\chapter{Overview of the proofs}\label{sec:overview} In this section, we outline the proofs of \cref{thm:intro-main,thm:intro-bounded,thm:intro-critical,thm:intro-outer}; the section also serves as a map of the paper. For each regime, we explain where the mixing profile comes from, what the main obstacle is, and which results overcome it, with references to the precise statements in the later sections. Some computations in this overview are heuristic; precise statements and proofs appear in the sections cited. \Cref{sec:ov-reductions} collects the reductions that are used in every regime. \Cref{sec:ov-hot} treats the temperatures above the critical one, \cref{sec:ov-cold} the temperatures below it, and \cref{sec:ov-critical} the critical window. \Cref{sec:ov-outer} explains how the critical mixing profiles are matched with the cutoff profiles of the mesoscopic regimes. \Cref{sec:ov-dependencies} records which sections each proof uses. \section{Common reductions}\label{sec:ov-reductions} \paragraph{Eigencoordinates and shifts.} Since $\nu_N$ is invariant under orthogonal transformations, we may work in an orthonormal eigenbasis of $W_N$, and $D_N(t)$ depends on $W_N$ only through its eigenvalues (\cref{sec:model-model}). Most of our dynamical estimates concern a general real symmetric matrix $A$ with eigenvalues $\alpha_1\ge\dots\ge\alpha_N$, the generator $L_A$, and the Gibbs measure $\pi_A\propto e^{x^\T Ax/2}\nu_N$; the model is the case $A=cW_N$. Because $|x|^2=N$ on $S_N$, adding a multiple of the identity to $A$ changes neither $L_A$ nor $\pi_A$ (see \eqref{eq:model-shift}). We use this freedom in two ways. At high temperature and in the critical window, we write the potential through the precisions $p_i=z-\alpha_i$ of \cref{def:model-multiplier}, which are positive and satisfy $\sum_ip_i^{-1}=N$, so that $\pi_A$ is proportional to $e^{-\sum_ip_ix_i^2/2}\nu_N$. At low temperature, we use the scaled eigenvalue gaps $a_i=c(\lambda_1-\lambda_i)/2$ of \eqref{eq:model-raw}, for which $\pi_{N,c}\propto e^{-F}\nu_N$ with $F=\sum_{i\ge2}a_ix_i^2$, and the first eigencoordinate carries no potential. Finally, we note that $L_A$, $\pi_A$, and $\nu_N$ are invariant under each reflection $x_i\mapsto-x_i$, and so is $\nu_NP^A_t$ (\cref{lem:model-symmetry}). \paragraph{Gibbs measures as conditioned Gaussian laws.} For positive numbers $v_1,\dots,v_N$ with $\sum_iv_i=N$, let $\mu_v$ be the law of a centered Gaussian vector with independent coordinates of variances $v_i$, conditioned on $|x|^2=N$ (\cref{def:cond-finite}). By \cref{prop:cond-bingham}, the Bingham law proportional to $\exp\{\frac12\sum_ib_ix_i^2\}\,\nu_N$ with $b_i=z-v_i^{-1}$ is $\mu_v$, for every $z\in\R$. In particular, we have $\nu_N=\mu_v$ with all $v_i=1$, and $\pi_A=\mu_q$ with $q_i=1/p_i$; this representation of Bingham laws is classical \cite{Bingham1974,KumeWood2005}. Under $\mu_v$, the coordinates behave like independent Gaussian variables when no coordinate carries a macroscopic share of the variance of the radial constraint, and in \cref{sec:conditioning} we measure this through \begin{equation}\label{eq:ov-eta} \eta_v=\frac{\max_iv_i}{\big(2\sum_jv_j^2\big)^{1/2}} . \end{equation} The main results of that section are a quantitative local limit theorem for the density of $\sum_iv_i(g_i^2-1)$, where the $g_i$ are independent standard Gaussian variables (\cref{gp-local-limit}); an exact formula for the density of a set of coordinates with respect to independent Gaussian variables (\cref{gp-head-density}), and a bound on this density when the set carries a small fraction of the variance (\cref{gp-growing-head}); moment bounds (\cref{gp-moments}); an exact formula for quadratic exponential moments (\cref{gp-real-tilt}); and moment bounds for quadratic statistics (\cref{gp-quadratic-lp}). For the model, $\eta_q$ is of order $N^{-1/2}$ at a fixed temperature $c<1$, and it is asymptotic to $M_N^{-1/2}$ along $c_N=1-\delta_N$ (\cref{sp-canonical-saddle}). In the critical window and at low temperature, the first eigencoordinates carry a share of the radial variance that does not vanish, and $\eta_q$ does not tend to zero. For the critical window, we also treat in \cref{sec:conditioning} countable variance sequences, for which the radial constraint must be renormalized (\cref{def:cond-countable,prop:cond-continuity}). \paragraph{The unnormalized process.} All our dynamical estimates use one exact representation of the diffusion. For quadratic potential coefficients $\rho_1,\dots,\rho_N\in\R$, let $Y$ solve \begin{equation}\label{eq:ov-unnormalized} dY_i=-2\rho_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i,\qquad Y(0)=g, \end{equation} where $g$ is a standard Gaussian vector independent of the Brownian motions $B_1,\dots,B_N$. We show in \cref{lem:proc-normalization} that $Y$ never vanishes and that $X=\sqrt N\,Y/|Y|$ is the Langevin diffusion with potential $\sum_i\rho_ix_i^2$ started from $\nu_N$, with the same time parameter. If we replace every $\rho_i$ by $\rho_i+\tau$, then $Y(t)$ is multiplied by $e^{-2\tau t}$ and $X$ does not change. We take $\rho_i=p_i/2$ at high temperature and in the critical window, which gives \eqref{eq:intro-unnormalized}, and $\rho_i=a_i$ at low temperature, in which case $Y_1$ is a martingale, since $\E|Y(0)|^2<\infty$ (\cref{rem:proc-choices}). The coordinates of $Y$ have linear drifts. By \eqref{eq:proc-squares}, the second moments $\E Y_i(t)^2$ solve the linear system with matrix $-4\diag(\rho)+(2/N)\ind\ind^\T$, and for $\rho_i=p_i/2$ we find that \begin{equation}\label{eq:ov-heat} \E Y_i(t)^2=h_i(t),\qquad h(t)=e^{tH}\ind,\qquad H=-2\diag(p)+\frac2N\,\ind\ind^\T, \end{equation} where $H$ is the heat matrix studied in \cref{sec:heat}. The only nonlinear feature of \eqref{eq:ov-unnormalized} is the random factor $|Y|$ in the noise. In \cref{sec:process}, we bound $|Y(t)|^2$ from above and from below by solutions of the linear system, in moments, in inverse moments, and along whole paths. We also write $Y=\Phi X_0$ there, where $\Phi$ is a matrix-valued process independent of the initial point $X_0$ (\cref{lem:proc-normalization}), and this is how the initial distribution enters the proofs at low temperature. \paragraph{Deterministic hypotheses.} As explained in \cref{sec:intro-ideas}, the random-matrix results of \cref{sec:spectral} enter only through deterministic hypotheses on the spectrum. These hold on events of probability at least $1-\eta$, for every fixed $\eta>0$ and with constants depending on $\eta$ (\cref{sec:model-quantifiers}), and \cref{lem:model-localization} removes the localization at the end of each proof. We list the hypotheses of each regime in \cref{tab:ov-hypotheses}. When a proof uses indexed bounds on the GOE edge gaps, \cref{sp-fixed-confidence-geometry} provides $\lambda_1-\lambda_i\asymp((i-1)/N)^{2/3}$ at every fixed confidence. The deterministic infinite-temperature comparison requires only the norm bound in \cref{wf-deterministic-theorem}. \begin{table}[ht] \centering \small \renewcommand{\arraystretch}{1.25} \begin{tabular}{@{}>{\raggedright\arraybackslash}p{0.2\textwidth}>{\raggedright\arraybackslash}p{0.5\textwidth}>{\raggedright\arraybackslash}p{0.24\textwidth}@{}} \toprule Regime & Deterministic hypotheses & For the GOE\\ \midrule $c_N\to0$ & $A=cG$ with $\|G\|\le M$ and $c\le c_0(M)$ (\cref{wf-deterministic-theorem}) & \cref{lem:inf-goe-elementary}; for the explicit centers, also \cref{sp-goe-rigidity}\\ $c\in(0,1)$ fixed & $\|A\|\le\Lambda$ and $Nd^2\ge C_{\mathrm w}(\Lambda)$ (\cref{ht-bingham-bridge}); eigenvalues within $o((\log N)^{-7/4})$ of the semicircle quantiles, uniformly (\cref{lem:hotf-transfer}) & \cref{sp-goe-rigidity,sp-canonical-saddle}\\ $c_N=1-\delta_N$ & $d\le d_0$, $K^{-1}((i-1)/N)^{2/3}\le p_i-d\le K((i-1)/N)^{2/3}$, and $Nd^{3/2}\ge M_0$; see \eqref{eq:hotm-geometry} & \cref{sp-fixed-confidence-geometry,sp-canonical-saddle}\\ $c_N=1+\sigma N^{-1/3}$ & $\sum_i(2r_i)^{-1}=N$, $c_0(i/N)^{2/3}\le r_i\le C_0(i/N)^{2/3}$, and $r_2-r_1\ge\delta N^{-2/3}$ (\cref{def:spec-critical-conditions}) & \cref{cg-canonical-weyl}\\ $c_N=1+\epsilon_N$ & $c_*(i-1)^{2/3}N^{-2/3}\le a_i\le C_*(i-1)^{2/3}N^{-2/3}$ and $a_3\ge(1+\varepsilon_*)a_2$ (\cref{def:coldm-admissible}); $\theta_N>0$, $\theta_N\to0$, and $m_N=\theta_NNa_2\to\infty$ (\cref{ccf-thm:full-profile}) & \cref{sp-condensed-geometry}\\ $c>1$ fixed & $\theta_N\ge\theta_*$, $a_N\le B$, $a_j\ge c_*(j-1)^{2/3}N^{-2/3}$, and $a_2\le C_*N^{-2/3}$ (\cref{def:coldf-hypotheses}); $\theta_N\to\theta\in(0,1)$ and $1+\varpi\le a_3/a_2\le D_3$ for fixed $\varpi>0$, $D_3<\infty$ (\cref{thm:coldf-deterministic}) & \cref{lem:coldf-inverse-gap-lln,lem:coldf-goe}\\ \bottomrule \end{tabular} \caption{The deterministic hypotheses on the spectrum in each regime, and the results that verify them for the GOE on events of probability close to one. Here $d=p_1$ is the smallest precision and $r_i=p_i/2$ are the half-precisions.} \label{tab:ov-hypotheses} \end{table} \paragraph{From pointwise limits to profiles.} The distance $D_N$ is nonincreasing by the contraction property of Markov semigroups (\cref{lem:model-tv}). Recall from \cref{cot-profile} that the profiles $\Psi$, $\cV$, and $\cV_\theta$ are continuous and decrease strictly from $1$ to $0$. Suppose that $T_N/w_N\to\infty$ and that $D_N(T_N+w_Nu)\to P(u)$ in probability for each fixed $u$, where only nonnegative times $T_N+w_Nu$ are considered, as in \cref{def:model-cutoff}. Then \cref{hp-monotone-profile-transfer} implies that the convergence is locally uniform in $u$, and that $(t_N(\eps)-T_N)/w_N\to P^{-1}(\eps)$ for every $\eps\in(0,1)$. Outside the critical window, it therefore suffices to prove convergence at each fixed offset $u$. We also use repeatedly that the total variation distance between two laws is at least the distance between the laws of any function of the configuration, for instance of finitely many eigencoordinates; this gives lower bounds on $D_N$. \section{High temperature}\label{sec:ov-hot} We first describe, for a fixed matrix $A$, the comparison on which cases (1)--(3) of \cref{thm:intro-main} and \cref{thm:intro-bounded} rest. We then explain how each regime evaluates it. \subsection{The Bingham comparison law}\label{sec:ov-hot-comparison} Let $A$ have precisions $p_i$, let $H$ and $h(t)$ be as in \eqref{eq:ov-heat}, and let $\bar h(t)=N^{-1}\sum_ih_i(t)$. We define the covariance evolution of $A$ by \begin{equation}\label{eq:ov-covariance} v_i(t)=\frac{h_i(t)}{\bar h(t)},\qquad\text{so that}\qquad \sum_{i=1}^Nv_i(t)=N,\qquad v_i(0)=1,\qquad \lim_{t\to\infty}v_i(t)=q_i . \end{equation} These are the variances of the linear system \eqref{eq:intro-covariance}, in which the spherical constraint holds only in expectation (\cref{sec:heat-covariance}), and the Bingham comparison law at time $t$ is $\mu_t=\mu_{v(t)}$ (\cref{def:heat-covariance}). We have $\mu_0=\nu_N$, and $\mu_t\to\pi_A$ as $t\to\infty$ (\cref{ht-covariance-system}). The mean $\bar h$ solves the renewal equation $\bar h=k+2k*\bar h$ with $k(t)=N^{-1}\sum_ie^{-2p_it}$ (\cref{lem:heat-algebra}); this is the finite-dimensional counterpart of the scalar Volterra equation of Cugliandolo and Dean \cite[(II.4)--(II.7)]{CugliandoloDean1995}. By the triangle inequality, we have \begin{equation}\label{eq:ov-triangle} \Big|D_N(t)-\|\mu_t-\pi_A\|_{\TV}\Big|\le\|\nu_NP^A_t-\mu_t\|_{\TV}. \end{equation} The proof at high temperature consists of two independent estimates: the right side of \eqref{eq:ov-triangle} is small at the relevant times, and $\|\mu_t-\pi_A\|_{\TV}$ converges to the profile. In general, $\mu_t$ need not coincide with the time-$t$ law, and $v_i(t)$ need not equal the second moment of $x_i$ under $\mu_t$ or under $\nu_NP^A_t$ (\cref{rem:heat-not-moments}). We compare the two laws only through the forward equation of the diffusion. \subsection{Likelihood ratios on a common sphere}\label{sec:ov-hot-lan} Both $\mu_t$ and $\pi_A$ are Gaussian laws conditioned on $S_N$, with variances $v_i(t)$ and $q_i$ of equal sum. We write $v_i(t)=q_i(1+e_i(t))$, so that $e_i(t)$ are the relative errors and $\cI_N(t)=\cI(q,v(t))=\frac12\sum_ie_i(t)^2$ is the information of \cref{def:model-information}. Since the two Gaussian densities differ by the exponential of a quadratic form, we have on $S_N$ \begin{equation}\label{eq:ov-loglik} \ell=\log\frac{d\mu_t}{d\pi_A}=\frac12\sum_{i=1}^N\frac{e_i(t)}{1+e_i(t)}\,Z_i^2+\text{const},\qquad Z_i=\frac{x_i}{\sqrt{q_i}} . \end{equation} We now explain heuristically why $\ell$ is approximately Gaussian under $\pi_A$. When $\eta_q$ is small, the variables $Z_i$ behave under $\pi_A=\mu_q$ like independent standard Gaussian variables, subject to the constraint $\sum_iq_i(Z_i^2-1)=0$. When the relative errors are small, $\ell$ is close to $\frac12\sum_ie_i(Z_i^2-1)$ up to a constant. For independent standard Gaussian variables, this sum has variance $\frac12\sum_ie_i^2=\cI_N$, and its covariance with the constraint is $\sum_iq_ie_i=\sum_i(v_i-q_i)=0$, so that conditioning on the constraint does not change its variance to leading order. As $\int e^\ell\,d\pi_A=1$, the mean of $\ell$ is then close to $-\cI_N/2$. In Le Cam's Gaussian experiment \cite{LeCam1960}, \cite[Chapters~6--7]{vanDerVaart1998}, if $\ell$ has law $\cN(-I/2,I)$ under $\pi_A$, then it has law $\cN(I/2,I)$ under $\mu_t$. In that case, we find that \begin{equation}\label{eq:ov-lan} \|\mu_t-\pi_A\|_{\TV}=\mu_t(\ell>0)-\pi_A(\ell>0)=2\Phi\big(\sqrt I/2\big)-1=\mathcal G(I), \end{equation} where $\mathcal G$ is the function of \eqref{eq:hotf-gaussian-tv}. We make this quantitative in \cref{gp-pinned-lan-quantitative-third}. Let $v$ and $\tilde v$ be variance sequences with equal sums, write $\tilde v_i=v_i(1+e_i)$, let $I=\cI(v,\tilde v)=\frac12\sum_ie_i^2$ be the information of $\tilde v$ relative to $v$, and let $\ell=\log(d\mu_{\tilde v}/d\mu_v)$. Suppose that $\max_i|e_i|\le1/4$, that $\eta_v$ is smaller than a universal constant, and that $I\le B$. Then the Wasserstein distance between the law of $\ell$ under $\mu_v$ and $\cN(-I/2,I)$ is at most $C_B(\eta_v+\max_i|e_i|)$, and so is the difference between $\|\mu_{\tilde v}-\mu_v\|_{\TV}$ and $\mathcal G(I)$. The proof combines an exact integration by parts formula for quadratic statistics under $\mu_v$, which accounts for the constraint (\cref{gp-pin-tangent-ibp-fourth}), a bound on inverse moments (\cref{gp-pin-radial-inverse-fourth}), and Stein's method \cite{Stein1972} (\cref{gp-scalar-stein-tests-fourth,gp-conditional-normal-third}). We deduce the corresponding limit statements in \cref{gp-pinned-lan}; in particular, the distance tends to one when the information diverges. We conclude that the high-temperature profile reduces to the following problem: find a center $T_N$ and a window $w_N$ such that, at the times $t=T_N+w_Nu$, we have $\cI_N(t)\to e^{-4u}$, $\max_i|e_i(t)|\to0$, and $\eta_q\to0$. Then $\|\mu_t-\pi_A\|_{\TV}$ tends to $\mathcal G(e^{-4u})=\Psi(u)$, which explains why the same profile $\Psi$ appears in cases (1)--(3). \subsection{The information at a fixed temperature}\label{sec:ov-hot-fixed} The information is determined by the mean of the heat vector (\cref{mc-exact-info}): \begin{equation}\label{eq:ov-info-formula} \cI_N(t)=\frac{N\,\bar h''(2t)}{8\,\bar h(t)^2},\qquad \bar h(t)=\vartheta+\int e^{-\lambda t}\,\rho(d\lambda), \end{equation} where $\vartheta=N/\sum_iq_i^2$, and $\rho$ is a positive measure on $[2d,2s]$: together with the atom $\vartheta$ at zero, it forms the spectral measure of $-H$ at the unit vector $N^{-1/2}\ind$ (\cref{lem:heat-algebra}). In particular, $\cI_N(t)$ decays at least like $e^{-4dt}$, and at large times its behavior is governed by the precisions near the smallest one, that is, by the eigenvalues near $\lambda_1$. At a fixed temperature $c\in(0,1)$, we compute $\cI_N(t)$ in \cref{sec:hot-fixed} by comparison with the same evolution for the semicircle law, which we solve exactly by Laplace transforms (\cref{prop:hotf-exact}), using the same scalar-resolvent method as \cite[Appendix~A]{CugliandoloDean1995}, whose appendix treats physical temperatures $T0$ be deterministic with $\delta_N^2t_N\to\infty$ and $\delta_N^2t_N=o(M_N^{2/3})$, as in \eqref{mc-time-hypotheses}. At such times, the relative error $e_i(t)$ is small unless $p_i-d_N$ is at most of order $1/t$, which is much smaller than $\delta_N^2$. The number of such indices is of order $Nt^{-3/2}$, and on this scale the information is governed by eigenvalues within distance $O(1/t_N)$ of the edge. In \cref{sec:hot-meso}, we therefore work with the measure $\rho=\rho_N$ of \eqref{eq:ov-info-formula} directly. We rescale it at the scale $1/t$, as in \eqref{mc-nu-definition}, and the rescaled measure converges vaguely in probability to $\pi^{-1}\sqrt y\,dy$, the image of the semicircle density near the edge (\cref{mc-measure-theorem}). The difficulty is that $\rho_N$ is not a simple function of the eigenvalue counts. However, its real resolvent is an explicit rational function of the precisions (\cref{lem:heat-algebra}). We find the limits of these resolvents at real arguments from the edge counts at a varying scale (\cref{sp-mesoscopic-counts,mc-raw-edge-measure}), together with the asymptotics of the Lagrange multiplier (\cref{sp-canonical-saddle}) and of the inverse gaps (\cref{sp-raw-gap-resolvent}). In \cref{pr-real-measure-transfer}, we prove that convergence of the real squared resolvents of positive measures implies their convergence against a large class of test functions; the proof uses the compact-interval moment method (see \cite[Chapter~III]{Widder1941}), a finite-difference calculation to recover the moments, and approximation by Bernstein polynomials \cite[pp.~1--2]{Bernstein1912}. The result is \cref{mc-information}: \begin{equation}\label{eq:ov-info-meso} \cI_N(t)=\frac{Nd_N^{3/2}}{128\sqrt\pi}\,w^{-3/2}e^{-4w}\big(1+o_{\PP}(1)\big),\qquad w=d_Nt . \end{equation} This is the right side of \eqref{eq:ov-info-fixed} with $c=1$ and $g$ replaced by $d_N$. If we set the right side of \eqref{eq:ov-info-meso} equal to $e^{-4u}$ and solve for $t$, we recover the center $T_N$ of case (3), with its correction $-\frac32\log\log M_N$, up to an error that is small compared with the window $\delta_N^{-2}$. To replace $d_N$ by $\delta_N^2$ in the center, we need that $\log(2+M_N)\lvert d_N/\delta_N^2-1\rvert\to0$ in probability, which follows from \cref{sp-canonical-saddle}. As $\eta_q=O_{\PP}(M_N^{-1/2})$, \cref{gp-pinned-lan-quantitative-third} applies, and $\|\mu_t-\pi_{N,c_N}\|_{\TV}$ converges to $\Psi(u)$ at the times $T_N+u/\delta_N^2$ (\cref{mc-proxy-profile}). \paragraph{The comparison with no condition on $Nd^2$.} The bound $C_\Lambda d^{-2}e^{-rt}$ of \cref{ht-bingham-bridge} requires $Nd^2$ to be large, and at the times $T_N+u\delta_N^{-2}$ it is about $\delta_N^{-4}M_N^{-1/4}$, which need not be small. The obstruction is the rate $r_0$ in \eqref{eq:ov-weighted-gradient}, which is positive only when $Nd^2$ is large. To overcome this, we prove in \cref{mgf-global-bridge} that for precisions that satisfy \eqref{eq:hotm-geometry} with $M=Nd^{3/2}\ge M_0$, \begin{equation}\label{eq:ov-global-bridge} \|\nu_NP^A_t-\mu_t\|_{\TV}\le C\,e^{-(d-2/N)t},\qquad 0\le t\le C_Td^{-1}\log M, \end{equation} where $C$ and $M_0$ depend only on the constants in \eqref{eq:hotm-geometry} and on $C_T$. At these times, the right side is $M^{-1/4}(\log M)^{O(1)}$, which tends to zero however slowly $M$ grows. The proof has three ingredients. The first is the third form of \cref{pc-centered-defect-transfer}. We solve the Poisson equation $-L_{B_u}\psi_u=R_u$ for the generator whose Gibbs measure is $\mu_u$ and take $j_u=\nabla_{S_N}\psi_u$ (\cref{lem:comp-poisson,cor:comp-poisson-flux}). In \cref{rp-general-flux}, we bound the energy $\mu_u\Gamma(\psi_u)$ by $C/M$ times explicit sums of the coefficients of $Q_C(u)$, under conditions on the variances at a scale $\delta$ (\cref{def:comp-variance-conditions}). The proof uses the lower bound on the spectral gap of \cref{ht-many-soft-sharp-gaps}, which is sharp when many precisions are comparable to the smallest one. Along the covariance evolution, we find that the energy is at most $C(1+u)^{-1}$ for $u\le d^{-1}$ and at most $Cd(1+ds)^2e^{-4ds}$ at $u=d^{-1}+s$ (\cref{mgf-exact-flow-envelopes}). The second ingredient is a bound on the gradient of $P^A_\tau f$ under $\mu_u$ with the rate $d-2/N$ in place of $r_0$. To obtain it, we keep the variable term of the weighted Bochner identity, which leads to the Feynman--Kac bound of \cref{hfk-pointwise-fk}: \begin{equation}\label{eq:ov-fk} \varphi(x)\,\Gamma(P^A_tf)(x)\le e^{-2(d-2/N)t}\,\E_x\Big[\exp\Big(2\int_0^t\Gamma(\log\varphi)(X_s)\,ds\Big)\varphi(X_t)\,\Gamma(f)(X_t)\Big]. \end{equation} By \cref{sgl-weighted-square}, the function $\Gamma(\log\varphi)$ is at most $(\kappa+\kappa^{-1}-2)/N$, which is not small compared with $d$ when $Nd^2$ is small. However, its time integral along the path is small: by \cref{iw-chemical-occupation}, its exponential moments are at most $1+C\lambda\sqrt{\log M/M}$, for every initial law. Together with a reverse variance argument along the path, this implies the bound of \cref{lem:hotm-gradient} on $\mu_u\Gamma(P^A_\tau f)$ for $0\le f\le1$. The third ingredient consists of the path estimates for the unnormalized process behind the second one. Under the initial law $\mu_u$ weighted by $1/\varphi$, we need moments of $\sup_t\varphi(X_t)$ along the whole time interval, and exponential moments of the occupation integral above (\cref{mgf-weighted-source-transfer}). We derive them from inverse moments of $|Y(t)|^2$ relative to $N\bar h(u+t)$, at fixed times (\cref{iw-backward-inverse}) and along the whole interval (\cref{hmr-main}), and from a comparison of the initial state with all future solutions of the linear system at once (\cref{iw-early-pin-simultaneous-centers}). The inverse bounds use \cref{pr-positive-martingale}, which requires the conditional relative variances to be bounded uniformly by a small constant; how small depends on the order of the inverse moment or on the exponential parameter. Its occupation bound additionally uses a deterministic bound on the quadratic variation of the logarithm. Inserting the first two ingredients into the Duhamel integral, we obtain \eqref{eq:ov-global-bridge}. The estimate holds for every $N$ and every matrix with $M\ge M_0$, with the same constants. We use it again in \cref{sec:outer-hot}, with $M$ fixed while $N\to\infty$. \subsection{The infinite-temperature crossover}\label{sec:ov-hot-infinite} When $A=cG$ with $\|G\|\le M$ and $c$ small, the distance is asymptotically determined by the information alone, uniformly over all times (\cref{wf-deterministic-theorem}). Indeed, \cref{ht-small-field-direct-comparison} implies that $\|\nu_NP^A_t-\mu_t\|_{\TV}\le Cc^2\min(t,1)$ for all $t$. The relative errors satisfy $\max_i|e_i(t)|\le Cce^{-t}$, and if $\cI_N(0)>0$, then the logarithmic derivative of $\cI_N(t)$ lies in $[-4-Cc,-4+Cc]$. Using \cref{lem:hotf-lan-uniform}, we