvectors. The next proposition confirms this for the observables above. Recall that these limits are taken after the critical limit, as explained after \zcref{d:intro-critical}. We set
\[
I_{\GUE}(z) = \frac{1 - e^{-2z}}{2z^2}, \qquad \Sigma^2_{\GUE}(L) = L - 2 \int_0^L (L - s) \Big( \frac{\sin \pi s}{\pi s} \Big)^2 \dd s,
\]
and we let $\delta_{\mathrm{Leb}}$ denote the point mass at the Lebesgue measure on $[0, 1]$, viewed as a law on $\mathcal P([0, 1])$.

\begin{proposition}\label{p:endpoints}
Let $L \ge 0$ and $0 < x < y < 1$. As $X \downarrow 0$ and as $X \to \infty$, the objects in the first column of the following table converge to the entries of the second and the third columns, respectively.
\begin{center}
\small\upshape
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{@{}lll@{}}
\toprule
Object & $X \downarrow 0$ & $X \to \infty$ \\
\midrule
pair measure $\nu_X(\dd s)$ & $\big( 1 - (\sin \pi s/(\pi s))^2 \big) \dd s$ & $\dd s$ \\
number variance $\Sigma_X^2(L)$ & $\Sigma^2_{\GUE}(L)$ & $L$ \\
law of $\mu_X$ under $\mathbb Q_X$ & $\delta_{\mathrm{Leb}}$ & law of $\delta_U$, with $U$ uniform on $[0, 1]$ \\
eigenvector correlation $\mathcal C_X(x, y)$ & $1$ & $0$ \\
kernel $\frac12 K_X(x, y; z)$ & $I_{\GUE}(z)$ & $0$ \\
\bottomrule
\end{tabular}
\end{center}
The pair measures converge vaguely, the laws converge weakly, and the kernels converge locally uniformly in $\Re z > 0$.
\end{proposition}

The proofs are in \zcref{s:coarse,s:scales}. The first two rows identify the pair measure and the number variance of $\Xi_X$ with those of the sine process (the local limit of the GUE) as $X \downarrow 0$, and with those of a Poisson process as $X \to \infty$. Recall from \zcref{c:intro-pair} that $\nu_X$ is determined by its Poisson transforms. One can check that $1 + \Re I_{\GUE}(\pi \eps - \iu \pi v)$ is the Poisson smoothing of the two-point function of the sine process, in the sense of \eqref{e:intro-pair-transform}. By \zcref{t:intensity}, the last two rows say that at separated sites the two-energy intensity tends to $1 + \Re I_{\GUE}(z)$ as $X \downarrow 0$ and to $1$ as $X \to \infty$, while the contribution of single eigenvalues tends to $P_{\eps_1 + \eps_2}(u - v)$ and to zero. We expect that $\Xi_X$ converges to the sine process as $X \downarrow 0$ and to a Poisson process as $X \to \infty$. However, \zcref{p:endpoints} concerns only the pair measures.

We now describe the first corrections to the metallic endpoint. We let
\[
G_{\mathrm N}(x, y) = \frac{x^2 + y^2}{2} - \max(x, y) + \frac13, \qquad x, y \in [0, 1],
\]
denote the Green kernel of the Neumann Laplacian on $[0, 1]$ acting on functions of mean zero. It satisfies $-\partial_x^2 G_{\mathrm N}(x, y) = \delta(x - y) - 1$, with $\partial_x G_{\mathrm N}(x, y) = 0$ at $x \in \{0, 1\}$ and $\int_0^1 G_{\mathrm N}(x, y) \dd x = 0$.

\begin{proposition}\label{p:metallic}
As $X \downarrow 0$,
\begin{equation}\label{e:intro-metallic-C}
\mathcal C_X(x, y) = 1 + 2X G_{\mathrm N}(x, y) + O(X^2)
\end{equation}
uniformly in $x, y \in [0, 1]$, and
\begin{equation}\label{e:intro-metallic-I}
I_X(z) = I_{\GUE}(z) + \frac{X^2}{45} \big( 1 - e^{-2z} \big) + O(X^3)
\end{equation}
uniformly for $z$ in compact subsets of $\{\Re z > 0\}$.
\end{proposition}

\zcref[S]{p:metallic} is proved in \zcref{s:metallic}. The first correction to the eigenvector correlation $\mathcal C_X$ of \zcref{c:coarse-cov} is linear in $X$, while the first correction to $I_X$ is quadratic. Both coefficients are spectral quantities of the Neumann Laplacian on $[0, 1]$, whose nonzero eigenvalues are $(\pi n)^2$, with normalized eigenfunctions $\sqrt 2 \cos(\pi n x)$ for $n \ge 1$. Indeed,
\[
G_{\mathrm N}(x, y) = \sum_{n \ge 1} \frac{2 \cos(\pi n x) \cos(\pi n y)}{(\pi n)^2}, \qquad \frac{1}{45} = 2 \sum_{n \ge 1} \frac{1}{(\pi n)^4} = 2 \Tr G_{\mathrm N}^2.
\]
By \eqref{e:intro-pair-transform}, the corresponding correction to the smoothed pair function $(P_\eps * \nu_X)(v)$ is
\[
\frac{X^2}{45} \Re \big( 1 - e^{-2\pi \eps + 2\pi \iu v} \big) \longrightarrow \frac{2X^2}{45} \sin^2(\pi v), \qquad \eps \downarrow 0.
\]
This has the form of the correction to the Wigner--Dyson two-point function of a metallic wire obtained by Kravtsov and Mirlin \cite[(15)]{kravtsov1994level}, and the spatial term $2X G_{\mathrm N}$ appears in \cite[(3.55), (3.74)]{mirlin2000statistics}; we compare both terms with the physics literature in \zcref{s:intro-physics}.

