\chapter{The infinite-temperature crossover}\label{sec:infinite} In this section, we let the rescaled inverse temperature $c=c_N$ tend to zero and prove \cref{thm:intro-bounded} and the first case of \cref{thm:intro-main}. The section has a deterministic part and a part about the GOE. In the deterministic part, \cref{wf-deterministic-theorem}, $G$ is a fixed symmetric matrix of bounded norm, and $A=cG$ with $c$ small. We show that the time-$t$ law $\nu_NP^A_t$ is within $O(c^2)$ of the Bingham comparison law $\mu_t$ uniformly in time, that the information $\cI_N(t)$ of the covariance evolution (\cref{def:model-information}) decays at rate $4+O(c)$, and that $D_N(t)$ is close to $2\Phi(\sqrt{\cI_N(t)}/2)-1$ uniformly in $t\ge0$. The distance is determined by the single function $\cI_N$. If $\cI_N(0)\to\infty$, there is cutoff at the time $\tau_N$ at which $\cI_N(\tau_N)=1$, with a window of order one. For the GOE, we show in \cref{wf-goe-classification} that $\cI_N(0)$ is asymptotic to $J_N/2$, where $J_N=Nc_N^2$. If $J_N$ stays bounded, the distance relaxes over a bounded time and there is no cutoff. If $J_N\to\infty$, there is cutoff at $\tau_N$. In \cref{wf-explicit-clocks}, we determine $\tau_N$ up to $o(1)$. In the regime $J_N\to\infty$, let $h_N=(c_N/4)\log J_N$ for all sufficiently large $N$. When $h_N$ is bounded, the information at times of order $\log J_N$ is governed by the orthogonal compression of multiplication by the eigenvalues to the subspace of vectors with mean zero, and its semicircle limit is computed with Chebyshev polynomials. The resulting center involves the modified Bessel function $I_1$. When $h_N\to\infty$, the relative errors are concentrated near the spectral edge. The computation of \cref{sec:hot-fixed} holds uniformly for small $c$, and it yields the center of \cref{hp-fixed-profile} evaluated at $c=c_N$. \Cref{tab:inf-regimes} summarizes these results. We use the covariance evolution of \cref{sec:heat} (\cref{ht-covariance-system}), \cref{ht-small-field-geometry,ht-small-field-direct-comparison}, and the results of \cref{sec:model-profiles}. From \cref{sec:hot-fixed}, we use the uniform form of the likelihood limit (\cref{lem:hotf-lan-uniform}), which rests on \cref{gp-pinned-lan}. About the GOE, the proof of \cref{wf-goe-classification} uses only the norm of $W_N$ and the traces of $W_N$ and $W_N^2$ (\cref{lem:inf-goe-elementary}). The proof of \cref{wf-explicit-clocks} uses in addition the rigidity estimate of \cref{sp-goe-rigidity}, through \cref{lem:hotf-grid}, the computations for the semicircle law (\cref{prop:hotf-edge,lem:hotf-transfer}), and the evolution on vectors with mean zero of \cref{sec:inf-bulk}. \paragraph{Notation.} We use the notation of \cref{sec:hot-fixed}: the function $\mathcal G(I)=2\Phi(\sqrt I/2)-1$ of \eqref{eq:hotf-gaussian-tv}, the semicircle quantile function $\gamma$, and the identification of vectors with step functions on $(0,1)$. For $f,f'\in\R^N$, we write $\langle f\rangle=N^{-1}\sum_if_i$, $\langle f,f'\rangle_N=N^{-1}\sum_if_if'_i$, and $\|f\|_{2,N}=\langle f,f\rangle_N^{1/2}$. Let $G$ be a real symmetric $N\times N$ matrix with eigenvalues $\lambda_1\ge\cdots\ge\lambda_N$, let $c\ge0$, and let $A=cG$. We write $D_N(t)=\|\nu_NP^A_t-\pi_A\|_{\TV}$ and define $t_N(\eps)$ by \eqref{model-distance} with this $D_N$. For $A=c_NW_N$, these are the quantities of \cref{sec:model}. We use the Lagrange multiplier $z$, the precisions $p_i$, the variances $q_i$, and the numbers $d$ and $s$ of \cref{def:model-multiplier}, and the covariance evolution of \cref{sec:heat}: the variances $v_i(t)$, the moving multiplier $z(t)$, the Bingham comparison law $\mu_t=\mu_{v(t)}$, the relative errors $e_i(t)=v_i(t)/q_i-1$, and the information $\cI_N(t)=\frac12\sum_ie_i(t)^2$. We write $e(t)=(e_i(t))_i$ and $\bar e(t)=\langle e(t)\rangle$. \paragraph{Adding a multiple of the identity.} Replacing $A$ by $A+\tau I$ replaces $z$ by $z+\tau$ and leaves the precisions $p_i$ and the variances $q_i$ unchanged. In \eqref{ht-covariance-ode}, it multiplies $m(t)$ by $e^{2\tau t}$, so that the variances $v_i(t)$ do not change, and neither do $e_i(t)$, $\cI_N(t)$ and $\mu_t=\mu_{v(t)}$. The quantities $D_N$ and $t_N$ do not change either, by \eqref{eq:model-shift}. In the proofs, we may therefore replace $G$ by $G-N^{-1}(\Tr G)I$, which has trace zero and norm at most $2\|G\|$. \section{Statements}\label{sec:inf-statements} \begin{theorem}[Small inverse temperature: comparison and information]\label{wf-deterministic-theorem} Let $M>0$. There exist $c_0\in(0,1)$, $N_0$ and $C$, depending only on $M$, such that the following holds for every $N\ge N_0$, every $c\in[0,c_0]$, and every real symmetric $N\times N$ matrix $G$ with $\|G\|\le M$, where $A=cG$. \begin{enumerate} \item For all $t\ge0$, \begin{equation}\label{wf-all-horizon-bridge} \|\nu_NP^A_t-\mu_t\|_{\TV}\le Cc^2\min(t,1). \end{equation} \item For all $t\ge0$, we have $\max_i|e_i(t)|\le Cce^{-t}$. If $\cI_N(0)=0$, then $\cI_N(t)=0$ for all $t\ge0$. If $\cI_N(0)>0$, then $\cI_N(t)>0$ for all $t\ge0$, and \begin{equation}\label{wf-information-decay} -4-Cc\le\frac d{dt}\log\cI_N(t)\le-4+Cc . \end{equation} \item For every $\kappa>0$, there exist $c_\kappa\in(0,c_0]$ and $N_\kappa\ge N_0$, depending only on $\kappa$ and $M$, such that if $c\le c_\kappa$ and $N\ge N_\kappa$, then \begin{equation}\label{wf-all-horizon-information} \sup_{t\ge0}\big|D_N(t)-\mathcal G(\cI_N(t))\big|\le\kappa . \end{equation} \item If $\cI_N(0)>1$, then there exists a unique $\tau_N>0$ with $\cI_N(\tau_N)=1$. It satisfies \begin{equation}\label{wf-universal-clock} \frac{\log\cI_N(0)}{4+Cc}\le\tau_N\le\frac{\log\cI_N(0)}{4-Cc},\qquad \big|\log\cI_N(\tau_N+u)+4u\big|\le Cc|u|\quad(u\ge-\tau_N). \end{equation} \item Let $c_N\to0$, and for each $N$ let $G_N$ be a real symmetric $N\times N$ matrix with $\|G_N\|\le M$, such that the information of $A=c_NG_N$ satisfies $\cI_N(0)\to\infty$. Then $\tau_N/\log\cI_N(0)\to1/4$, and for every compact set $K\subset\R$ and every compact interval $I\subset(0,1)$, \[ \sup_{u\in K}\big|D_N(\tau_N+u)-\Psi(u)\big|\to0,\qquad \sup_{\eps\in I}\big|t_N(\eps)-\tau_N-u_\eps\big|\to0 . \] \end{enumerate} \end{theorem} In the last part, the window is of order one while $\tau_N\to\infty$, so there is cutoff for every sequence along which the initial information diverges. The relative bound $\tau_N=\frac14\log\cI_N(0)(1+O(c_N))$ given by \eqref{wf-universal-clock} does not determine $\tau_N$ up to $o(1)$; this is done for the GOE in \cref{wf-explicit-clocks}. \begin{theorem}[Relaxation and cutoff near infinite temperature for the GOE]\label{wf-goe-classification} Let $c_N\ge0$ be deterministic with $c_N\to0$, let $J_N=Nc_N^2$, and consider the model with $A=c_NW_N$. All limits below are in probability. \begin{enumerate} \item Suppose that $\sqrt N\,c_N\to a\in[0,\infty)$. Then \begin{equation}\label{wf-bounded-information-profile} \sup_{t\ge0}\Big|D_N(t)-\Big(2\Phi\Big(\frac{ae^{-2t}}{2\sqrt2}\Big)-1\Big)\Big|\to0, \end{equation} and for every $\eps\in(0,1)$, \begin{equation}\label{wf-bounded-information-time} t_N(\eps)\to\max\Big\{\frac12\log\frac a{2\sqrt2\,\Phi^{-1}((1+\eps)/2)},\,0\Big\}, \end{equation} where the limit is zero if $a=0$. If $\eps>2\Phi(a/(2\sqrt2))-1$, then $\PP(t_N(\eps)=0)\to1$. If $c_N=0$, then $\pi_{N,0}=\nu_N$ and $D_N(t)=0$ for all $t\ge0$. \item Suppose that $J_N\to\infty$. Define \begin{equation}\label{eq:inf-first-crossing} \tau_N=\begin{cases}\inf\{t\ge0:\cI_N(t)\le1\}&\text{if }\cI_N(0)>1,\\ 1&\text{if }\cI_N(0)\le1.\end{cases} \end{equation} Then \begin{equation}\label{eq:inf-crossing-probability} \PP\big(\cI_N(0)>1\text{ and }\tau_N\text{ is the only solution of }\cI_N(t)=1\text{ in }[0,\infty)\big)\longrightarrow1 . \end{equation} Moreover, $\tau_N/\log J_N\to1/4$, and for every compact set $K\subset\R$ and every compact interval $I\subset(0,1)$, \begin{equation}\label{wf-goe-cutoff} \sup_{\substack{u\in K\\ \tau_N+u\ge0}}\big|D_N(\tau_N+u)-\Psi(u)\big|\to0,\qquad \sup_{\eps\in I}\big|t_N(\eps)-\tau_N-u_\eps\big|\to0 . \end{equation} In particular, $D_N$ has cutoff profile $\Psi$ with center $\tau_N$ and window $1$, and there is cutoff. \end{enumerate} \end{theorem} The time $\tau_N$ in \eqref{eq:inf-first-crossing} is defined for every realization of $W_N$, and $\cI_N(\tau_N)=1$ whenever $\cI_N(0)>1$. Indeed, for every real symmetric matrix $A$, the variances $v_i(t)$ are continuous in $t$, and they converge to $q_i$ as $t\to\infty$ by \eqref{ht-covariance-variance-range}, because $d>0$ (\cref{def:model-multiplier}). Then $\cI_N$ is continuous on $[0,\infty)$, and $\cI_N(t)\to0$ as $t\to\infty$. If $\cI_N(0)>1$, the set $\{t\ge0:\cI_N(t)\le1\}$ is nonempty and closed, and its infimum $\tau_N$ belongs to it. Then $\tau_N>0$ and $\cI_N>1$ on $[0,\tau_N)$, and continuity gives $\cI_N(\tau_N)\ge1$. We conclude that $\cI_N(\tau_N)=1$, and that $\tau_N$ is the smallest solution of $\cI_N(t)=1$ on the event $\{\cI_N(0)>1\}$. The theorem asserts that it is the only solution with probability tending to one; we do not claim uniqueness on all of $\{\cI_N(0)>1\}$. Under the assumptions of the fourth part of \cref{wf-deterministic-theorem}, the solution is unique, and $\tau_N$ is the time given there. The time $\tau_N$ is a Borel function of $W_N$; this is needed for the limits in probability in the second part. To see this, fix $N$ and $c_N$, and write $\cI_N(W,t)$ for the information at time $t$ when $W$ is a real symmetric $N\times N$ matrix and $A=c_NW$. This function is jointly continuous in $(W,t)$. Indeed, the ordered eigenvalues of $W$ are continuous functions of $W$ \cite[Corollary~A.6]{AndersonGuionnetZeitouni2010}. The Lagrange multiplier is the unique solution $z>c_N\lambda_1$ of \eqref{eq:model-multiplier}, whose left side is continuous in $(z,W)$ and strictly decreasing in $z$ there; bracketing the solution between $z-\epsilon$ and $z+\epsilon$ shows that it depends continuously on $W$. Finally, the solution of \eqref{ht-covariance-ode} depends continuously on the eigenvalues of $A$ and on $t$, and its normalization $\sum_jm_j(t)$ is positive by \cref{ht-covariance-system}. We conclude that $q_i$, $v_i(t)$, and $\cI_N(t)$ are continuous in $(W,t)$. For $r\ge0$, the function $W\mapsto\min_{0\le s\le r}\cI_N(W,s)$ is Borel, because by continuity it is the infimum over a countable dense subset of $[0,r]$. On $\{\cI_N(0)>1\}$, we have $\tau_N\le r$ if and only if $\min_{0\le s\le r}\cI_N(s)\le1$, and on $\{\cI_N(0)\le1\}$, we have $\tau_N\le r$ if and only if $r\ge1$. This proves that $\tau_N$ is Borel. The event in \eqref{eq:inf-crossing-probability} is the intersection of $\{\cI_N(0)>1\}$ with the event that $\cI_N(t)=1$ has exactly one solution, and it is Borel as well. For $m,k\ge1$, the sets of matrices $W$ for which $\cI_N(W,t)=1$ for some $t\in[0,m]$, or for two times $s,t\in[0,m]$ with $|s-t|\ge1/k$, are closed, by joint continuity and the compactness of $[0,m]$. The set of $W$ for which $\cI_N(W,t)=1$ has exactly one solution in $[0,\infty)$ is the union over $m$ of the sets of the first kind, minus the union over $m$ and $k$ of the sets of the second kind. In \eqref{wf-goe-cutoff}, the supremum is over $u\in K$ with $\tau_N+u\ge0$, and an empty supremum is zero, as in \cref{def:model-cutoff}. Since $\tau_N\to\infty$ in probability, the restriction is void with probability tending to one. The same convention applies to the other suprema of this kind in this section. At equality $\eps=2\Phi(a/(2\sqrt2))-1$ with $a>0$, the first part gives $t_N(\eps)\to0$ in probability, but not that $t_N(\eps)=0$ with probability tending to one. Along a sequence with $J_N\to J\in[0,\infty]$, the first part applies if $J<\infty$, with $a=\sqrt J$, and the second if $J=\infty$. A cutoff profile with a diverging center therefore appears exactly when $J_N\to\infty$. For $h\in\R$, let \begin{equation}\label{eq:inf-bessel} \mathcal B(h)=\frac12\int_{-2}^2e^{4h\lambda}\rho_{\mathrm{sc}}(\lambda)\,d\lambda=\frac{I_1(8h)}{8h},\qquad \mathcal B(0)=\frac12, \end{equation} where $I_1$ is the modified Bessel function of the first