find that $\sup_{t\ge0}|D_N(t)-\mathcal G(\cI_N(t))|\to0$ as $c\to0$ and $N\to\infty$. If $\cI_N(0)\to\infty$, it follows that there is cutoff with a window of order one, at the time $\tau_N$ defined by $\cI_N(\tau_N)=1$. For the GOE, we have $\cI_N(0)=(J_N/2)(1+o_{\PP}(1))$ with $J_N=Nc_N^2$ (\cref{wf-goe-classification}). Here $e_i(0)=p_i-1$ is close to $-c(\lambda_i-N^{-1}\sum_j\lambda_j)$, and the centered second moment $N^{-1}\Tr(W_N^2)-(N^{-1}\Tr W_N)^2$ tends to $1$ in probability. The proof of the classification uses only the norm of $W_N$ and the traces of $W_N$ and $W_N^2$, and not the rigidity estimate (\cref{lem:inf-goe-elementary}). If $\sqrt Nc_N\to a$, the information is asymptotic to $a^2e^{-4t}/2$, and $\mathcal G(a^2e^{-4t}/2)$ is the limit in \cref{thm:intro-bounded}. If $J_N\to\infty$, there is cutoff at the first time $\tau_N$ at which $\cI_N\le1$. With probability tending to one, $\cI_N(0)>1$ and $\tau_N$ is the only solution of $\cI_N(t)=1$. In \cref{wf-explicit-clocks}, we determine $\tau_N$ up to $o(1)$. Here we need the rigidity estimate, through the approximation of the eigenvalues by the semicircle quantiles in \cref{lem:hotf-grid}. On the time scale $\tau_N\approx\frac14\log J_N$, the product $c_N\tau_N$ is close to $h_N=(c_N/4)\log J_N$. When $h_N$ is bounded, the rescaled relative errors $c^{-1}e^{2t}e(t)$ are close to $-e^{2ct\mathcal T}\lambda$, where $\mathcal T$ is the multiplication by the eigenvalues compressed to vectors of mean zero, and $\lambda$ is the vector of eigenvalues of $G$, which we may take to have mean zero (\cref{lem:inf-compressed}). For the semicircle law, we compute this operator with Chebyshev polynomials (\cref{lem:inf-chebyshev}), which gives $\cI_N(t)\approx J_Ne^{-4t}\mathcal B(c_Nt)$ with $\mathcal B(h)=I_1(8h)/(8h)$ (\cref{lem:inf-bulk}). Solving $\cI_N(\tau_N)=1$, we have $\tau_N=\frac14[\log J_N+\log\mathcal B(h_N)]+o_{\PP}(1)$. When $h_N\to\infty$, the relative errors concentrate near the edge of the spectrum. The computation of \cref{sec:hot-fixed} holds uniformly for $c\in(0,c_1]$, and it yields the center of case (2) evaluated at $c=c_N$. \section{Low temperature}\label{sec:ov-cold} \Cref{sec:cold-fixed,sec:cold-meso} prove cases (5) and (4) of \cref{thm:intro-main}, at fixed $c>1$ and along $c_N=1+\epsilon_N$. We use the scaled eigenvalue gaps $a_i$ and the top fraction $\theta_N$ of \cref{sec:model-equilibrium}, with $R_N=Na_2$ and $m_N=\theta_NR_N$. At fixed $c>1$, $a_2$ is of order $N^{-2/3}$ and $R_N$ of order $N^{1/3}$. Along $c_N=1+\epsilon_N$, we have $\theta_N/\epsilon_N\to1$, and $m_N$ is of order $(N\epsilon_N^3)^{1/3}$ (\cref{sp-condensed-geometry,lem:coldm-indexed}). Throughout, the dynamics is represented by the unnormalized process \eqref{eq:ov-unnormalized} with $\rho_i=a_i$, whose first coordinate $Y_1$ is a martingale while $Y_i$ is pulled toward zero at rate $2a_i$ for $i\ge2$. At a fixed temperature, we take as its initial value a point $X_0$ with law $\nu_N$, which also starts the diffusion from $\nu_N$ (\cref{lem:proc-normalization}). \subsection{The mechanism}\label{sec:ov-cold-mechanism} We describe the mechanism at a fixed temperature, with $\theta_N\to\theta\in(0,1)$, following the outline in \cref{sec:coldf-outline}. It has three steps. \paragraph{Gaussian approximation at time $t_{\mathrm p}$.} At the time $t_{\mathrm p}=(\log N)/100$, the pair $(Y_1(t_{\mathrm p}),Y_2(t_{\mathrm p}))$ is close in total variation to $\theta^{-1/2}(H_1,H_2)$, where $H_1,H_2$ are independent standard Gaussian variables (\cref{prop:coldf-initial-coordinates}). These variables are built from the initial point and from the noise up to time $t_{\mathrm p}$. To see why, we write $Y(t)=\Phi(t)X_0$ as in \cref{lem:proc-normalization}, with $\Phi$ independent of $X_0$. Conditionally on $\Phi$, finitely many coordinates of $Y(t_{\mathrm p})$ are linear functions of a uniform point of $S_N$. By the rotation invariance of $\nu_N$, they are close to a centered Gaussian vector whose covariance is the corresponding block of $\Phi\Phi^\T$; this is the classical Gaussian approximation of the coordinates of a uniform point \cite[(1)]{DiaconisFreedman1987}. The block concentrates around $\theta^{-1}$ times the identity. Its diagonal entries have means $h_i(t_{\mathrm p})=\E Y_i(t_{\mathrm p})^2$, which are close to $\theta_N^{-1}$. Indeed, the mean of $|Y|^2$ solves a renewal equation whose kernel $(2/N)\sum_{i\ge2}e^{-4a_it}$ is a defective probability density with total mass $1-\theta_N$, the mean square of $Y_1$ satisfies $\lvert h_1(t)-\theta_N^{-1}\rvert\le C[(1+t)^{-1/2}+t/N]$ for $1\le t\le N$, so that $h_1(t_{\mathrm p})=\theta_N^{-1}+o(1)$, and $h_i(t_{\mathrm p})$ is close to $h_1(t_{\mathrm p})$ for each fixed $i$, since $a_it_{\mathrm p}\to0$ (\cref{lem:coldf-renewal}). \paragraph{Growth of the top coordinate.} After $t_{\mathrm p}$, the martingale $Y_1$ barely moves, by the second part of \cref{prop:coldf-initial-coordinates}, while $|Y(t)|^2$ relaxes to $Y_1(t_{\mathrm p})^2/\theta_N$ (\cref{prop:coldf-normalization}). In particular, at the times $t_u=\log R_N/(4a_2)+u/a_2$ we have $X_1(t_u)^2/N\to\theta$: the first eigencoordinate grows from order one to $\sqrt{\theta N}$. \paragraph{The ratio of the first two coordinates.} By Duhamel's formula, we have for $t\ge t_{\mathrm p}$ \[ Y_2(t)=e^{-2a_2(t-t_{\mathrm p})}Y_2(t_{\mathrm p})+\sqrt{2/N}\int_{t_{\mathrm p}}^te^{-2a_2(t-s)}\,|Y(s)|\,dB_2(s). \] We now argue heuristically. During the last part of $[t_{\mathrm p},t_u]$, of length of order $a_2^{-1}$, we may replace $|Y(s)|$ by $|Y_1(t_{\mathrm p})|/\sqrt{\theta_N}$. Conditionally on $(Y_1(t_{\mathrm p}),Y_2(t_{\mathrm p}))$, the stochastic integral is approximately centered Gaussian with variance $Y_1(t_{\mathrm p})^2/(2\theta_NNa_2)$. After division by $|Y_1(t_{\mathrm p})|$, its Gaussian approximation is independent of this early pair. Recall that $X_2=\sqrt NY_2/|Y|$, and note that $a_2t_{\mathrm p}\to0$ and $\sqrt{2a_2N\theta_N}\,e^{-2a_2t_u}=\sqrt{2\theta_N}\,e^{-2u}$, since $Na_2=R_N$. We find that \begin{equation}\label{eq:ov-ratio} \sqrt{2a_2}\,X_2(t_u)\approx\sqrt{2\theta_N}\,e^{-2u}\,\frac{Y_2(t_{\mathrm p})}{|Y_1(t_{\mathrm p})|}+G, \end{equation} where $G$ is a standard Gaussian variable independent of $(Y_1(t_{\mathrm p}),Y_2(t_{\mathrm p}))$. By the first step, the ratio is close in law to $H_2/|H_1|$, which has the standard Cauchy law \cite{Marsaglia1965}. Under $\pi_{N,c}$, the variable $\sqrt{2a_2}\,x_2$ is approximately standard Gaussian. As $\cV_\theta(u)$ is the total variation distance between the laws of $G+\sqrt{2\theta}e^{-2u}C$ and $G$, this identifies the profile, provided that the other coordinates do not contribute. More precisely, for every fixed $k$, the vector $(\sqrt{2a_j}X_j(t_u))_{j=2}^k$ converges in distribution to $(G_2+\sqrt{2\theta}e^{-2u}C,G_3,\dots,G_k)$ by \cref{thm:coldf-finite-coordinates}, under the hypothesis $\mathrm H_k(\varpi,D_k)$ of \cref{def:coldf-hypotheses} when $k\ge3$. In particular, $a_3\ge(1+\varpi)a_2$ for a constant $\varpi>0$. The coordinates $j\ge3$ forget their values at time $t_{\mathrm p}$, since this separation of the gaps makes their inherited part negligible. As the total variation distance decreases under projections, we conclude that $\liminf_ND_N(t_u)\ge\cV_\theta(u)$. \subsection{The upper bound}\label{sec:ov-cold-upper} The main difficulty is the upper bound, since $D_N$ involves all $N$ coordinates and weak convergence of finitely many of them does not control it. We first reduce to a hemisphere: by \cref{lem:model-symmetry}, $D_N(t)$ equals the distance between the laws folded onto $\{x_1>0\}$. There we use