\subsection{Fluctuations across scales}

We finally consider the fluctuations of $\Xi_X$ in windows of length $L \to \infty$, at fixed $X$. Recall from \eqref{e:intro-variance} that $\Sigma_X^2(L)$ is the variance of the number of points of $\Xi_X$ in a window of length $L$. We set $f_L(s) = f(s/L)$ and $\Xi_X(f) = \sum_{x \in \Xi_X} f(x)$ for functions $f$ on $\R$ and $L > 0$. We use the Fourier transform $\hat f(k) = \int_\R e^{-2\pi \iu k s} f(s) \dd s$, and for functions $f$ and $g$ such that the integral converges absolutely we set
\[
\langle f, g \rangle_{\dot H^{1/4}} = \int_\R |k|^{1/2} \hat f(k) \overline{\hat g(k)} \dd k.
\]

\begin{theorem}\label{t:largecov}
Let $X \in (0, \infty)$, and let $f, g \colon \R \to \C$ be compactly supported functions of bounded variation, or Schwartz functions. Then, as $L \to \infty$,
\begin{equation}\label{e:intro-largecov}
\begin{aligned}
\frac{\Cov\big( \Xi_X(f_L), \Xi_X(g_L) \big)}{\sqrt{XL}} \longrightarrow {} & \frac{1}{2\sqrt\pi} \langle f, g \rangle_{\dot H^{1/4}} \\
= {} & \frac{1}{16 \pi^{3/2}} \int_\R \int_\R \frac{\big( f(s) - f(s') \big) \overline{\big( g(s) - g(s') \big)}}{|s - s'|^{3/2}} \dd s \dd s',
\end{aligned}
\end{equation}
and both integrals converge absolutely. In particular, $\Sigma_X^2(L) \sim \pi^{-3/2} \sqrt{XL}$ as $L \to \infty$.
\end{theorem}

\zcref[S]{t:largecov} is proved in \zcref{s:scales}. For the sine process, the variance of $\Xi(f_L)$ stays bounded for smooth $f$, and the number variance grows logarithmically; for a Poisson process of intensity one, these variances grow linearly. However, at every finite $X$, the process $\Xi_X$ lies strictly between these cases. Its variances grow like $\sqrt L$, with the homogeneous Sobolev seminorm of order $1/4$ in place of the seminorm of order $1/2$ that describes the sine process. The asymptotics $\pi^{-3/2} \sqrt{XL}$ confirms the law of Altshuler and Shklovskii \cite[(6)--(6a)]{altshuler1986repulsion}, which they derived for a metal, $X \ll 1$. Physics predicts it at every $X$ for windows much longer than $X$ \cite[(2), (25)--(26)]{micklitz2016spectral}, and for band matrices a version on mesoscopic scales was proved by Erd\H{o}s and Knowles, also in the localized regime \cite[Section 1.6]{erdos2015altshulerI} (see \zcref{s:intro-physics}).

By \zcref{p:endpoints}, we have $\Sigma_X^2(L) \to L$ as $X \to \infty$ for all $L \ge 0$, while \zcref{t:largecov} implies that $\Sigma_X^2(L) = o(L)$ as $L \to \infty$ for all $X > 0$. In other words,
\[
\lim_{X \to \infty} \lim_{L \to \infty} \frac{\Sigma_X^2(L)}{L} = 0, \qquad \lim_{L \to \infty} \lim_{X \to \infty} \frac{\Sigma_X^2(L)}{L} = 1.
\]
The next theorem resolves this by describing the number variance on windows whose length is of order $X$.

\begin{theorem}\label{t:crossover}
Let $I_0$ denote the modified Bessel function of order zero. Let $m_\infty(u) = e^{-2/|u|} I_0(2/|u|)$ for $u \in \R \setminus \{0\}$, let $m_\infty(0) = 0$, and set
\begin{equation}\label{e:intro-crossover}
V(\ell) = \int_\R m_\infty(u) \Big( \frac{\sin \pi \ell u}{\pi u} \Big)^2 \dd u, \qquad \ell \ge 0.
\end{equation}
Then $\Sigma_X^2(X \ell)/X \to V(\ell)$ as $X \to \infty$, locally uniformly in $\ell \in [0, \infty)$. Further, $V(\ell) \sim \ell$ as $\ell \downarrow 0$ and $V(\ell) \sim \pi^{-3/2} \sqrt \ell$ as $\ell \to \infty$, and for $\ell > 0$ we have $0 < V(\ell) < \ell$, and $V(\ell)/\ell$ is strictly decreasing in $\ell$.
\end{theorem}