kind (\cref{rem:inf-bessel}). The proofs use only the integral. \begin{theorem}[Explicit centers]\label{wf-explicit-clocks} In the setting of the second part of \cref{wf-goe-classification}, let $h_N=(c_N/4)\log J_N$ for all $N$ large enough that $J_N>0$. \begin{enumerate} \item Suppose that $(h_N)$ is bounded, and let \begin{equation}\label{wf-bulk-center} T^{\mathrm{bulk}}_N=\frac14\big[\log J_N+\log\mathcal B(h_N)\big]. \end{equation} Then $\tau_N-T^{\mathrm{bulk}}_N\to0$ in probability, and \eqref{wf-goe-cutoff} holds with $T^{\mathrm{bulk}}_N$ in place of $\tau_N$. If moreover $h_N\to0$, then $T^{\mathrm{bulk}}_N-\frac14\log(J_N/2)\to0$. \item Suppose that $h_N\to\infty$. Let $g_N=(1-c_N)^2$ and let $T_N(c_N)$ be the center \eqref{hp-fixed-center} of \cref{hp-fixed-profile} evaluated at $c=c_N$, that is, \begin{equation}\label{wf-edge-center} T_N(c_N)=\frac{\log N-\frac32\log\log N+\log A_{c_N}}{4g_N},\qquad A_c=\frac{\sqrt c\,(1-c)^3}{4\sqrt\pi\,(1+c)^2}. \end{equation} Then $\tau_N-T_N(c_N)\to0$ in probability, and for every compact set $K\subset\R$ and every compact interval $I\subset(0,1)$, in probability, \begin{equation}\label{wf-edge-profile} \sup_{u\in K}\big|D_N(T_N(c_N)+u/g_N)-\Psi(u)\big|\to0,\qquad \sup_{\eps\in I}\big|g_N\big(t_N(\eps)-T_N(c_N)\big)-u_\eps\big|\to0 . \end{equation} \end{enumerate} \end{theorem} In the second part, no condition such as $c_N^2\log N\to0$ is imposed, and $g_N$ cannot be replaced by $1$ in the center. Indeed, $g_N^{-1}-1\ge2c_N$ and $c_N\log N\ge4h_N\to\infty$, so replacing $4g_N$ by $4$ in the denominator of \eqref{wf-edge-center} changes the center by an amount that tends to infinity. Sequences along which $h_N$ has several limit points in $[0,\infty]$ are still covered by \cref{wf-goe-classification}, with the center $\tau_N$. \begin{table}[ht] \centering \caption{The regimes of \cref{wf-goe-classification,wf-explicit-clocks}, with $J_N=Nc_N^2$. The parameter $h_N=(c_N/4)\log J_N$ is used only in the regime $J_N\to\infty$, for $N$ large.}\label{tab:inf-regimes} \renewcommand{\arraystretch}{1.3} \begin{tabular}{@{}llll@{}} \toprule Regime & Center & Window & Statement\\ \midrule $\sqrt N\,c_N\to a<\infty$ & no cutoff & --- & \cref{wf-goe-classification}, part 1\\ $J_N\to\infty$ & $\tau_N$ as defined in \eqref{eq:inf-first-crossing} & $1$ & \cref{wf-goe-classification}, part 2\\ $J_N\to\infty$, $h_N$ bounded & $\frac14[\log J_N+\log\mathcal B(h_N)]$ & $1$ & \cref{wf-explicit-clocks}, part 1\\ $J_N\to\infty$, $h_N\to\infty$ & $T_N(c_N)$ in \eqref{wf-edge-center} & $(1-c_N)^{-2}$ & \cref{wf-explicit-clocks}, part 2\\ \bottomrule \end{tabular} \end{table} \begin{remark}[The Bessel function]\label{rem:inf-bessel} By \cite[(10.32.2)]{OlverLozierBoisvertClark2010} with $\nu=1$, we have $I_1(x)=\pi^{-1}x\int_{-1}^1\sqrt{1-y^2}\,e^{xy}\,dy$. The substitution $\lambda=2y$ yields $\int e^{s\lambda}\rho_{\mathrm{sc}}(\lambda)\,d\lambda=I_1(2s)/s$ for $s\ne0$, and with $s=4h$ we obtain the second expression in \eqref{eq:inf-bessel}. We also note that $\mathcal B(h)\ge\frac12e^{4h\int\lambda\rho_{\mathrm{sc}}}=\frac12$ by Jensen's inequality, and that $\mathcal B(h)\le\frac12e^{8|h|}$ and $|\mathcal B'(h)|\le4e^{8|h|}$, since $|\lambda|\le2$ on the support of $\rho_{\mathrm{sc}}$. \end{remark} \section{Small inverse temperature}\label{sec:inf-small} The following lemmas concern a matrix $A=cG$ with $\Tr G=0$, $\|G\|\le M$, and $cM\le1/8$. Then $|\lambda_i|\le M$ for all $i$, and $\sum_i\lambda_i=0$. We write $\lambda=(\lambda_i)_i$ for the vector of eigenvalues. \begin{lemma}[The Lagrange multiplier and the covariance evolution]\label{lem:inf-small} Let $\Tr G=0$, $\|G\|\le M$, and $cM\le1/8$. Then \begin{equation}\label{wf-small-field-saddle} |z-1|\le8c^2M^2,\qquad |q_i-1-c\lambda_i|\le8c^2M^2,\qquad |p_i-1|\le2cM,\qquad |q_i-1|\le2cM, \end{equation} and $3/4\le d\le1\le s\le5/4$. For all $t\ge0$ and all $i$, \begin{equation}\label{wf-small-field-proxy} |v_i(t)-q_i|\le4cMe^{-2dt},\qquad |e_i(t)|\le5cMe^{-3t/2} . \end{equation} \end{lemma} \begin{proof} Since $N^{-1}\Tr A=0$ and $\|A\|\le cM\le1/8$, \cref{ht-small-field-geometry} with $\epsilon=\|A\|$ implies that $|z-1|\le8c^2M^2$, $|q_i-1-c\lambda_i|\le8c^2M^2$, and $|d-1|,|s-1|\le2cM\le1/4$. Then $p_i=z-c\lambda_i$ satisfies $|p_i-1|\le8c^2M^2+cM\le2cM$, and similarly $|q_i-1|\le cM+8c^2M^2\le2cM$. We have $|v_i(t)-q_i|\le(s-d)d^{-2}e^{-2dt}$ by \eqref{ht-covariance-variance-range}, and the first bound in \eqref{wf-small-field-proxy} follows from $s-d=c(\lambda_1-\lambda_N)\le2cM$ and $d^{-2}\le16/9$. Finally, $|e_i(t)|=p_i|v_i(t)-q_i|\le\frac54\cdot4cMe^{-2dt}$, and $2d\ge3/2$. \end{proof} \begin{lemma}[Relative errors and information]\label{lem:inf-relative} Let $A$ be a real symmetric matrix. For all $t\ge0$, \begin{equation}\label{wf-relative-error-ode} e_i'(t)=-2p_ie_i(t)+2\bar e(t)\big(1+e_i(t)\big),\qquad \sum_iq_ie_i(t)=0,\qquad z(t)=z-\bar e(t), \end{equation} and \begin{equation}\label{eq:inf-info-derivative} \cI_N'(t)=-2\sum_ip_ie_i(t)^2+2N\bar e(t)^2+2\bar e(t)\sum_ie_i(t)^2 . \end{equation} If moreover $A=cG$ with $\Tr G=0$, $\|G\|\le M$, and $cM\le1/8$, then $|\bar e(t)|\le2cM\|e(t)\|_{2,N}$ and $|\cI_N'(t)+4\cI_N(t)|\le16cM\,\cI_N(t)$ for all $t\ge0$. \end{lemma} \begin{proof} We have $\sum_iq_ie_i=\sum_iv_i-\sum_iq_i=N-N=0$. Recall that $z(t)=1+N^{-1}\sum_i\alpha_iv_i(t)$ with $\alpha_i=z-p_i$, and that $p_iv_i=1+e_i$. Using $\sum_iv_i=N$, we find $z(t)=1+z-N^{-1}\sum_i(1+e_i)=z-\bar e$. Recall also from \eqref{ht-covariance-identities} that $v_i'=2+2(\alpha_i-z(t))v_i$, where now $\alpha_i-z(t)=-p_i+\bar e$. Multiplying by $p_i$, we have \[ e_i'=p_iv_i'=2p_i-2p_i(1+e_i)+2\bar e(1+e_i)=-2p_ie_i+2\bar e(1+e_i), \] which proves \eqref{wf-relative-error-ode}. Equation \eqref{eq:inf-info-derivative} follows from $\cI_N'=\sum_ie_ie_i'$ and $\sum_ie_i=N\bar e$. Now let $A=cG$ as in the second statement. The second identity in \eqref{wf-relative-error-ode} implies that $\bar e=N^{-1}\sum_i(1-q_i)e_i$. We obtain $|\bar e|\le2cM\|e\|_{2,N}$ from \eqref{wf-small-field-saddle} and