the graph coordinates $y=(x_2,\dots,x_N)$, with $x_1=(N-|y|^2)^{1/2}$. By \cref{mesocold-prop:stationary-graph}, the Gibbs measure in these coordinates is close to the Gaussian product law $G=\bigotimes_{j\ge2}\cN(0,(2a_j)^{-1})$, with an error at most $C/m_N+Ce^{-c_0m_N}$ that includes the factor $(1-|y|^2/N)^{-1/2}$ of the parametrization. The bound is uniform in $\theta_N$, which we need in \cref{sec:cold-meso}. We then split $y$ into its first coordinates $y_{\mathrm z}$ (the retained coordinates $x_2,\dots,x_{J_N}$ at a fixed temperature, described below, and $x_2,\dots,x_J$ in the approach from below) and the remaining coordinates $y_{\mathrm v}$. By \cref{ccf-lem:finite-product-closure}, we have \begin{equation}\label{eq:ov-closure} \Big|\|\rho-\pi\|_{\TV}-\|\mu-G_{\mathrm z}\|_{\TV}\Big|\le2(1-Z)+\sqrt{H_{\mathrm v}/2}+\|\rho_{\mathrm z}-\mu\|_{\TV}+\|\pi-G\|_{\TV}. \end{equation} Here $\rho$ is the time-$t$ law, and $\sigma\le\rho$ is the endpoint subprobability measure weighted by a path weight with values in $[0,1]$ and by a cutoff function, with mass $Z>0$. The measure $\sigma/Z$ is a normalized cutoff-weighted law in the sense of \cref{sec:scores}, with normalizing mass $Z$. Further, $H_{\mathrm v}$ is the relative entropy of $\sigma/Z$ with respect to the product of its marginal on $y_{\mathrm z}$ and $G_{\mathrm v}$, and $\mu$ is the limiting law of $y_{\mathrm z}$. We now explain how we bound the four terms on the right side. \paragraph{Weighting at the entrance into a cap.} At a fixed temperature, we weight the path law by a smooth function of the position at the time $T_{\mathrm e}=\chi_N/a_2$, with $\chi_N\to\infty$ slowly. This function takes values in $[0,1]$, equals one on a cap $\{x_1^2\ge\sigma N\}$, and vanishes outside a larger cap. By \cref{prop:coldf-entrance}, the mean of the weight is at least $1-\eps$ for large $N$, where $\eps>0$ is small and fixed. By \cref{prop:coldf-residence}, the path then stays in a slightly larger cap up to times of order $\log R_N/a_2$, outside an event whose probability is exponentially small in $R_N/\log R_N$. We also multiply the endpoint measure by a cutoff function that depends only on $x_1^2$, equals one on $\{x_1^2\ge rN\}$, and vanishes on $\{x_1^2\le rN/2\}$, for a fixed $r>0$. We bound the derivatives of the cutoff and the mass of the region outside which it equals one (\cref{lem:coldf-collar,lem:coldf-cutoff}). \paragraph{The conditional relative entropy of the remaining coordinates.} In \cref{sec:scores}, we compute the time derivative of the conditional relative entropy of the remaining coordinates given the first ones exactly (\cref{prop:score-entropy-identity}) and prove a logarithmic Sobolev inequality for this conditional relative entropy (\cref{prop:score-conditional-lsi}). It follows from the Gaussian logarithmic Sobolev inequality \cite{Gross1975} after a comparison of the metric of the remaining coordinates (\cref{lem:score-schur}). The entropy therefore decays at a rate close to $4b$, where $b$ is the smallest of the gaps $a_j$ of the remaining coordinates, up to errors from the cutoff function. At a fixed temperature, the upper bound on the density of the weighted law gives an initial value of order $N$ at the time $T_{\mathrm e}+a_2^{-1}$ (\cref{lem:coldf-initial-entropy}). Heuristically, if only $x_2$ is retained, decay at rate $4a_3$ during a time close to $\log R_N/(4a_2)$ multiplies the entropy by about $R_N^{-a_3/a_2}$. Since $N$ is of order $R_N^3$, this does not beat the initial value unless $a_3>3a_2$. To overcome this, we retain all coordinates with $a_j\le16a_2/r$, and their number is bounded (\cref{lem:coldf-bounded-head}). On the support of the cutoff function, the metric of the sphere in graph coordinates is at least $r/2$ times the Euclidean metric. The Gaussian logarithmic Sobolev inequality for the remaining coordinates then bounds the entropy by $(32a_2)^{-1}$ times its dissipation (\cref{lem:coldf-coarse-lsi}). After the error terms are absorbed, the entropy decays at rate $16a_2$, which multiplies it by about $R_N^{-4}$ during this time. This beats the initial value, and the entropy tends to zero at the times $t_u$ (\cref{prop:coldf-conditional-entropy}). \paragraph{The first coordinates in total variation.} We upgrade weak convergence to convergence in total variation by Gaussian smoothing (\cref{csh-thm:weak-to-tv}). For this, we need that the limit $\mu$ has finite Fisher information, and that the laws $\rho_{\mathrm z}$ are close in total variation to laws with bounded Fisher information. The limit is a Gaussian convolution, whose Fisher information is at most the dimension; no moment of the Cauchy law is needed (\cref{csh-lem:convolution-fisher}). For the time-$t$ law, we bound the score with a Bismut-type formula over the final time interval of length $a_2^{-1}$ (\cref{csr-thm:reset,csr-cor:cold-scale}). This formula controls the derivative, along a rotation in the plane of two eigencoordinates, of the logarithm of the density of an endpoint law weighted by a functional of the path. The bound uses only the final interval of the original uniformly initialized diffusion, with no score bound before that interval; its left-endpoint law may be evolved and prefix-weighted. Such formulas go back to Bismut \cite{Bismut1984} and to Elworthy and Li \cite[Theorems~2.1 and~3.1]{ElworthyLi1994}. With the Fisher information of the coordinates in the graph chart (\cref{csh-lem:graph-fisher}), this leads to \cref{csh-cor:cold-head}, which we use in the approach from below. At a fixed temperature, we apply the formula over final intervals of length $a_j^{-1}\le a_2^{-1}$, bound the Fisher information of finitely many coordinates directly in the graph chart, and apply \cref{csh-thm:weak-to-tv}. This gives \cref{prop:coldf-finite-head-tv}, and \cref{cq-hc-head-smoothing} is its case of the second coordinate. \paragraph{The Gibbs measure.} The last term in \eqref{eq:ov-closure} is small by \cref{mesocold-prop:stationary-graph}. The parameters of the proof are chosen in the order of \cref{tab:coldf-parameters}. Since $J_N$ is bounded, \cref{sec:coldf-assembly} combines the estimates along subsequences on which $J_N$ is constant. \subsection{The approach from below}\label{sec:ov-cold-meso} Along $c_N=1+\epsilon_N$, the top fraction $\theta=\theta_N$ tends to zero, and $m=\theta Na_2$ may tend to infinity arbitrarily slowly. The main difficulty compared with a fixed temperature is uniformity: the time scales $\theta^{-2}$ and $a_2^{-1}$ and the size $\theta^{-1/2}$ of the first unnormalized coordinate all depend on $\theta$, and every error must be bounded by a negative power of $m$. As we describe in \cref{sec:coldm-outline}, the mean $\E|Y(t)|^2$ decreases like $N(1+t)^{-1/2}$ up to the time $\theta^{-2}$ and like $N\theta^{-2}t^{-3/2}$ afterwards, until a time of order $a_2^{-1}$; after that time, it is of order $\theta^{-2}$, with its radial part determined by the first coordinate through $\lvert Y(t)\rvert^2\approx Y_1(t)^2/\theta$. In \cref{lem:coldm-renewal,cpm-prop:all-moments}, we prove renewal estimates and moment bounds of all fixed orders with constants that do not depend on $\theta$. Let $\tau=m^{\mathfrak a}\theta^{-2}$ with $\mathfrak a=1/10$, let $H_i=\sqrt\theta\,Y_i(\tau)$, and let $t_m(u)=\log(2m)/(4a_2)+u/a_2$. Then the following hold. \begin{itemize} \item The vector $(H_1,\dots,H_J)$ is close to a standard Gaussian vector (\cref{cpm-lem:finite-seed}). \item The process $\theta^2|Y(t)|^2$ stays close to $H_1^2$ along a time interval of length of order $a_2^{-1}\log m$ that contains the times $t_m(u)$ for $u$ in a compact set, with an error bounded by a negative