\zcref[S]{t:crossover} is proved in \zcref{s:scales}, and \zcref{f:intro-crossover} shows the curve. The function $m_\infty$ is the limit on the scale $X$ of the Bartlett spectrum $m_X$ of $\Xi_X$ (the structure factor of physics), the positive measure whose Fourier transform is the reduced covariance measure $\delta_0 + \nu_X - \dd s$ \cite[Section 8.2]{daley2003introduction}; we show in \zcref{s:scales} that its Laplace transform $\int_\R e^{-2a|u|} m_\infty(u) \dd u$ is the limit of $X I_X(Xa)$ as $X \to \infty$. In physics, $m_\infty$ is the spectral form factor of a long unitary wire, with $u$ the time in units of the Heisenberg time of a localization volume. In cold-atom quench experiments, it is the contrast of the peak of coherent forward scattering \cite[(13)]{micklitz2014strong}, and we compare \zcref{t:crossover} with the physics literature in \zcref{s:intro-physics}. \zcref[S]{t:crossover} proves these predictions for the limiting processes $\Xi_X$. We have $m_\infty(u) \to 1$ as $|u| \to \infty$, which gives Poisson behavior in short windows, and $m_\infty(u) \sim |u|^{1/2}/(2\sqrt\pi)$ as $u \to 0$, which gives the law of \zcref{t:largecov} in long windows. The crossover has the following heuristic explanation. By \zcref{t:profile}, an eigenvector occupies about $N/X$ sites when $X$ is large, and the eigenvalues of eigenvectors localized in a common region of this size have unfolded spacings of order $X$. A window of length much smaller than $X$ contains at most one eigenvalue from each region, and these eigenvalues are nearly independent. A window of length much larger than $X$ contains many eigenvalues from each region, and these repel each other.

\begin{figure}[htbp]
\centering
\begin{tikzpicture}
\begin{loglogaxis}[
  width=0.78\linewidth, height=0.48\linewidth,
  xmin=1e-3, xmax=1e3, ymin=3e-3, ymax=2,
  xlabel={$\ell$}, ylabel={$V(\ell)/\ell$},
  tick label style={font=\small}, label style={font=\small},
  axis lines=left, clip=true]
\addplot[thick] table[x=ell, y=ratio, row sep=\\] {
  ell ratio \\
  1.000000e-03 9.783191e-01 \\
  1.467799e-03 9.704160e-01 \\
  2.154435e-03 9.598552e-01 \\
  3.162278e-03 9.458687e-01 \\
  4.641589e-03 9.275412e-01 \\
  6.812921e-03 9.038274e-01 \\
  1.000000e-02 8.736062e-01 \\
  1.467799e-02 8.357897e-01 \\
  2.154435e-02 7.895049e-01 \\
  3.162278e-02 7.343566e-01 \\
  4.641589e-02 6.707500e-01 \\
  6.812921e-02 6.002021e-01 \\
  1.000000e-01 5.254954e-01 \\
  1.467799e-01 4.504814e-01 \\
  2.154435e-01 3.793881e-01 \\
  3.162278e-01 3.157343e-01 \\
  4.641589e-01 2.613375e-01 \\
  6.812921e-01 2.160895e-01 \\
  1.000000e+00 1.787012e-01 \\
  1.467799e+00 1.477355e-01 \\
  2.154435e+00 1.220722e-01 \\
  3.162278e+00 1.008316e-01 \\
  4.641589e+00 8.326808e-02 \\
  6.812921e+00 6.875312e-02 \\
  1.000000e+01 5.676225e-02 \\
  1.467799e+01 4.685920e-02 \\
  2.154435e+01 3.868195e-02 \\
  3.162278e+01 3.193059e-02 \\
  4.641589e+01 2.635696e-02 \\
  6.812921e+01 2.175589e-02 \\
  1.000000e+02 1.795782e-02 \\
  1.467799e+02 1.482269e-02 \\
  2.154435e+02 1.223485e-02 \\
  3.162278e+02 1.009877e-02 \\
  4.641589e+02 8.335606e-03 \\
  6.812921e+02 6.880268e-03 \\
  1.000000e+03 5.679015e-03 \\
};
\addplot[dashed, gray, domain=1e-3:1e3, samples=2] {1};
\addplot[densely dotted, thick, gray, domain=1e-2:1e3, samples=2] {pi^(-1.5)*x^(-0.5)};
\end{loglogaxis}
\end{tikzpicture}
\caption{The variance per unit length $V(\ell)/\ell$ of \zcref{t:crossover} (solid line), computed by numerical quadrature with the supplementary script \texttt{crossover\_ratio.py}, with the Poisson value $1$ (dashed gray line) and the large-window asymptote $\pi^{-3/2} \ell^{-1/2}$ (dotted gray line).}
\label{f:intro-crossover}
\end{figure}