the Cauchy--Schwarz inequality. We also have $|\bar e|\le2cM\max_i|e_i|\le10c^2M^2$ by \eqref{wf-small-field-proxy}. Recall that $\|e\|_{2,N}^2=2\cI_N/N$. We bound the terms of \eqref{eq:inf-info-derivative} as follows: \[ \begin{gathered} \Big|2\sum_ip_ie_i^2-4\cI_N\Big|\le2\max_i|p_i-1|\sum_ie_i^2\le8cM\,\cI_N,\\ 2N\bar e^2\le16c^2M^2\cI_N,\qquad 2|\bar e|\sum_ie_i^2\le40c^2M^2\cI_N . \end{gathered} \] As $56c^2M^2\le7cM$, we conclude that $|\cI_N'+4\cI_N|\le15cM\,\cI_N$. \end{proof} At bounded times, the relative errors are close to those of a linear evolution with rate two. \begin{lemma}[Relative errors at bounded times]\label{lem:inf-bounded-time} Let $\Tr G=0$, $\|G\|\le M$, and $cM\le1/8$. For all $t\ge0$, \[ \big\|e(t)+ce^{-2t}\lambda\big\|_{2,N}\le24c^2M^2(1+t)e^{-t}. \] \end{lemma} \begin{proof} We first note that \cref{lem:inf-relative} implies that $\cI_N(s)\le\cI_N(0)e^{-(4-16cM)s}$, that is, $\|e(s)\|_{2,N}\le\|e(0)\|_{2,N}e^{-(2-8cM)s}$. We have $e(0)=(p_i-1)_i$, because $v_i(0)=1$, and $\|e(0)\|_{2,N}\le2cM$ by \eqref{wf-small-field-saddle}. We write \eqref{wf-relative-error-ode} as $e'=-2e+F$ with $F_i=-2(p_i-1)e_i+2\bar e(1+e_i)$. Using \cref{lem:inf-small,lem:inf-relative} and $1+5cM\le2$, we have $\|F\|_{2,N}\le(4cM+8cM)\|e\|_{2,N}=12cM\|e\|_{2,N}$. By variation of constants, \[ e(t)=e^{-2t}e(0)+\int_0^te^{-2(t-s)}F(s)\,ds . \] We have $e(0)=(z-1)\ind-c\lambda$, and therefore $\|e^{-2t}(e(0)+c\lambda)\|_{2,N}=e^{-2t}|z-1|\le8c^2M^2e^{-2t}$. The integral has norm at most \[ \int_0^te^{-2(t-s)}\,12cM\cdot2cMe^{-(2-8cM)s}\,ds=24c^2M^2e^{-2t}\int_0^te^{8cMs}\,ds\le24c^2M^2\,t\,e^{-(2-8cM)t}. \] Adding the two bounds and using $8cM\le1$, we obtain the claim. \end{proof} \section{Proof of the deterministic theorem}\label{sec:inf-proof-det} \begin{proof}[Proof of \cref{wf-deterministic-theorem}] As explained at the beginning of the section, we may assume that $\Tr G=0$ and $\|G\|\le M'$, with $M'=2M$. Let $\epsilon_0$, $N_0'$ and $C_1$ be the universal constants of \cref{ht-small-field-direct-comparison}. We set $C=\max\{C_1M'^2,16M',5M'\}$, $c_0=\min\{1/2,1/(8M'),\epsilon_0/M',2/C\}$, and $N_0=N_0'$. Let $c\le c_0$ and $N\ge N_0$. For the first part, we have $\|A-N^{-1}(\Tr A)I\|=\|A\|\le cM'\le\epsilon_0$, and \eqref{wf-all-horizon-bridge} is the first bound of \cref{ht-small-field-direct-comparison}. For the second part, the bound on $e_i(t)$ is \eqref{wf-small-field-proxy}. We have $|\cI_N'|\le6\cI_N$ by \cref{lem:inf-relative}. If $\cI_N(0)=0$, we obtain $\cI_N\equiv0$ from Gronwall's inequality. If $\cI_N(0)>0$, the same inequality gives $\cI_N(t)\ge\cI_N(0)e^{-6t}>0$, and dividing $|\cI_N'+4\cI_N|\le16cM'\cI_N$ by $\cI_N$ yields \eqref{wf-information-decay}. For the third part, fix $\kappa>0$, and let $\delta>0$ be given by \cref{lem:hotf-lan-uniform} with $\kappa/2$ in place of $\kappa$. Let $c_\kappa\le c_0$ satisfy $C_1c_\kappa^2M'^2\le\kappa/2$ and $5c_\kappa M'\le\delta$, and $N_\kappa\ge N_0$ with $(4/3)(2N_\kappa)^{-1/2}\le\delta$. Let $c\le c_\kappa$, $N\ge N_\kappa$ and $t\ge0$. Using \eqref{wf-all-horizon-bridge} and the triangle inequality, we have \[ \big|D_N(t)-\|\mu_t-\pi_A\|_{\TV}\big|\le\|\nu_NP^A_t-\mu_t\|_{\TV}\le\kappa/2 . \] We have $\mu_t=\mu_{v(t)}$ and $\pi_A=\mu_q$, with $v_i(t)=q_i(1+e_i(t))$ and $\max_i|e_i(t)|\le5cM'\le\delta$. Note that $\eta_q\le(4/3)(2N)^{-1/2}\le\delta$, because $q_i\le1/d\le4/3$ and $\sum_iq_i^2\ge N$. Then \cref{lem:hotf-lan-uniform} implies that $|\|\mu_t-\pi_A\|_{\TV}-\mathcal G(\cI_N(t))|\le\kappa/2$, and \eqref{wf-all-horizon-information} follows. For the fourth part, \eqref{wf-information-decay} and $Cc\le2$, which holds since $c\le c_0$, show that $\cI_N$ is continuous and strictly decreasing, with $\cI_N(t)\le\cI_N(0)e^{-2t}\to0$. If $\cI_N(0)>1$, it follows that there exists a unique $\tau_N>0$ with $\cI_N(\tau_N)=1$. We integrate \eqref{wf-information-decay} from $0$ to $\tau_N$ to obtain the first bound in \eqref{wf-universal-clock}, and from $\tau_N$ to $\tau_N+u$ to obtain the second, using $\log\cI_N(\tau_N)=0$. For the fifth part, \eqref{wf-universal-clock} implies that $\tau_N/\log\cI_N(0)\to1/4$, and in particular that $\tau_N\to\infty$. Fix $\kappa>0$ and a compact set $K$, and let $U=\sup\bigl(\{0\}\cup\{|u|:u\in K\}\bigr)$. For $N$ large, we have $c_N\le c_\kappa$, $N\ge N_\kappa$, and $\tau_N\ge U$. For $u\in K$, we have $\cI_N(\tau_N+u)=e^{-4u+r}$ with $|r|\le Cc_NU$ by \eqref{wf-universal-clock}, and then \[ \big|\cI_N(\tau_N+u)^{1/2}-e^{-2u}\big|\le e^{2U}\big(e^{Cc_NU/2}-1\big). \] Combining this with \eqref{wf-all-horizon-information} and \eqref{eq:hotf-G-lipschitz}, we find $|D_N(\tau_N+u)-\Psi(u)|\le\kappa+e^{2U}(e^{Cc_NU/2}-1)$ for all $u\in K$. As the parameter $\kappa$ is arbitrary, the first limit follows. The function $D_N$ is nonincreasing and tends to zero as $t\to\infty$, by the arguments in the proofs of \cref{lem:model-tv} and \cref{cor:model-profile-cutoff}, which apply to any symmetric matrix $A$. The second limit then follows from the deterministic version of \cref{hp-monotone-profile-transfer}, with $b_n=\tau_N$, $w_n=1$ and $V=\Psi$, and from $\Psi^{-1}(\eps)=u_\eps$ (\cref{cot-profile}). \end{proof} \section{Proof of the classification for the GOE}\label{sec:inf-proof-goe} Throughout the rest of this section, $\lambda_1\ge\cdots\ge\lambda_N$ are the eigenvalues of $W_N$. Let $\bar\lambda=N^{-1}\sum_i\lambda_i=N^{-1}\Tr W_N$, let $\tilde\lambda_i=\lambda_i-\bar\lambda$, and let $\tilde G=W_N-\bar\lambda I$, which has eigenvalues $\tilde\lambda_i$. Let $\sigma_N\ge0$ be defined by \[ \sigma_N^2=\|\tilde\lambda\|_{2,N}^2=\int_0^1(\ell_N-\bar\lambda)^2\,d\omega . \] The proof of \cref{wf-goe-classification} uses the following elementary facts about the GOE, which concern only the norm of $W_N$ and the traces of $W_N$ and $W_N^2$. The rigidity estimate of \cref{sp-goe-rigidity} is used only for the explicit centers, from \cref{sec:inf-bulk} on. \begin{lemma}[Norm and traces of the GOE]\label{lem:inf-goe-elementary} Let $m_{2,N}=N^{-1}\Tr(W_N^2)$. For