power of $m$ in every fixed $L^p$ norm (\cref{cpm-thm:radial-path}). \item Let $E=\{\ell_-\le\lvert H_1\rvert\le\ell_+,\ \lvert H_i\rvert\le\ell_+\ (2\le i\le J)\}$. On $E$, outside an event of probability at most $Cm^{-B_0}$ for every fixed $B_0>0$, the path does not leave the domain where $X_1^2/(\theta N)\in[1/4,4]$ during the time interval of the previous item (\cref{cpm-cor:actual-residence}). The losses from killing the path on leaving this domain and from the cutoff function are controlled by \cref{cpm-cor:killed-collars}. We choose the bounds $\ell_-,\ell_+$ so that $\PP(E)$ is arbitrarily close to one and condition the path law on $E$. \end{itemize} At the times $t_m(u)$, the analog of \eqref{eq:ov-ratio} is $\sqrt{2a_2}X_2\approx e^{-2u}H_2/|H_1|+G_2$, which gives the profile $\cV$ and the lower bound (\cref{mesocold-thm:weak-profile}). For the upper bound, the initial value of the conditional relative entropy comes from a different argument. \Cref{mc-renyi-thm} rests on the density bounds of \cref{sec:density}, which we describe in \cref{sec:ov-critical-density}: at times $t\ge\theta^{-2}$, every moment of the density of the time-$t$ law with respect to the Gibbs measure is at most $e^{O(N\theta^3)}=e^{O(m^3)}$. The initial conditional relative entropy is then of order $m^3$. Heuristically, decay at rate $4a_3$ over an interval of length $(1/4-1/100)a_2^{-1}\log m$ would multiply the entropy by $m^{-(24/25)(a_3/a_2)}$. If $a_3/a_2$ is close to one, this does not beat the initial factor $m^3$. To overcome this, we keep the first $J$ coordinates, with $J$ so large that $b=a_{J+1}\ge6a_2$. Then the conditional relative entropy of the remaining coordinates decays at a rate at least $4\cdot\frac78\cdot b$, up to errors of order $m^{-1}$ (\cref{cce-thm:consumer}). Over the interval from $\mathfrak b\,a_2^{-1}\log(em)$ to $t_m(u)$, of length about $(\frac14-\frac1{100})a_2^{-1}\log m$, the entropy is multiplied by at most $m^{-5}$, which beats $m^3$ (\cref{cce-cor:quantitative}). We choose the parameters in the order of \cref{tab:coldm-parameters}. Finally, the convergence is uniform over all admissible gap vectors in $\mathcal A_N$ (\cref{def:coldm-admissible}) with $\theta\le1/M$ and $m\ge M$, in the iterated limit $\lim_{M\to\infty}\limsup_{N\to\infty}$ (\cref{ccf-cor:uniform-envelope}). We deduce it from the statement along sequences by a diagonal argument (\cref{lem:coldm-transfer}), and this is the form that we use in \cref{sec:outer-cold}. \section{The critical window}\label{sec:ov-critical} In the critical window $c_N=1+\sigma N^{-1/3}$, the half-precisions $r_i=p_i/2$ are of order $(i/N)^{2/3}$ for every $i$, including $i=1$ (\cref{cg-canonical-weyl}). For every fixed $i$, the inverse drift rate $(2r_i)^{-1}$ is of order $N^{2/3}$, and the limiting dynamics retains the individual scaled rates. To prove \cref{thm:intro-critical}, we construct a countable limit of the dynamics in \cref{sec:crit-process}, and in \cref{sec:crit-profile} we show that the distance converges to the distance between the normalized limit and the limiting Gibbs law. We write $b=N^{2/3}$. \subsection{The critical limit process}\label{sec:ov-critical-process} Let $R_{i,N}=br_{i,N}$ be the scaled half-precisions. We rescale the unnormalized process \eqref{eq:ov-unnormalized} with $\rho_i=r_{i,N}$ by \begin{equation}\label{eq:ov-critical-scaling} U_{i,N}(s)=N^{-1/6}Y_i(bs),\qquad Z_N(s)=b^{-1}|Y(bs)|^2,\qquad N^{-1/3}X_i(bs)=\frac{U_{i,N}(s)}{\sqrt{Z_N(s)}}, \end{equation} so that $dU_{i,N}=-2R_{i,N}U_{i,N}\,ds+\sqrt{2Z_N}\,dB_i^{(N)}$ for independent Brownian motions $B_i^{(N)}$; see \eqref{eq:cproc-scaling}. Each coordinate starts at $U_{i,N}(0)=N^{-1/6}g_i$, which tends to zero. The rescaled squared norm $Z_N=N^{-1/3}\sum_iU_{i,N}^2$ is a sum over all coordinates, and the corresponding series for the limiting coordinates diverges. We describe the limit of $Z_N$ instead through its mean and its fluctuation. Let $\psi_N(s)=\E Z_N(s)=N^{1/3}\bar h(bs)$. Then $\psi_N$ converges to the mean function $\psi_R$ (\cref{cd-mean}), which is determined by its Laplace transform for $p>0$: \[ \int_0^\infty e^{-ps}\psi_R(s)\,ds=\frac1{2\Lambda_R(p)},\qquad \Lambda_R(p)=\frac12\sum_{i\ge1}\Big(\frac1{R_i}-\frac1{R_i+p/4}\Big). \] The mean function solves a Volterra equation that is the rescaled limit of the renewal equation for $\bar h$ (\cref{sec:ov-hot-comparison}), and in this sense it is the rescaled limit of the scalar equation of Cugliandolo and Dean. Note that $\sqrt s\,\psi_R(s)\to\psi_*\in(0,\infty)$ as $s\downarrow0$. The fluctuation $Z_N-\psi_N$ converges as well, and the limit $(U,Z)$ solves the countable system \eqref{cd-limiting-system}: \[ \begin{gathered} U_i(s)=\sqrt2\int_0^se^{-2R_i(s-v)}\sqrt{Z(v)}\,dB_i(v)\quad(i\ge1),\qquad Z=\psi_R+V,\\ V(s)=\sqrt8\sum_{i\ge1}\int_0^sk_i(s-v)\sqrt{Z(v)}\,U_i(v)\,dB_i(v), \end{gathered} \] with the explicit kernels $k_i$ of \eqref{cd-H-limit}. This is the critical limit process of \cref{cd-functional}. We show there that its law is unique and that finitely many rescaled eigencoordinates converge to the coordinates $\xi_i=U_i/\sqrt Z$ of the normalized critical limit process; see \eqref{cd-angular-functional}. Uniqueness comes from a Picard iteration for a system in which the noise amplitude $\sqrt{2Z}$ and a weighted norm of the coordinates are truncated (\cref{cd-clipped-picard}), and the moment bounds that we need for tightness come from \cref{cd-heat-profiles,cd-lift-moments}. A stochastic integral is not a continuous function of its integrand and its driving path. We identify the stochastic integrals of a limit point through the joint laws of the rescaled processes, by approximating them with Riemann sums (\cref{lem:cproc-joint-law-identification}). The start is singular, and this is how the uniform initial distribution enters the limit. Every coordinate starts at zero, while $Z(s)\sim\psi_*s^{-1/2}$ and the weighted norm $\sum_iU_i(s)^2/R_i$ tends to $2$ as $s\downarrow0$ (\cref{rem:cproc-entrance}). In finite dimension, this weighted norm is a martingale whose initial value $N^{-1/3}\sum_ig_i^2/R_{i,N}$ is spread over all coordinates, as it is for a uniform point of $S_N$. The rescaled Gibbs measures converge as well. By \cref{cd-pin}, the first coordinates of $N^{-1/3}x$ under $\pi_{N,c_N}$ converge in total variation to those of $\rho_R$, the Gaussian sequence with variances $(2R_i)^{-1}$ conditioned on the renormalized radial constraint of \cref{def:cond-countable}. At the critical temperature, this law is due to Du and Huang \cite[Section~2.2]{DuHuang2026}. The proof of \cref{cd-profile} compares the normalized critical entrance process with this limiting Gibbs law. \subsection{Density bounds}\label{sec:ov-critical-density} The second ingredient of the critical window, which we also use along $c_N=1+\epsilon_N$, bounds the density of the time-$t$ law with respect to the Gibbs measure in every space $L^p$, at times when the distance is still close to one. By \cref{thm:dens-main}, under the hypotheses of \cref{def:dens-hyp} on a time interval $[0,S]$, we have for every fixed $p>1$ \begin{equation}\label{eq:ov-density} \log\int_{S_N}\Big(\frac{d(\nu_NP^r_t)}{d\pi_r}\Big)^p\,d\pi_r\le C_pN(1+t)^{-3/2},\qquad 0\le t\le S . \end{equation} Heuristically, the right side counts the coordinates that have not equilibrated at time $t$, namely those with $r_it\lesssim1$, and each of them contributes a bounded amount. In the critical window, the hypotheses hold on $[0,KN^{2/3}]$, and the right side is bounded at times of order $N^{2/3}$ (\cref{cl3-actual-source-density}). With the Poincar\'e constant of order $N^{2/3}$ of \cref{cg-critical-lsi}, we