\begin{remark}\label{r:intro-scope}
We list the main questions that our results leave open. The results are proved only for $|E| \le E_*$. The endpoint statements of \zcref{p:endpoints} concern pair measures, number variances, and laws of the mass measure, and we do not prove that $\Xi_X$ itself converges to the sine process or to a Poisson process. The crossover of \zcref{t:crossover} is a statement about the limiting processes $\Xi_X$, not about matrices with $X_N(E) \to \infty$. The two-energy intensities of \zcref{t:intensity} relate the eigenvalues to the eigenvectors, but we do not identify the joint limit of the eigenvalues and the mass measures as a marked point process. The eigenvector results describe one eigenvector selected by its eigenvalue; they do not give the law of a single coordinate $N w_\alpha(i)$, or the joint law of several eigenvectors. Finally, \zcref{c:intro-pair,t:largecov} are second-moment statements, and we do not prove central limit theorems for linear statistics of $\Xi_X$. A point process is called number rigid if, for every bounded interval, the configuration outside the interval determines almost surely the number of points inside. Since the variance of $\Xi_X(f_L)$ grows like $\sqrt L$, the variance criterion for number rigidity of Ghosh and Peres \cite[Theorem 6.1]{ghosh2017rigidity}, which is stated for processes on $\C$ and whose proof applies on $\R$, cannot be met by dilations of a test function, and we do not know whether $\Xi_X$ is number rigid.
\end{remark}

\section{Previous work and methods}\label{s:intro-methods}

\subsection{Relation to the physics literature}\label{s:intro-physics}

The physics predictions for band matrices at the critical bandwidth come in two steps. The first is the reduction of the matrix model to the one-dimensional sigma model, which is a saddle-point approximation for wide bands without control of the errors. The second is the evaluation of observables within the sigma model, which is in many cases an exact computation and in others a perturbation expansion in $X$ (see \cite{mirlin2000statistics}). Our interface theorem, \zcref{t:interface}, makes the first step rigorous for the matrix of \zcref{d:model} at $W^2 \asymp N$, and Part III carries out the second step rigorously for the observables of \zcref{s:intro-results}. (For the review \cite{mirlin2000statistics}, we use the equation numbers of the arXiv version.)

In the physics literature, a disordered wire in the unitary class is described by the ratio of the length of the wire to the length $\xi = 2\pi \nu A D_{\mathrm{diff}}$, which is given by the density of states $\nu$, the cross-section $A$, and the diffusion constant $D_{\mathrm{diff}}$ \cite[(52)--(53)]{efetov1983kinetics} (see \cite[Section 3.2.1]{mirlin2000statistics}). For band matrices, Fyodorov and Mirlin wrote the sigma model on the lattice with a coupling equal to the square of the imaginary part of the saddle point times $\sum_r J(r) r^2$ \cite[(7)]{fyodorov1995statistical}. Recall from \eqref{e:scaling} that $s_E = \sqrt{1 - E^2/4}$, so that $4 s_E^2 = 4 - E^2$. For our profile, the saddle point, which solves the self-consistent equation of the semicircle law (see \zcref{s:ov-scales}), has imaginary part $s_E$, and $\sum_r J(r) r^2 = 2W^2$ away from the ends of the chain, so that the coupling equals $2 s_E^2 W^2$. Comparing it with the continuum normalization of \cite[(2.10)]{mirlin2000statistics}, we find that $\xi$ is twice the coupling, and
\[
\xi = (4 - E^2) W^2, \qquad X_N(E) = \frac{N}{\xi}.
\]
The localization length given after \cite[(7)]{fyodorov1995statistical} is $2\xi$, four times the coupling; it is the decay length of $|\psi|$, while $\xi$ is that of $|\psi|^2$. This identification is heuristic, but it agrees with the typical decay of the mass in \zcref{t:shape}, at rate one in units of $(4 - E^2) W^2$ sites, and with the parameter of the critical result \cite[Theorem 1.2]{shcherbina2020characteristic} on characteristic polynomials.

Several of our results are rigorous versions of predictions from the sigma model. The coarse law of \zcref{t:coarse} is the law of the envelope of an eigenfunction of a finite wire, which was derived by Fyodorov and Mirlin \cite{fyodorov1994statistical} and written by Mirlin as a path integral \cite[(3.32)--(3.34)]{mirlin2000statistics}, following the method of Kolokolov for strictly one-dimensional systems \cite{kolokolov1993method}. Using the Girsanov theorem, one can check that the law \cite[(3.34)]{mirlin2000statistics} of the logarithm of the envelope is the law of $\log(e^{g_X}/Z_X)$ under $\mathbb Q_X$, with $g_X$, $Z_X$, and $\mathbb Q_X$ as in \zcref{d:brownian}; we write out this translation after \eqref{e:coarse-mirlin}. In the same description, an eigenfunction is the product of the envelope and a fast factor with Gaussian statistics \cite[(3.33)]{mirlin2000statistics}, which accounts for the factor $1 + \1_{i = j}$ in \zcref{t:intensity,t:profile}. The law of \zcref{t:coarse} is also the eigenvector law of the critical one-dimensional random Schr\"odinger operator of Rifkind and Vir\'ag, by \zcref{r:coarse-rv}. Band matrices are not among the operators of their universality conjectures \cite[Conjectures 1.2--1.3]{rifkind2018eigenvectors}, and \zcref{t:coarse} adds them to this class. The eigenvalue processes of the two models differ. As $X \downarrow 0$, the process $\Xi_X$ tends to the sine process, in the sense of \zcref{p:endpoints}, while the eigenvalues of the weakly disordered Schr\"odinger operator are locally close to periodic \cite{kritchevski2012scaling}.