every $N\ge2$, \[ \begin{gathered} \E\bar\lambda=0,\qquad\E\bar\lambda^2=\frac2{N^2},\\ \E(m_{2,N}-1)^2\le\frac{18}{N^2} . \end{gathered} \] Moreover, $\PP(\|W_N\|\le3)\to1$. In particular, with $b_N=N^{-1/4}$, the event \begin{equation}\label{eq:inf-elementary-event} Q_N=\big\{\|W_N\|\le3,\ |\bar\lambda|\le b_N,\ |m_{2,N}-1|\le b_N\big\} \end{equation} satisfies $\PP(Q_N)\to1$. On $Q_N$, we have \begin{equation}\label{eq:inf-centered-moment} \Tr\tilde G=0,\qquad\|\tilde G\|\le4,\qquad\sigma_N^2=m_{2,N}-\bar\lambda^2,\qquad|\sigma_N-1|\le2b_N . \end{equation} \end{lemma} \begin{proof} For a centered Gaussian variable $Y$ with variance $v$, we have $\E Y^2=v$, $\E Y^4=3v^2$, and $\Var(Y^2)=2v^2$. The trace $\Tr W_N=\sum_iW_{ii}$ is a sum of $N$ independent Gaussian variables with mean zero and variance $2/N$. This gives $\E\bar\lambda=0$ and $\E\bar\lambda^2=N^{-2}\cdot N\cdot2/N=2/N^2$. Next, \[ m_{2,N}=\frac1N\sum_{i=1}^NW_{ii}^2+\frac2N\sum_{i5/2)\to0$. Let $Z_1,\dots,Z_N$ be standard Gaussian variables, independent of each other and of $X_N$, and let $Y_N=\diag(Z_i/\sqrt N)$. Then $X_N+Y_N$ has the law of $W_N$, since its diagonal entries have variance $2/N$. By a union bound and Markov's inequality, \[ \PP\big(\|Y_N\|>1/2\big)\le\sum_{i=1}^N\PP\big(Z_i^4>N^2/16\big)\le N\cdot\frac{16\cdot3}{N^2}=\frac{48}N . \] We conclude that $\PP(\|W_N\|>3)\le\PP(\|X_N\|>5/2)+48/N\to0$. By Chebyshev's inequality and the moment bounds above, we have $\PP(|\bar\lambda|>b_N)\le2N^{-2}b_N^{-2}=2N^{-3/2}$ and $\PP(|m_{2,N}-1|>b_N)\le18N^{-2}b_N^{-2}=18N^{-3/2}$. Together with the bound on the norm, this gives $\PP(Q_N^c)\le\PP(\|W_N\|>3)+20N^{-3/2}\to0$. Finally, on $Q_N$, we have $\Tr\tilde G=0$ and $\|\tilde G\|\le\|W_N\|+|\bar\lambda|\le3+b_N\le4$. Since $N^{-1}\sum_i\lambda_i^2=m_{2,N}$, expanding the square gives $\sigma_N^2=m_{2,N}-\bar\lambda^2$. Then $|\sigma_N^2-1|\le|m_{2,N}-1|+\bar\lambda^2\le b_N+b_N^2\le2b_N$. As $\sigma_N\ge0$, we have $|\sigma_N-1|=|\sigma_N^2-1|/(\sigma_N+1)\le|\sigma_N^2-1|$, which completes the proof. \end{proof} We now work on the event $Q_N$, on which \cref{wf-deterministic-theorem} applies with $G=W_N$ and $M=3$. We write $C$, $c_0$, $c_\kappa$, and $N_\kappa$ for its constants with $M=3$. As explained at the beginning of the section, we may compute $e_i(t)$, $\cI_N(t)$, and $\tau_N$ with the matrix $c_N\tilde G$ in place of $c_NW_N$. By \eqref{eq:inf-centered-moment}, on $Q_N$, \cref{lem:inf-small,lem:inf-relative,lem:inf-bounded-time} apply to $c_N\tilde G$ with $M=4$ once $32c_N\le1$, which holds for all $N$ larger than a threshold depending only on the sequence $(c_N)$. Let $\tilde z$ be the Lagrange multiplier of $c_N\tilde G$. \paragraph{The initial information.} We have $e_i(0)=\tilde z-1-c_N\tilde\lambda_i$, because $v_i(0)=1$. Expanding the square and using $\sum_i\tilde\lambda_i=0$, we find \begin{equation}\label{eq:inf-initial-information} \cI_N(0)=\frac{c_N^2}2\sum_i\tilde\lambda_i^2+\frac N2(\tilde z-1)^2=\frac{J_N}2\sigma_N^2+\frac N2(\tilde z-1)^2 . \end{equation} By \eqref{wf-small-field-saddle} with $M=4$, we have $|\tilde z-1|\le128c_N^2$, and hence \begin{equation}\label{eq:inf-initial-bounds} \frac{J_N}2\sigma_N^2\le\cI_N(0)\le\frac{J_N}2\big(\sigma_N^2+2^{14}c_N^2\big). \end{equation} These formulas also hold when $c_N=0$. \begin{proof}[Proof of \cref{wf-goe-classification}] If $c_N=0$, then $L=\Delta_{S_N}$ and $\pi_{N,0}=\nu_N$ is invariant, so $D_N\equiv0$. We now prove the two parts. \paragraph{The first part.} Let $\sqrt N\,c_N\to a\in[0,\infty)$. On $Q_N$ and for $N$ large, we use \cref{lem:inf-bounded-time} for $c_N\tilde G$ with $M=4$, together with $(1+t)e^{-t}\le1$, and find that for all $t\ge0$ \[ \Big|\sqrt{2\cI_N(t)}-\sqrt N\,c_N\sigma_Ne^{-2t}\Big|\le\sqrt N\,\big\|e(t)+c_Ne^{-2t}\tilde\lambda\big\|_{2,N}\le384\sqrt N\,c_N^2 . \] Here we used $\sqrt{2\cI_N(t)}=\sqrt N\|e(t)\|_{2,N}$ and $\|c_Ne^{-2t}\tilde\lambda\|_{2,N}=c_N\sigma_Ne^{-2t}$. Using also $|\sigma_N-1|\le2b_N$ from \eqref{eq:inf-centered-moment}, we find that, on $Q_N$ and for $N$ large, \[ \sup_{t\ge0}\big|\sqrt{2\cI_N(t)}-ae^{-2t}\big|\le R_N,\qquad R_N=384\sqrt N\,c_N^2+\big|\sqrt N\,c_N-a\big|+2b_N\sqrt N\,c_N . \] The number $R_N$ does not depend on $W_N$, and it tends to zero, because $\sqrt Nc_N$ is bounded and $\sqrt Nc_N^2=(\sqrt Nc_N)c_N\to0$. Let $D_\infty(t)=\mathcal G(a^2e^{-4t}/2)=2\Phi(ae^{-2t}/(2\sqrt2))-1$. By \eqref{eq:hotf-G-lipschitz}, \[ \sup_{t\ge0}\big|\mathcal G(\cI_N(t))-D_\infty(t)\big|\le2^{-1/2}\sup_{t\ge0}\big|\sqrt{2\cI_N(t)}-ae^{-2t}\big|\le2^{-1/2}R_N . \] Fix $\kappa>0$. For $N\ge N_\kappa$ with $c_N\le c_\kappa$, we have $\sup_t|D_N(t)-\mathcal G(\cI_N(t))|\le\kappa$ on $Q_N$, by \eqref{wf-all-horizon-information}. Let $\delta_N=\sup_{t\ge0}|D_N(t)-D_\infty(t)|$. Since $D_N$ is continuous in $t$ (see the proof of \cref{cor:model-profile-cutoff}), the supremum may be taken over rational $t$, and $\delta_N$ is a random variable. Then $\delta_N\le\kappa+2^{-1/2}R_N$ on $Q_N$, for all $N$ larger than a threshold depending only on $\kappa$ and the sequence $(c_N)$. Since $R_N\to0$, we have $\PP(\delta_N>2\kappa)\le\PP(Q_N^c)$ for $N$ large, and $\PP(Q_N^c)\to0$ by \cref{lem:inf-goe-elementary}. As $\kappa$ is arbitrary, $\delta_N\to0$ in probability, which is \eqref{wf-bounded-information-profile}. We now turn to the mixing times. Fix $\eps\in(0,1)$, and let $s_*$ be the right side of \eqref{wf-bounded-information-time}. If $a>0$, then $D_\infty$ is continuous and strictly decreasing, with $D_\infty(0)=2\Phi(a/(2\sqrt2))-1$ and $D_\infty(t)\to0$, and $D_\infty(s)=\eps$ exactly when $s=\frac12\log(a/(2\sqrt2\Phi^{-1}((1+\eps)/2)))$. Recall that $D_N$ is nonincreasing (\cref{lem:model-tv}). We consider three cases. \begin{itemize} \item If $\eps0$ and $s_*>0$. For $h\in(0,s_*)$, we have $D_\infty(s_*-h)>\eps>D_\infty(s_*+h)$. When $\delta_N$ is smaller than both differences, we have $D_N(s_*+h)<\eps$, which gives $t_N(\eps)\le s_*+h$, and $D_N(t)\ge D_N(s_*-h)>\eps$ for all $t\le s_*-h$, which gives $t_N(\eps)\ge s_*-h$. \item If $\eps>D_\infty(0)$, then $s_*=0$. When $\delta_N<\eps-D_\infty(0)$, we have $D_N(0)<\eps$ and $t_N(\eps)=0$. This also covers the case $a=0$, in which $D_\infty\equiv0$. \item If $\eps=D_\infty(0)$ and $a>0$, then $s_*=0$. For $h>0$, we have $D_\infty(h)<\eps$, and when $\delta_N<\eps-D_\infty(h)$, we have $t_N(\eps)\le h$. \end{itemize} In each case, the convergence $\delta_N\to0$ in probability implies that $t_N(\eps)\to s_*$ in probability, and in the second case $\PP(t_N(\eps)=0)\to1$. \paragraph{The second part.