The limit $2 + 2X/3$ of \zcref{t:profile} agrees with the prediction of Fyodorov and Mirlin \cite{fyodorov1992analytical,fyodorov1994statistical}; its expansion to second order in $X$ is in \cite[(12), (17)]{fyodorov1995mesoscopic}. Its slope $2X/3$ for large $X$ agrees with \cite[(3.48)]{mirlin2000statistics}. The tail \eqref{e:intro-tail} of the mean profile, with the constant $\pi^{7/2}/32$, was obtained by Efetov and Larkin for the density correlator of a long wire \cite[(56)]{efetov1983kinetics}, and the whole curve $\bar p_\infty$ agrees with the curve of \cite[(75)--(76)]{ivanov2009correlations}, up to a factor two that comes from the normalization there (see \zcref{r:coarse-mean-physics}). The two-sided shape $p_\infty$ of \zcref{t:shape} is implicit in \cite[(3.30)--(3.34)]{mirlin2000statistics}, and its counterparts for random Schr\"odinger operators are described after \zcref{t:shape}. The distinction between coincident and adjacent sites in \zcref{t:intensity}, with the weight $B/(B - F)$ of the contact kernel, is the finite-wire counterpart of the local correlations of the infinite wire of Skvortsov and Ostrovsky \cite[(2), (7)--(8)]{skvortsov2007local}.

For the metallic corrections of \zcref{p:metallic}, the spatial term $2X G_{\mathrm N}(x, y)$ corresponds to the diffusion propagator of a wire with reflecting ends \cite[(2.22)--(2.23)]{mirlin2000statistics}, once the conductance in the normalization of \cite{mirlin2000statistics} is identified with $1/X$ (see \zcref{r:met-traces}). It appears in the first-order correlations of eigenfunction amplitudes \cite[(3.55), (3.74)]{mirlin2000statistics}, derived in \cite{fyodorov1995mesoscopic,blanter1997correlations}. The spectral term has the form of the correction of order $g_{\mathrm{KM}}^{-2}$ to the Wigner--Dyson two-point function that Kravtsov and Mirlin obtained for a metallic sample of dimensionless conductance $g_{\mathrm{KM}}$ \cite[(15)]{kravtsov1994level} (see also \cite[(2.25), (2.27)]{mirlin2000statistics}). The two corrections agree if $g_{\mathrm{KM}} = 1/(2\pi X)$. This identification is heuristic, and the agreement of the coefficients serves only to check consistency; we explain both in \zcref{r:met-km}.

The asymptotics $\Sigma_X^2(L) \sim \pi^{-3/2} \sqrt{XL}$ of \zcref{t:largecov}, including the constant, is the formula of Altshuler and Shklovskii \cite[(6)--(6a)]{altshuler1986repulsion} in dimension one and in the unitary class, since a window of $L$ spacings corresponds to the ratio $2\pi X L$ of energy to Thouless energy in their notation. They derived it by perturbation theory for a good metal, which corresponds to $X \ll 1$, and for windows that are large compared with the Thouless energy. In long wires, physics predicts the same law for energy differences much larger than the level spacing of a localization volume \cite[(2)]{micklitz2016spectral}, which in our units means windows of length much larger than $X$, and the formula \cite[(25)--(26)]{micklitz2016spectral} describes such windows at every value of the ratio of length to localization length. For band matrices, a rigorous version on mesoscopic energy scales, with the same seminorm $\int |k|^{1/2} \hat f(k) \overline{\hat g(k)} \dd k$, was proved by Erd\H{o}s and Knowles \cite[Theorem 2.3]{erdos2015altshulerI} (see also \cite{erdos2015altshulerII}); as they note, it also holds in the localized regime \cite[Section 1.6]{erdos2015altshulerI}.

For the eigenvalue process, physics predicts a family of limits indexed by $X$, which tends to the GUE statistics as $X \to 0$, by Efetov's computation of the zero mode \cite{efetov1983supersymmetry} (see \cite[(2.11)]{mirlin2000statistics}), and to Poisson statistics as $X \to \infty$. \zcref[S]{t:process} identifies this family through its sources, and \zcref{p:endpoints} confirms its two endpoints at the level of pair measures. For real symmetric block band matrices, the Poisson limit was proved directly in the localized regime in \cite{fan2026poisson}. The sigma-model expression for the correlations of eigenfunction amplitudes at two energies separates the contribution of single eigenfunctions as a singular part \cite[(3.4), (3.69)--(3.70)]{mirlin2000statistics}, which is the formal counterpart of the residues \eqref{e:intro-residue} and of the extraction in \zcref{s:extract}.

The structure factor $m_\infty$ of \zcref{t:crossover} and the stationary pairing $\mathcal I$ of \eqref{e:scales-pairing}, which satisfies $\mathcal I(a) = \int_\R e^{-2a|u|} m_\infty(u) \dd u$ by \zcref{l:scales-pairing}, were found in physics, for long unitary wires. The function $m_\infty$ is the spectral form factor of such a wire. Micklitz, M\"uller, and Altland computed it as the contrast of the forward-scattering peak \cite[(13)]{micklitz2014strong}, and Marinho and Micklitz as the form factor itself \cite[(19)]{marinho2018spectral}. The level correlation function of a long wire, whose transform is $\mathcal I$, was derived by Micklitz \cite[(8)--(9)]{micklitz2016spectral} and by Marinho and Micklitz \cite[(12)--(13)]{marinho2018spectral}. In our variables, the latter formula is $X^{-1} \Re \mathcal I(-\iu \pi s/X)$ at a separation of $s$ mean level spacings, by \eqref{e:scales-physics}, and \zcref{r:scales-physics} compares the conventions and normalizations of these works with each other and with ours. We did not find the number variance $V$ in these works. All of them use the exact zero mode of the singlet transfer operator of an infinite unitary wire, found by Skvortsov and Ostrovsky \cite[(14)]{skvortsov2007local}, which is also the stationary state in our proof in \zcref{s:scales}.

\subsection{Methods}\label{s:intro-sources}

We give an overview of the proofs in \zcref{s:overview}. Here we describe the sources of their ingredients.

The supersymmetric method was introduced by Efetov \cite{efetov1982supersymmetry,efetov1983supersymmetry} (see \cite{efetov1997supersymmetry,verbaarschot1985grassmann}). Accounts accessible to probabilists are \cite{spencer2012susy}, \cite[Chapter 11]{bauerschmidt2019introduction}, and \cite{wegner2016supermathematics}, and \zcref{s:grassmann} collects what we use. Supersymmetric transfer matrices were used early in rigorous work on one-dimensional random Schr\"odinger operators \cite{campanino1986supersymmetric,klein1986rigorous} (see \cite[Section 1]{disertori2021density}). For band matrices, the supersymmetric transfer-matrix formalism was suggested by Efetov and adapted at a heuristic level by Fyodorov and Mirlin \cite{fyodorov1994statistical} (see \cite[Section 1]{shcherbina2021universality}). Rigorous supersymmetric analyses of band matrices with the profile $J$ of \zcref{d:model} go back to the work of Disertori, Pinson, and Spencer on the density of states in three dimensions \cite{disertori2002density} (see also \cite{disertori2004density,disertori2017density}). In one dimension, a supersymmetric transfer operator was developed by M.~Shcherbina and T.~Shcherbina \cite{shcherbina2016transfer,shcherbina2017characteristic,shcherbina2018universality,shcherbina2020characteristic,shcherbina2021universality} (see the survey \cite{shcherbina2019transfer}) and by Disertori, Lohmann, and Sodin \cite{disertori2021density}, who used a supersymmetry that commutes with the transfer operator and attribute it to Constantinescu \cite{constantinescu1988supersymmetric}. Bao and Erd\H{o}s \cite{bao2017delocalization} combined supersymmetry with comparison arguments for block band matrices. Our exact representation of $T_N^{(k)}$ as a transfer chain in \zcref{s:rep} follows this line.

Our argument has three new steps, and \zcref{s:ov-strategy} explains what each of them replaces and where it enters. First, we control the whole chain at $W^2 \asymp N$, including its noncompact sector, at every source rank $k$ and with all coefficients in the Grassmann variables. At this bandwidth the chain converges to the heat semigroup of the sigma model over the whole interval $[0, 1]$. We prove this by averaging over the fast variables of the chain, which relax over about $W$ steps, much fewer than the $N$ steps of the chain, with a normalization that an identity coming from supersymmetry (a Ward identity) makes exact. In the earlier work \cite{shcherbina2019transfer}, the transfer operator is analyzed through its spectrum, above the threshold \cite{shcherbina2021universality} and, at the threshold, for the compact sector alone \cite{shcherbina2020characteristic}. Second, we attach both ends of the chain to its bulk up to errors $O(e^{-\gamma W})$, for some $\gamma > 0$. Third, we extract eigenvector moments from local densities, as described below.