} Let $J_N\to\infty$. Then $c_N>0$ for $N$ large. On $Q_N$, \eqref{eq:inf-initial-bounds} and \eqref{eq:inf-centered-moment} imply that \[ \cI_N(0)\ge\frac{J_N}2(1-2b_N)^2,\qquad\Big|\frac{2\cI_N(0)}{J_N}-1\Big|\le b_N+b_N^2+2^{14}c_N^2 . \] Since the right sides do not depend on $W_N$, we have $\cI_N(0)\to\infty$ and $\log\cI_N(0)-\log(J_N/2)\to0$ uniformly on $Q_N$. Let $N$ be so large that $\cI_N(0)>1$ on $Q_N$, $c_N\le c_0$, and $N\ge N_0$. For every matrix in $Q_N$, the fourth part of \cref{wf-deterministic-theorem} then implies that the equation $\cI_N(t)=1$ has exactly one solution in $[0,\infty)$, which is positive and satisfies \eqref{wf-universal-clock}. By the discussion after \cref{wf-goe-classification}, the time $\tau_N$ of \eqref{eq:inf-first-crossing} is a solution of this equation, and it is the only one. We have shown that $Q_N$ is contained in the event in \eqref{eq:inf-crossing-probability} for $N$ large. As $\PP(Q_N)\to1$, this proves \eqref{eq:inf-crossing-probability}. By \eqref{wf-universal-clock}, the ratio $\tau_N/\log\cI_N(0)$ lies between $(4+Cc_N)^{-1}$ and $(4-Cc_N)^{-1}$ on $Q_N$, for $N$ large. Since $Cc_N\le2$ for $N$ large, we have \[ \Big|\frac{\tau_N}{\log\cI_N(0)}-\frac14\Big|\le Cc_N\qquad\text{on }Q_N . \] Together with $\log\cI_N(0)-\log(J_N/2)\to0$ uniformly on $Q_N$ and $\log J_N\to\infty$, this gives $\sup_{Q_N}|\tau_N/\log J_N-1/4|\to0$. As $\PP(Q_N)\to1$, we have $\tau_N/\log J_N\to1/4$ in probability, and in particular $\tau_N\to\infty$ in probability. Next, fix a compact set $K\subset\R$ and $\kappa>0$, and let $U=\sup\bigl(\{0\}\cup\{|u|:u\in K\}\bigr)$. For $N$ large, on $Q_N$, we have $\tau_N\ge U$, $c_N\le c_\kappa$, and $N\ge N_\kappa$. For $u\in K$, we have $\cI_N(\tau_N+u)=e^{-4u+r}$ with $|r|\le Cc_NU$, by \eqref{wf-universal-clock}, and then $|\cI_N(\tau_N+u)^{1/2}-e^{-2u}|\le e^{2U}(e^{Cc_NU/2}-1)$. Using \eqref{wf-all-horizon-information} and \eqref{eq:hotf-G-lipschitz} as in the proof of the fifth part of \cref{wf-deterministic-theorem}, we find that, on $Q_N$ and for $N$ large, \[ \sup_{u\in K}\big|D_N(\tau_N+u)-\Psi(u)\big|\le\kappa+e^{2U}\big(e^{Cc_NU/2}-1\big) . \] Here the restriction $\tau_N+u\ge0$ is void, because $\tau_N\ge U$. The left side is a random variable: $\tau_N$ is Borel, $D_N$ is jointly measurable and continuous in $t$ (see the proof of \cref{cor:model-profile-cutoff}), and the supremum may be taken over a countable dense subset of $K$. As $\kappa$ is arbitrary and $\PP(Q_N)\to1$, the first limit in \eqref{wf-goe-cutoff} holds in probability. The center $\tau_N$ is a positive random variable, by the discussion after \cref{wf-goe-classification}, and $1/\tau_N\to0$ in probability. Then $D_N$ has cutoff profile $\Psi$ with center $\tau_N$ and window $1$ in the sense of \cref{def:model-cutoff}, and \cref{cor:model-profile-cutoff} gives the second limit in \eqref{wf-goe-cutoff} and cutoff. \end{proof} \section{The center when \texorpdfstring{$h_N$}{h\_N} is bounded}\label{sec:inf-bulk} For the explicit centers, we use the approximation of the eigenvalues by the semicircle quantiles, in addition to \cref{lem:inf-goe-elementary}. In this section and the next, fix $C_{\mathrm{gr}}\ge1$ and work on the event $F_{C_{\mathrm{gr}}}$ of \cref{sp-goe-rigidity}. Put $E_N=(C_{\mathrm{gr}}+7)N^{-5/12}$. For every $\eta\in(0,1)$, the constant $C_{\mathrm{gr}}$ can be chosen so that $\PP(F_{C_{\mathrm{gr}}})\ge1-\eta$ for all sufficiently large $N$. We keep $C_{\mathrm{gr}}$ fixed until the deterministic estimates are complete, and then let $\eta\downarrow0$. Recall the notation $\bar\lambda$ and $\tilde\lambda_i$ of \cref{sec:inf-proof-goe}. By \cref{lem:hotf-grid}, we have, on $F_{C_{\mathrm{gr}}}$ and for $N$ large, \begin{equation}\label{eq:inf-rigidity} \|\ell_N-\gamma\|_\infty\le E_N,\qquad \max_i|\lambda_i|\le3,\qquad |\bar\lambda|\le E_N,\qquad \max_i|\tilde\lambda_i|\le4 . \end{equation} The event $F_{C_{\mathrm{gr}}}$ is contained in the event $Q_N$ of \cref{lem:inf-goe-elementary} for $N$ large. Indeed, $\int_0^1\gamma^2\,d\omega=\int\lambda^2\rho_{\mathrm{sc}}(\lambda)\,d\lambda=1$ and $m_{2,N}=\int_0^1\ell_N^2\,d\omega$. Then \eqref{eq:inf-rigidity} implies \[ |m_{2,N}-1|\le\int_0^1|\ell_N-\gamma|\,|\ell_N+\gamma|\,d\omega\le(4+E_N)E_N\le5E_N . \] For $N$ large, we have $5E_N\le b_N$, and also $|\bar\lambda|\le E_N\le b_N$ and $\|W_N\|=\max_i|\lambda_i|\le3$. Every conclusion of the proof of \cref{wf-goe-classification} that holds on $Q_N$ for $N$ large then also holds on $F_{C_{\mathrm{gr}}}$ for $N$ large. When $c\,t$ is bounded, the relative errors $e(t)$ are of order $ce^{-2t}$, and their profile is obtained from a linear evolution on vectors with mean zero. Let $\Lambda=\diag(\lambda_i)$ act on $\R^N$, let $\Pi f=f-\langle f\rangle\ind$ be the orthogonal projection onto the subspace $\mathcal H_0=\{f:\langle f\rangle=0\}$, and let $\mathcal T=\Pi\Lambda$, restricted to $\mathcal H_0$. For $f,f'\in\mathcal H_0$, we have $\langle f,\mathcal Tf'\rangle_N=\langle f,\Lambda f'\rangle_N=\langle\Lambda f,f'\rangle_N$, so that $\mathcal T$ is self-adjoint on $\mathcal H_0$, with $\|\mathcal T\|\le\max_i|\lambda_i|$. When $\Tr G=0$, the vector $\lambda$ lies in $\mathcal H_0$. The operator $\mathcal T$ is applied only to vectors in $\mathcal H_0$; equations in $\R^N$ are first projected by $\Pi$. \begin{lemma}[Evolution on vectors with mean zero]\label{lem:inf-compressed} Let $\Tr G=0$, $\|G\|\le M$, and $00$ and every $t\ge0$ with $ct\le H$, \begin{equation}\label{wf-compressed-evolution} \big\|\hat e(t)+e^{2ct\mathcal T}\lambda\big\|_{2,N}\le C_{H,M}\,c,\qquad C_{H,M}=\big(54HM^3+3M^2\big)e^{9MH}. \end{equation} \end{lemma} \begin{proof} We write $\hat e=\hat e^0+\langle\hat e\rangle\ind$ with $\hat e^0=\Pi\hat e$, derive an equation for $\hat e^0$, and compare it with $\partial_tf=2c\mathcal Tf$. \paragraph{Step 1: the equation for $\hat e$.