For the noncompact sector, we use the hyperbolic Hubbard--Stratonovich transformation on the domain of Sch\"afer and Wegner \cite{schafer1980disordered}, which was made rigorous by Zirnbauer \cite[Section III.E, Lemma 3.2 and Theorem 3.3]{zirnbauer1996riemannian}. Other hyperbolic domains were treated by Fyodorov \cite{fyodorov2005hubbard}, by Fyodorov, Wei, and Zirnbauer \cite{fyodorov2008hyperbolic}, and by M\"uller-Hill and Zirnbauer \cite{mullerhill2011equivalence}, who also gave an explicit deformation of the Sch\"afer--Wegner domain into the Euclidean one \cite[Appendix B]{mullerhill2011equivalence}. These works concern a single matrix variable. The product contour for the chain, with all coefficients in the Grassmann variables, is constructed in \zcref{s:contour}.

The normalization of the sigma model and the evaluation of the rank-one observables rest on supersymmetric localization (see \zcref{s:grass-susy}), which goes back to Parisi and Sourlas \cite{parisi1979random}. Schwarz and Zaboronsky proved a general localization theorem on compact supermanifolds \cite[Theorem 1]{schwarz1997supersymmetry}. For the noncompact target space of our sigma model, we follow the argument of Disertori, Spencer, and Zirnbauer \cite[Appendix C]{disertori2010quasi} (see also \cite[Theorem 11.4.5]{bauerschmidt2019introduction}). On noncompact supermanifolds, changes of variables produce boundary terms, which were described by Rothstein \cite{rothstein1987integration} (see \cite[Section II.A]{zirnbauer1996riemannian}) and are known in physics as Efetov--Wegner terms \cite{kieburg2011efetov}. We work in global coordinates in which they do not arise; they reappear in the polar coordinates $(B, F)$ as the mass that the singlet diffusion of \zcref{r:intro-singlet} carries into the corner.

The extraction of eigenvector moments from the part of Poisson-smoothed local densities that comes from coincident eigenvalues (\zcref{s:extract}) has, to our knowledge, no rigorous antecedent; its counterpart in physics is described in \zcref{s:intro-physics}. Besides \zcref{t:interface} and elementary properties of Gaussian matrices, Part III uses two known results about $H$, the factorial moment bounds of Peled, Schenker, Shamis, and Sodin \cite[Theorem 3]{peled2019wegner} and the density of states of \cite[Theorem 1 and Remark 1.3]{disertori2021density}. The law of $\Xi_X$ is determined through its Laplace functionals \cite{daley2008introduction}, which the ratios in \eqref{e:intro-source-law} approximate by Fej\'er's theorem. The Brownian motion of \zcref{d:brownian} arises from the noncompact sector. On the scale where $B$ is large, the singlet diffusion moves in $B$ approximately as a geometric Brownian motion. The physics counterpart of this scaling is the transfer equation \cite[(3.22)--(3.23)]{mirlin2000statistics}, and our computation of the residues in \zcref{s:residue} is a rigorous version of it. The resulting computations use Dufresne's identity \cite{dufresne1990distribution}, in the form of \cite[Theorem 6.2]{matsumoto2005exponentialI}, the Kontorovich--Lebedev transform \cite[Section 10.43(v)]{dlmf}, and the Liouville operator $-\partial_y^2 + e^y + 1/4$, whose use in one-dimensional localization goes back to Kolokolov \cite{kolokolov1993method} (see \cite[(3.28)--(3.31)]{mirlin2000statistics}, and \cite{comtet1998exponential} for exponential functionals of Brownian motion in disordered systems). The function $2\sqrt q K_1(2\sqrt q)$ that describes each side of a long wire is the zero mode of \cite[(55)]{efetov1983kinetics}. In Part IV, the passage from the chain to the continuum semigroup uses the Trotter--Kurtz approximation theorem \cite{trotter1958approximation,kurtz1969extensions}, in the form of \cite[Chapter 1, Section 6]{ethier1986markov}, and the a priori bounds on partial chains use Hua's determinant inequality \cite{hua1955inequalities} (see \cite{paige2008hua}) and the Minami-type estimate of Graf and Vaghi \cite{graf2007remark}.

\subsection{Computations confirmed by computer}\label{s:intro-unverified}

The calculations used in the proofs are written out explicitly in the text. Supplementary scripts check selected identities numerically or symbolically. Each script is named where it is used. We list four of the longer ones. The first two belong to the rank-one radial reduction of \zcref{s:sigma-radial}. They are the radial density $d^{-2}$ of the invariant integral, which \zcref{l:sigma-polar-density} establishes, and the radial part of $\mathsf C_1$ in \zcref{l:sigma-radial-part}, which the proof of that lemma derives from the values of the Casimir operator in \zcref{l:sigma-casimir-values}. \zcref[S]{r:sigma-scripts} names the two scripts that confirm them, the first through the divergence identity \eqref{e:sigma-polar-divergence}. This reduction is used in all of Part III, through $K_X$, $\widetilde K_X$, and $I_X$, but not in the proof of \zcref{t:interface}. The third is the form \eqref{e:res-polar} of $\mathsf C_1$ in the polar chart, which \zcref{l:res-polar} proves from \zcref{l:res-stabilizer,l:res-entries}, and which \zcref{r:res-polar-check} confirms. The fourth is the second-order metallic coefficient $I^{[2]}(z) = (1 - e^{-2z})/45$ in \zcref{l:met-coefficients}, the sum of the three integrals over the time simplex in \eqref{e:met-second-terms}, which \zcref{r:met-script} confirms.

\subsection{Organization of the paper}\label{s:intro-organization}

\zcref[S]{s:overview} gives an overview of the proof and ends, in \zcref{s:ov-organization}, with a description of what each section of the four parts is for. \zcref[S]{s:tools} collects general tools, \zcref{s:contour} constructs the product contour, and \zcref{s:regularity} proves the construction of the semigroup of the sigma model stated in \zcref{p:sigma-regular}.

\section{Notation}\label{s:intro-notation}

We collect the conventions used throughout the paper. Symbols that are specific to one section are defined there.

Unless stated otherwise, $c, C > 0$ denote constants that may depend on the fixed parameters of the statement at hand, such as the source rank $k$, the damping $\eps$, the margin $\delta$, the bounds $X_\pm$ in \zcref{d:intro-critical}, and test functions, but not on $N$, $W$, or $E$. Their values may change from line to line. When the dependence on a parameter matters, we indicate it by a subscript, as in $C_k$. Let $f$ be a complex-valued and $g$ a nonnegative function of the parameters at hand. We write $f = O(g)$ if there exists a constant $C$ such that $|f| \le Cg$, and $f = o(g)$ if $|f|/g \to 0$ in the limit under consideration; a subscript, as in $O_k(g)$, indicates a parameter on which the constant depends. For positive $f$ and $g$, we write $f \asymp g$ if there exist constants $c, C > 0$ such that $cg \le f \le Cg$, and $f \sim g$ if $f/g \to 1$. In heuristic statements and in descriptions of other works, $f \ll g$ and $g \gg f$ mean that $f$ is much smaller than $g$, in the sense of the work cited. Limits of quantities that depend on $N$ are taken in the critical limit at $X$, or uniformly in the critical regime, as specified in \zcref{d:intro-critical}.

We set $[n] = \{1, \dots, n\}$ for positive integers $n$. As in \zcref{s:intro-results}, $\iu$ denotes the imaginary unit, and we reserve the letters $i$ and $j$ for sites in $[N]$. We let $\Id$ denote the identity matrix or operator, $\1_A$ the indicator function of a set $A$, and $\delta_{ij}$ the Kronecker delta. For $a \in \C$, we let $I_a$ and $K_a$ denote the modified Bessel functions of order $a$; the functions $I_X$, $K_X$, and $\widetilde K_X$, indexed by the parameter $X$, are the kernels of \zcref{d:kernels}.

The matrix $H$ is defined on a probability space with probability measure $\Prob$ and expectation $\E$. We use the same symbols for the law and the expectation of the Brownian motion $\beta$ of \zcref{d:brownian}, which is defined on a separate probability space, and we let $\E_{\mathbb Q_X}$ denote the expectation under $\mathbb Q_X$. For complex random variables $\zeta_1$ and $\zeta_2$ with finite second moments, we set $\Cov(\zeta_1, \zeta_2) = \E[(\zeta_1 - \E \zeta_1) \overline{(\zeta_2 - \E \zeta_2)}]$ and $\Var(\zeta_1) = \Cov(\zeta_1, \zeta_1)$.

Recall that $G(z) = (H - z)^{-1}$ denotes the resolvent of $H$ for $z \in \C$ with $\Im z \neq 0$, with entries $G_{ij}(z)$. By the spectral theorem,
\[
G_{ij}(z) = \sum_{\alpha = 1}^N \frac{u_\alpha(i) \overline{u_\alpha(j)}}{\lambda_\alpha - z},
\]
and in particular $\Im G_{ii}(z) > 0$ for $\Im z > 0$. The supersymmetric variables of Parts II and IV have their own sign conventions, and \zcref{l:rep-dictionary} relates them to $G$. Grassmann algebras, supermatrices, the supertrace $\Str$, and the superdeterminant $\Sdet$ are introduced in \zcref{s:grassmann}.

Recall the Poisson kernel $P_\eps(v) = \eps/(\pi(v^2 + \eps^2))$ and the Fourier transform $\hat f(k) = \int_\R e^{-2\pi \iu k s} f(s) \dd s$ of \zcref{s:intro-results}. We have $P_{\eps_1} * P_{\eps_2} = P_{\eps_1 + \eps_2}$. Point processes on $\R$ carry the vague topology, and spaces of probability measures carry the topology of weak convergence.

Supplementary scripts accompany the paper. They check identities numerically or in exact arithmetic, among them the computations listed in \zcref{s:intro-unverified}, and each script is named where it is used. The scripts are distributed with the source files of this paper, in the folder \texttt{anc}, and the paths of the scripts named in the text, such as \texttt{polar/check\_polar\_density.py}, are relative to that folder.

The sections of the four parts are numbered consecutively, and the appendices are lettered. Theorems, propositions, lemmas, corollaries, definitions, and remarks share one counter within each section, and equations are numbered within each section. A reference such as \zcref{t:interface}(b) refers to part (b) of a numbered statement.