} Since $p_i=1+(z-1)-c\lambda_i$, \eqref{wf-relative-error-ode} reads $e_i'=-2e_i-2(z-1)e_i+2c\lambda_ie_i+2\bar e+2\bar ee_i$. Multiplying by $c^{-1}e^{2t}$ and using $c^{-1}e^{2t}\bar e=\langle\hat e\rangle$, we find \[ \hat e'=2c\Lambda\hat e+2\langle\hat e\rangle\ind+R_1,\qquad R_1=\big(2\bar e-2(z-1)\big)\hat e . \] Recall that $|z-1|\le8c^2M^2$ and $|\bar e|\le10c^2M^2$ (\cref{lem:inf-small,lem:inf-relative}), which give $\|R_1\|_{2,N}\le36c^2M^2\|\hat e\|_{2,N}$. Next, $\sum_iq_i\hat e_i=0$ by \eqref{wf-relative-error-ode}, so that $\langle\hat e\rangle=\langle(\ind-q)\hat e\rangle$, where $q=(q_i)_i$. Using \eqref{wf-small-field-saddle}, we find $|\langle\hat e\rangle|\le2cM\|\hat e\|_{2,N}\le\frac14\|\hat e\|_{2,N}$. Together with $\|\hat e\|_{2,N}\le\|\hat e^0\|_{2,N}+|\langle\hat e\rangle|$, this gives \begin{equation}\label{eq:inf-mean-part} |\langle\hat e\rangle|\le2cM\|\hat e\|_{2,N},\qquad \|\hat e\|_{2,N}\le\frac43\|\hat e^0\|_{2,N}. \end{equation} \paragraph{Step 2: the projected equation.} Applying $\Pi$, and using $\Pi\ind=0$ and $\Pi\Lambda\ind=\Pi\lambda=\lambda$, we obtain \[ (\hat e^0)'=2c\mathcal T\hat e^0+R_2,\qquad R_2=2c\langle\hat e\rangle\lambda+\Pi R_1,\qquad \hat e^0(0)=c^{-1}\Pi\big((z-1)\ind-c\lambda\big)=-\lambda . \] Using Step~1 and $\|\lambda\|_{2,N}\le M$, we find $\|R_2\|_{2,N}\le40c^2M^2\|\hat e\|_{2,N}\le54c^2M^2\|\hat e^0\|_{2,N}$. Then $\frac d{dt}\|\hat e^0\|_{2,N}^2\le(4cM+108c^2M^2)\|\hat e^0\|_{2,N}^2\le18cM\|\hat e^0\|_{2,N}^2$, where we used $cM\le1/8$, and therefore \[ \|\hat e^0(t)\|_{2,N}\le Me^{9cMt}. \] \paragraph{Step 3: comparison.} By variation of constants on $\mathcal H_0$, \[ \hat e^0(t)+e^{2ct\mathcal T}\lambda=\int_0^te^{2c(t-s)\mathcal T}R_2(s)\,ds . \] Since $\|e^{2c(t-s)\mathcal T}\|\le e^{2cM(t-s)}$, the right side has norm at most \[ \int_0^te^{2cM(t-s)}54c^2M^3e^{9cMs}\,ds\le54c^2M^3te^{9cMt}\le54cM^3He^{9MH}\qquad(ct\le H). \] Finally, $\|\hat e-\hat e^0\|_{2,N}=|\langle\hat e\rangle|\le\frac83cM\|\hat e^0\|_{2,N}\le3cM^2e^{9MH}$ by \eqref{eq:inf-mean-part}. Adding the two bounds proves \eqref{wf-compressed-evolution}. \end{proof} The limit of $\mathcal T$ for the semicircle law is identified with Chebyshev polynomials. \begin{lemma}[The compressed multiplication for the semicircle law]\label{lem:inf-chebyshev} Let $L^2_0(\rho_{\mathrm{sc}})=\{f\in L^2(\rho_{\mathrm{sc}}):\int f\rho_{\mathrm{sc}}=0\}$, and let $\mathcal T_{\mathrm{sc}}f=\lambda f-(\int\lambda f\rho_{\mathrm{sc}})\ind$ for $f\in L^2_0(\rho_{\mathrm{sc}})$. Then $\mathcal T_{\mathrm{sc}}$ is a bounded self-adjoint operator on $L^2_0(\rho_{\mathrm{sc}})$, and for every $h\in\R$, \[ \big\|e^{2h\mathcal T_{\mathrm{sc}}}\lambda\big\|_{L^2(\rho_{\mathrm{sc}})}^2=\int e^{4h\lambda}\rho_{\mathrm{sc}}(\lambda)\,d\lambda=2\mathcal B(h). \] \end{lemma} \begin{proof} Self-adjointness follows as for $\mathcal T$. Let $U_n$ be the Chebyshev polynomials of the second kind, and let $u_n(\lambda)=U_n(\lambda/2)$. The $U_n$ are orthogonal on $(-1,1)$ with weight $(1-x^2)^{1/2}$ and $\int_{-1}^1U_n^2(1-x^2)^{1/2}dx=\pi/2$, and they satisfy $U_0=1$, $U_1(x)=2x$ and $U_{n+1}(x)=2xU_n(x)-U_{n-1}(x)$ \cite[Table~18.3.1 and (18.9.1), Table~18.9.1]{OlverLozierBoisvertClark2010}. Substituting $\lambda=2x$, we find that $(u_n)_{n\ge0}$ is orthonormal in $L^2(\rho_{\mathrm{sc}})$, with \[ u_0=1,\qquad u_1=\lambda,\qquad \lambda u_0=u_1,\qquad \lambda u_n=u_{n+1}+u_{n-1}\quad(n\ge1). \] Polynomials are dense in $L^2(\rho_{\mathrm{sc}})$, so $(u_n)_{n\ge0}$ is an orthonormal basis, and $(u_n)_{n\ge1}$ is an orthonormal basis of $L^2_0(\rho_{\mathrm{sc}})$. In these bases, $\mathcal T_{\mathrm{sc}}u_1=u_2$ and $\mathcal T_{\mathrm{sc}}u_n=u_{n+1}+u_{n-1}$ for $n\ge2$, because the projection removes only the multiple of $u_0$. Let $S:L^2(\rho_{\mathrm{sc}})\to L^2_0(\rho_{\mathrm{sc}})$ be the unitary map with $Su_n=u_{n+1}$. Comparing the two recursions, we find $\mathcal T_{\mathrm{sc}}S=S\Lambda_{\mathrm{sc}}$, where $\Lambda_{\mathrm{sc}}$ is multiplication by $\lambda$ on $L^2(\rho_{\mathrm{sc}})$. Hence $e^{4h\mathcal T_{\mathrm{sc}}}=Se^{4h\Lambda_{\mathrm{sc}}}S^{-1}$, and, since $\lambda=u_1=Su_0$, \[ \big\|e^{2h\mathcal T_{\mathrm{sc}}}\lambda\big\|^2=\langle u_1,e^{4h\mathcal T_{\mathrm{sc}}}u_1\rangle=\langle u_0,e^{4h\Lambda_{\mathrm{sc}}}u_0\rangle=\int e^{4h\lambda}\rho_{\mathrm{sc}}(\lambda)\,d\lambda . \] In the first step, we used that $\mathcal T_{\mathrm{sc}}$ is self-adjoint. \end{proof} For the GOE, we compare $\mathcal T$ with $\mathcal T_{\mathrm{sc}}$ on the common space $L^2(0,1)$. We write $\mathcal T_N$ for the operator $\mathcal T$ built from the centered eigenvalues $\tilde\lambda_i$. \begin{lemma}[The information when $c\,t$ is bounded]\label{lem:inf-bulk} Let $H>0$. On $F_{C_{\mathrm{gr}}}$, for $N$ large, all $c\in(0,1/32]$ and all $h\in[0,H]$, \begin{equation}\label{eq:inf-bulk-operator} \Big|\big\|e^{2h\mathcal T_N}\tilde\lambda\big\|_{2,N}-\sqrt{2\mathcal B(h)}\Big|\le2(2+4H)e^{8H}E_N . \end{equation} Moreover, there exists $C_H'$ depending only on $H$ such that, on $F_{C_{\mathrm{gr}}}$, for $N$ large, all $c\in(0,1/32]$ and all $t\ge0$ with $ct\le H$, the information of $A=cW_N$ satisfies \begin{equation}\label{wf-bulk-information} \Big|\frac{2e^{4t}}{Nc^2}\,\cI_N(t)-2\mathcal B(ct)\Big|\le C_H'\,(c+E_N). \end{equation} \end{lemma} \begin{proof} Let $\hat\Pi$ be the orthogonal projection onto functions with mean zero in $L^2(0,1)$, and let $\mathcal T_N'=\hat\Pi\Lambda_N\hat\Pi$ and $\mathcal T'=\hat\Pi\Lambda_\gamma\hat\Pi$, where $\Lambda_N$ and $\Lambda_\gamma$ are multiplication by $\ell_N-\bar\lambda$ and by $\gamma$. These are self-adjoint operators on $L^2(0,1)$ of norm at most $4$ and $2$, respectively, by \eqref{eq:inf-rigidity}. The identification of $(\R^N,\langle\cdot,\cdot\rangle_N)$ with step functions is an isometry that commutes with $\Pi$ and $\Lambda$, so that $\|e^{2h\mathcal T_N}\tilde\lambda\|_{2,N}=\|e^{2h\mathcal T_N'}(\ell_N-\bar\lambda)\|_{L^2(0,1)}$. Similarly, the map $F\mapsto F\circ\gamma$ is an isometry from $L^2(\rho_{\mathrm{sc}})$ into $L^2(0,1)$, which maps $L^2_0(\rho_{\mathrm{sc}})$ into functions with mean zero and intertwines $\mathcal T_{\mathrm{sc}}$ with $\mathcal T'$. Using \cref{lem:inf-chebyshev}, we find $\|e^{2h\mathcal T'}\gamma\|_{L^2(0,1)}=\sqrt{2\mathcal B(h)}$. Next, \eqref{eq:inf-rigidity} implies that $\|\mathcal T'_N-\mathcal T'\|\le\|\ell_N-\bar\lambda-\gamma\|_\infty\le2E_N$. Using the identity $e^X-e^Y=\int_0^1e^{sX}(X-Y)e^{(1-s)Y}ds$ for bounded operators, we find $\|e^{2h\mathcal T_N'}-e^{2h\mathcal T'}\|\le4hE_Ne^{8h}$. Using also $\|\ell_N-\bar\lambda-\gamma\|_{L^2(0,1)}\le2E_N$ and $\|\gamma\|_{L^2(0,1)}=1$, we obtain \[ \Big|\big\|e^{2h\mathcal T_N'}(\ell_N-\bar\lambda)\big\|-\big\|e^{2h\mathcal T'}\gamma\big\|\Big|\le (2+4h)e^{8h}E_N\le2(2+4H)e^{8H}E_N, \] which proves \eqref{eq:inf-bulk-operator}. For \eqref{wf-bulk-information}, we work with $c\tilde G$, $\tilde G=W_N-\bar\lambda I$, which has the same information, and we apply \cref{lem:inf-compressed} with $M=4$. Since $2\cI_N(t)=N\|e(t)\|_{2,N}^2=Nc^2e^{-4t}\|\hat e(t)\|_{2,N}^2$, the left side of \eqref{wf-bulk-information} equals $|\|\hat e(t)\|_{2,N}^2-2\mathcal B(ct)|$. Combining \eqref{wf-compressed-evolution} with \eqref{eq:inf-bulk-operator} for $h=ct$, we have $|\|\hat e(t)\|_{2,N}-\sqrt{2\mathcal B(ct)}|\le C_{H,4}c+2(2+4H)e^{8H}E_N$. This proves \eqref{wf-bulk-information}, because $\sqrt{2\mathcal B(ct)}\le e^{4H}$. \end{proof} \section{Proof of the explicit centers}\label{sec:inf-proof-explicit} \begin{proof}[Proof of \cref{wf-explicit-clocks}] Fix $\eta\in(0,1)$ and choose $C_{\mathrm{gr}}$ as in \cref{sec:inf-bulk}, so that $\PP(F_{C_{\mathrm{gr}}})\ge1-\eta$ for all sufficiently large $N$. We work on $F_{C_{\mathrm{gr}}}$, with the notation of \cref{sec:inf-proof-goe,sec:inf-bulk}. Since $F_{C_{\mathrm{gr}}}\subset Q_N$ for $N$ large, the proof of \cref{wf-goe-classification} shows that, on $F_{C_{\mathrm{gr}}}$ and for $N$ large, $\cI_N(0)>1$ and $\tau_N$ is the unique solution of $\cI_N(\tau_N)=1$. The function $\log\cI_N$ is decreasing with derivative at most $-(4-Cc_N)$, by \eqref{wf-information-decay}, and it vanishes at $\tau_N$. This implies that, for every $s\ge0$, \begin{equation}\label{eq:inf-root-bracket} |s-\tau_N|\le\frac{|\log\cI_N(s)|}{4-Cc_N} . \end{equation} In both regimes, we show that the information at the proposed center plus $u$ (or plus $u/g_N$) tends to $e^{-4u}$, uniformly for $u$ in compact sets. \paragraph{Bounded $h_N$.} Let $H=\sup_Nh_N+1$, fix a compact set $K\subset\R$, and let $U=\sup\bigl(\{0\}\cup\{|u|:u\in K\}\bigr)$. For $u\in K$, let $t=T^{\mathrm{bulk}}_N+u$. Using $\frac12\le\mathcal B(h)\le\frac12e^{8h}$ (\cref{rem:inf-bessel}), we have \[ |c_Nt-h_N|\le\frac{c_N}4|\log\mathcal B(h_N)|+c_NU\le c_N(2H+1+U)\to0 . \] For $N$ large, we have $0\le c_Nt\le H$. Note that $J_Ne^{-4t}=e^{-4u}/\mathcal B(h_N)$ by \eqref{wf-bulk-center}. Then \eqref{wf-bulk-information} gives \[ \cI_N(t)=\frac{J_Ne^{-4t}}2\Big(2\mathcal B(c_Nt)+O_H(c_N+E_N)\Big)=e^{-4u}\,\frac{\mathcal B(c_Nt)}{\mathcal B(h_N)}+O_{H,U}(c_N+E_N). \] We also have $|\mathcal B(c_Nt)-\mathcal B(h_N)|\le4e^{8H}|c_Nt-h_N|\to0$ and $\mathcal B(h_N)\ge1/2$, by \cref{rem:inf-bessel}. We conclude that $\sup_{u\in K}|\cI_N(T^{\mathrm{bulk}}_N+u)-e^{-4u}|\to0$ on $F_{C_{\mathrm{gr}}}$. With $u=0$, \eqref{eq:inf-root-bracket} gives $|T^{\mathrm{bulk}}_N-\tau_N|\to0$ on $F_{C_{\mathrm{gr}}}$, which proves the convergence in probability after letting $\eta\downarrow0$. Using \eqref{wf-all-horizon-information} and \eqref{eq:hotf-G-lipschitz}, we also find that $\sup_{u\in K}|D_N(T^{\mathrm{bulk}}_N+u)-\Psi(u)|\to0$ in probability. Because $T^{\mathrm{bulk}}_N\to\infty$, \cref{cor:model-profile-cutoff} then gives the second limit in \eqref{wf-goe-cutoff} with $T^{\mathrm{bulk}}_N$ in place of $\tau_N$. Finally, if $h_N\to0$, then $\mathcal B(h_N)\to\mathcal B(0)=1/2$, and $T^{\mathrm{bulk}}_N-\frac14\log(J_N/2)=\frac14\log(2\mathcal B(h_N))\to0$. \paragraph{Diverging $h_N$.} Let $h_N\to\infty$. For $N$ large, $c_N\le1/2$, and since $J_N\le N$, we have $c_N\log N\ge c_N\log J_N=4h_N\to\infty$. For $N$ large, it follows that $c_N\ge1/\log N$ and $|\log c_N|\le\log\log N$. Recalling that $\log A_c=\frac12\log c+3\log(1-c)-2\log(1+c)-\log(4\sqrt\pi)$, we find $|\log A_{c_N}|\le\frac12\log\log N+C_0$ with a universal $C_0$. Fix a compact set $K$, and for $u\in K$ let $t=t_N(u)=T_N(c_N)+u/g_N$. By \eqref{wf-edge-center}, \[ 4g_Nt=\log N-\frac32\log\log N+\log A_{c_N}+4u,\qquad Ne^{-4g_Nt}=\frac{(\log N)^{3/2}e^{-4u}}{A_{c_N}} . \] The first identity gives $4g_Nt/\log N\to1$ uniformly in $u\in K$. As $g_N\in[1/4,1]$, for $N$ large we have $t\le2\log N$ and $c_Nt\ge c_N\log N/5\to\infty$. We now use \cref{lem:hotf-transfer,prop:hotf-edge} with $c_1=1/2$. On $F_{C_{\mathrm{gr}}}$, for $N$ large, we have $\|\ell_N-\gamma\|_\infty\le E_N\le1/16$ and $c_NE_Nt\le E_N\log N\le1/4$ for all $u\in K$. Let $U=\sup\bigl(\{0\}\cup\{|u|:u\in K\}\bigr)$. By the second identity in the last display, $\sqrt N\,e^{-2g_Nt}=A_{c_N}^{-1/2}(\log N)^{3/4}e^{-2u}$. Then the first bound in \eqref{eq:hotf-information-transfer}, with $C_2$ the constant of \cref{lem:hotf-transfer} for $c_1=1/2$, together with $1+t\le3\log N$, implies that \[ \sup_{u\in K}\Big|\sqrt{2\cI_N(t)}-\Big(N\int\eta_t^2\rho_{\mathrm{sc}}\Big)^{1/2}\Big|\le3C_2e^{2U}\,\frac{c_N}{\sqrt{A_{c_N}}}\,E_N(\log N)^{7/4}\le36C_2e^{2U}E_N(\log N)^{7/4}, \] where $\eta_t$ is the continuum relative error of \cref{sec:hotf-continuum} with $c=c_N$. In the last inequality, we used that $c\,A_c^{-1/2}=2\pi^{1/4}c^{3/4}(1+c)(1-c)^{-3/2}\le6\sqrt2\,\pi^{1/4}c^{3/4}\le12$ for $0