deduce from the $L^2$ bound at time $b$ that $D_N(b+u)\le Ce^{-u/(K_{\mathrm P}b)}$, where $K_{\mathrm P}b$ is an upper bound on the Poincar\'e constant (see \eqref{eq:cprof-l2-upper}). This bound gives the behavior of the critical profile at infinity (\cref{cg-source-l2-decay}). In the proof, we differentiate the R\'enyi divergence of the time-$t$ law with respect to a moving reference measure, the Bingham comparison law $\mu_{\gamma t}$ at an earlier time $\gamma t$ (\cref{lem:dens-derivative}). R\'enyi divergences were introduced in \cite{Renyi1961}, and Vempala and Wibisono computed their derivative along the Langevin dynamics when the reference measure is stationary \cite[Lemmas~5 and~6]{VempalaWibisono2019}; here the reference measure moves. A logarithmic Sobolev inequality for $\mu_{\gamma t}$ with constant of order $1+t$ (\cref{cg-critical-lsi}) bounds the dissipation from below. The error terms are quadratic forms in the coordinates with small coefficients, because a term common to all coordinates cancels exactly (\cref{lem:dens-coefficients}). We compute their exponential moments under $\mu_{\gamma t}$ exactly from \cref{gp-real-tilt} (\cref{lem:dens-tilt}). A scalar differential inequality closes the argument (\cref{cg-renyi-differential-inequality}). The logarithmic Sobolev inequalities come from \cref{sec:inequalities}. Extending functions on the sphere to Gaussian space, we find logarithmic Sobolev inequalities with an additive error term (\cref{cg-canonical-tube-entropy}). Combined with Rothaus's lemma \cite{Rothaus1985} and a Poincar\'e inequality, they imply a logarithmic Sobolev inequality as soon as three variances are comparable to the largest one (\cref{cg-three-soft-lsi}). The Poincar\'e inequalities are proved separately when many precisions are comparable to the smallest one (\cref{cg-many-soft-pi}), and when only a few precisions are close to the smallest one, which is the case of the Gibbs measure in the critical window (\cref{cg-folded-pi,cg-full-sphere-bridge}). \subsection{Convergence of the distance}\label{sec:ov-critical-profile} Let $\mu^\infty_s$ be the law of $\xi(s)$ on $\R^{\N}$, and let $F_R(s)=\|\mu^\infty_s-\rho_R\|_{\TV}$. Let $(r_{\cdot,N})$ be a deterministic sequence of half-precisions that satisfies the critical spectral conditions and converges to $R$. We prove in \cref{cd-profile} that $D_N(bs)\to F_R(s)$ uniformly on compact subsets of $(0,\infty)$, and that $F_R$ is continuous and strictly decreasing, with $F_R(0+)=1$ and $F_R(\infty)=0$. The obstacle is again that $D_N$ involves all coordinates: the weak convergence of finitely many coordinates of \cref{cd-functional} controls neither the remaining coordinates nor the total variation distance of the first ones. We first prove convergence at every fixed positive time (\cref{cd-pointwise-profile}), in two steps. \paragraph{The remaining coordinates.} By \cref{cd-averaged-tail}, for every $s_0>0$ we have $0\le D_N(bs)-D_N^{\le J}(bs)\le CJ^{-1/3}$ for all $s\ge s_0$, where $D_N^{\le J}$ is the distance between the laws of the first $J$ coordinates and $C$ depends on $s_0$. The argument is as follows. The density of the time-$t$ law with respect to the Gibbs measure is bounded in $L^2$ at $t=s_0b/2$ (\cref{cl3-actual-source-density}). The smoothing of the finite-dimensional semigroup then bounds its energy at every time $bs$ with $s\ge s_0$ (\cref{cd-finite-smoothing}). Next, the conditional law of the remaining coordinates given the first $J$ satisfies a logarithmic Sobolev inequality with constant $Cb/(\gamma J^{2/3})$, uniformly over all conditioning values (\cref{cd-tail-lsi}); we deduce it from \cref{cg-many-soft-pi,cg-three-soft-lsi} on spheres of arbitrary radius (\cref{lem:cprof-radius}). The resulting Poincar\'e inequality bounds the conditional variance of the density by a multiple of $J^{-2/3}$ times its energy, and we obtain the bound $J^{-1/3}$ by comparing the $L^1$ and $L^2$ norms. Here $J$ is fixed before $N\to\infty$. \paragraph{The first coordinates.} For fixed $J$ and $s>0$, the total variation error between the first $J$ rescaled coordinates under $\nu_NP_{bs}$ and their limiting law tends to zero (\cref{cd-head-current}). Weak convergence holds by \cref{cd-functional}. To upgrade it, we integrate by parts under the Gibbs measure along vector fields that move the first $J$ coordinates and compensate in the remaining ones. The divergences of these fields are bounded in $L^2(\pi_N)$ through inverse moments of a block of coordinates, which follow from the Gaussian domination of the Gibbs measure (\cref{cd-pin,cd-stationary-fields}). Together with the density and energy bounds above, this gives a bound on translations of the first-coordinate law that is uniform in $N$ and in $s\ge s_0>0$. Mollification and weak convergence then give convergence in total variation at each fixed $s$. Since total variation on $\R^{\N}$ is the limit of the distances between the first $J$ coordinates (\cref{lem:cprof-product-tv}), we combine this convergence with the bound on the remaining coordinates, let $N\to\infty$, and then let $J\to\infty$. This proves $D_N(bs)\to F_R(s)$ for every $s>0$ (\cref{cd-pointwise-profile}). \paragraph{Continuity, positivity, and strict decrease.} We prove these properties of $F_R$ with estimates for the finite-dimensional semigroups that are uniform in $N$, and we do not need a semigroup for the limiting process. By the spectral theorem, $\|(P_{bh}-I)P_{bt}u\|_{L^2(\pi_N)}\le h\|u\|_{L^2(\pi_N)}/(et)$ for $t>0$ and $h\ge0$ (\cref{cd-finite-smoothing}). We apply this bound to the density at time $s_0b/2$, and we find that $\|\nu_NP_{bt}-\nu_NP_{bs}\|_{\TV}\le L|t-s|$ for $s,t\ge s_0$, where $L$ does not depend on $N$. This bound passes to the limit through the weak convergence of the first $J$ coordinates and \cref{lem:cprof-product-tv}, and $F_R$ is locally Lipschitz (\cref{lem:cprof-limit-continuity}). The exponential lower bound on $D_N$ of \cref{sec:ov-critical-goe} passes to $F_R$ by the convergence at every fixed time, and $F_R$ is positive. For strict decrease, suppose that $F_R(s_1)=F_R(s_2)$ for some $s_10$ (\cref{cd-finite-correlation}). On the time scale $b$, the spectral gap of \cref{cg-critical-lsi} is bounded below uniformly in $N$, and the sums of the absolute values of the other spectral coefficients are bounded. An argument with Chebyshev polynomials then forces $F_R(s_1)=0$, which contradicts positivity (\cref{cd-vanishing-correlations}). Since $D_N(bs)$ is nonincreasing in $s$ and converges at every $s$ to the continuous function $F_R$, a finite grid of times gives the uniform convergence on compact sets (\cref{cd-profile}). We record the order in which the parameters are chosen in \cref{tab:cprof-order}. \subsection{Lower bounds and the GOE}\label{sec:ov-critical-goe} The lower bounds on the critical time scale do not use the profile. A finite-dimensional estimate at a short critical time shows that $D_N(s_*b)\ge3/4$ (\cref{cg-entrance-lower}). For a reversible semigroup, a lower bound at time $2\ell$ and a bound on the $L^4$ norm of the density at time $\ell$ imply an exponential lower bound at all times (\cref{cl3-density-two-time-lower}). With \cref{cl3-actual-source-density}, we find that $D_N(t)\ge c_1e^{-C_1t/b}$ for all $t\ge0$. Together with the upper bound \eqref{eq:cprof-l2-upper}, this gives $c_2b\log(1/\eps)\le t_N(\eps)\le C_2b\log(1/\eps)$ for every $\eps\in(0,1/2)$, including accuracies that depend on $N$ (\cref{cl3-density-clock}). Combining the lower bound with the pointwise convergence \eqref{eq:cprof-pointwise} proved in \cref{cd-profile}, we have $F_R(s)\ge c_1e^{-C_1s}$ (\cref{critical-finite-time-positivity}). For the GOE, the half-precisions satisfy the critical spectral conditions on events of probability at least $1-\eta$, uniformly for $\sigma$ in compact sets, and the spectral data converge jointly in distribution to their limits, which are functions of the reflected GOE Airy point process $(\chi_i)$, the limit of $N^{2/3}(2-\lambda_i)$, for finitely many values of $\sigma$ at once (\cref{cg-canonical-weyl}). The finite and limiting profiles and mixing times are Borel functions of the spectral data (\cref{lem:cprof-finite,lem:cprof-measurable}). The maps from the spectral data to the finite profiles depend on $N$. We control them only along convergent sequences of spectral data that satisfy the critical spectral conditions with fixed constants (\cref{cd-profile,cd-quantiles}). A localized form of the continuous mapping theorem (\cref{lem:cprof-varying-map}), applied on the events where these conditions hold, then gives \cref{cd-goe-profile}, jointly with the spectral data and for finitely many temperatures at once. The limit is a function of the Airy point process arising from the distributional edge limit, and the critical mixing profile is random. Since $F_\sigma^{-1}(\eps)/F_\sigma^{-1}(1-\eps)>1$ almost surely, there is no cutoff. \section{The critical mixing profile far from the critical temperature}\label{sec:ov-outer} \Cref{thm:intro-outer} concerns the iterated limit in which $N\to\infty$ first, at fixed $\sigma=\mp S$, and then $S\to\infty$. The mesoscopic theorems give no information here, since they assume $N\delta_N^3\to\infty$ or $m_N\to\infty$, and the critical theorem holds at fixed $\sigma$. The link between the regimes is that several estimates of \cref{part:tools,part:hot,part:cold} hold at every finite $N$, with constants that depend only on the spacing of the precisions, and not on $N$ or on the size of the effective parameter, once this parameter exceeds a fixed threshold. We use them for finite sequences that approximate the limiting rates of the critical window: we let $N\to\infty$ with the effective parameter fixed, and only afterwards let it tend to infinity. \paragraph{Finite approximations.} Fix a realization of the Airy point process outside a null set. For $\xi>0$ and all sufficiently large $N$ depending on $\xi$, we construct in \cref{oc-approximants} the precisions \[ p_{i,N}=N^{-2/3}(\Delta_i+\xi)\quad(i\le N/2),\qquad p_{i,N}=\gamma_NN^{-2/3}(\Delta_i+\xi)\quad(i>N/2), \] with $\gamma_N\in(1,3]$ chosen such that $\sum_ip_{i,N}^{-1}=N$. The smallest precision is $\xi N^{-2/3}$, so that $M=Np_{1,N}^{3/2}=\xi^{3/2}$ does not depend on $N$, and the spacing constants do not depend on $N$ or $\xi$. The half-precisions $p_{i,N}/2$ satisfy the critical spectral conditions of \cref{def:spec-critical-conditions} and converge to $R=R^{\mathscr S(\xi)}$, and by \cref{cd-profile} their distances converge to $F_{\mathscr S(\xi)}$. \paragraph{Far above the critical temperature.} In \cref{sec:outer-hot}, we take $\xi=\alpha^2$ with $\alpha$ large. For these approximations, we bound the three errors of \cref{sec:ov-hot} at finite $N$, with constants that do not depend on $N$ or $\alpha$. At the time $t=w/d_N$, the comparison error is at most $Ce^{-w}$ by \eqref{eq:ov-global-bridge}, and the relative errors are at most $Cwe^{-2w}$ by \cref{hi3-information-envelopes}. Further, $\eta_q$ is of order $\alpha^{-3/2}$, so that \cref{gp-pinned-lan-quantitative-third} applies. The information converges as $N\to\infty$: by \eqref{eq:ov-info-formula} and \cref{cd-mean}, we have $\cI_N(N^{2/3}s)\to\psi''(2s)/(8\psi(s)^2)$, where $\psi$ is the mean function of the limit (\cref{oc-information-limit}). As $\alpha\to\infty$, the transient part of $\psi$ is the Laplace transform of a positive measure with an explicit real resolvent, whose asymptotics follow from \cref{pr-real-measure-transfer} (\cref{oc-derivative-heat}). At the times $(w^*_\alpha+u)/\alpha^2$ with $w^*_\alpha=\frac14(\log\alpha^3-\frac32\log\log\alpha^3-\log(16\sqrt\pi))$, the limiting information tends to $e^{-4u}$. We let $N\to\infty$ at fixed $\alpha$, then $\alpha\to\infty$, and finally pass from $\alpha$ to $S$ through $\alpha^2=\varrho_{-S}$. This proves \cref{oc-full-hot}; the order of the choices is in \cref{t:outh-choices}. \paragraph{Far below the critical temperature.} In \cref{sec:outer-cold}, we take $\xi=\varrho_S$, which tends to zero as $S\to\infty$. By \cref{cot-deficit}, the scaled gaps $a_{i,N}=(p_{i,N}-p_{1,N})/2$ are admissible in the sense of \cref{def:coldm-admissible}, with constants that do not depend on $S$, while $N^{2/3}a_{2,N}=\Delta_2/2$, $N^{1/3}\theta_N\to S-\Xi$, and $m_N\to(S-\Xi)\Delta_2/2$. At fixed $S$, the parameter $m_N$ has a finite limit, and \cref{ccf-thm:full-profile} does not apply, but the uniform statement \cref{ccf-cor:uniform-envelope} does. For every sufficiently large $L$ and all $S$ with $(S-\Xi)\Delta_2/2>L$, it implies that the error at fixed $S$ is at most $E_I(L)$ in the limit $N\to\infty$, where $E_I(L)\to0$ as $L\to\infty$. The critical theorem identifies the limit of the distance at the corresponding times with $F_S(\tau^{\mathrm c}_S+2u/\Delta_2)$. Letting $S\to\infty$ and then $L\to\infty$ proves \cref{cot-transfer}. \section{Dependencies}\label{sec:ov-dependencies} In \cref{tab:ov-dependencies}, we list for each section its main results and the earlier sections whose results it uses. Every section uses the conventions of \cref{sec:model}, which we do not list. \begin{table}[ht] \centering \small \renewcommand{\arraystretch}{1.2} \begin{tabular}{@{}l>{\raggedright\arraybackslash}p{0.5\textwidth}>{\raggedright\arraybackslash}p{0.27\textwidth}@{}} \toprule Section & Main results & Uses Sections\\ \midrule \ref{sec:spectral} & random-matrix inputs; the Lagrange multiplier & ---\\ \ref{sec:conditioning} & Gaussian laws conditioned on a sphere & ---\\ \ref{sec:likelihood} & \cref{gp-pinned-lan-quantitative-third,gp-pinned-lan} & \ref{sec:conditioning}\\ \ref{sec:heat} & covariance evolution; \cref{mc-exact-info,hi3-information-envelopes,cd-heat-profiles} & \ref{sec:conditioning}\\ \ref{sec:inequalities} & \cref{sgl-semigroup-gradient,ht-many-soft-sharp-gaps,cg-critical-lsi} & \ref{sec:spectral}, \ref{sec:conditioning}, \ref{sec:heat}\\ \ref{sec:comparison} & \cref{pc-centered-defect-transfer,ht-bingham-bridge,rp-general-flux} & \ref{sec:conditioning}, \ref{sec:heat}, \ref{sec:inequalities}\\ \ref{sec:process} & \cref{lem:proc-normalization,iw-chemical-occupation,hmr-main,cd-lift-moments} & \ref{sec:conditioning}, \ref{sec:heat}, \ref{sec:inequalities}\\ \ref{sec:scores} & \cref{csr-thm:reset,csh-thm:weak-to-tv,cce-thm:consumer} & ---\\ \ref{sec:density} & \cref{thm:dens-main,cl3-actual-source-density} & \ref{sec:conditioning}, \ref{sec:heat}, \ref{sec:inequalities}\\ \midrule \ref{sec:hot-fixed} & \cref{thm:intro-main}(2) & \ref{sec:spectral}, \ref{sec:likelihood}, \ref{sec:heat}, \ref{sec:comparison}\\ \ref{sec:hot-meso} & \cref{thm:intro-main}(3) & \ref{sec:spectral}--\ref{sec:heat}, \ref{sec:comparison}, \ref{sec:process}\\ \ref{sec:infinite} & \cref{thm:intro-main}(1), \cref{thm:intro-bounded} & \ref{sec:spectral}, \ref{sec:heat}, \ref{sec:comparison}, \ref{sec:hot-fixed}\\ \midrule \ref{sec:cold-fixed} & \cref{thm:intro-main}(5) & \ref{sec:spectral}, \ref{sec:process}, \ref{sec:scores}\\ \ref{sec:cold-meso} & \cref{thm:intro-main}(4) & \ref{sec:spectral}, \ref{sec:inequalities}, \ref{sec:process}--\ref{sec:density}, \ref{sec:cold-fixed}\\ \midrule \ref{sec:crit-process} & the critical limit process; \cref{cd-pin} & \ref{sec:spectral}, \ref{sec:conditioning}, \ref{sec:heat}, \ref{sec:process}\\ \ref{sec:crit-profile} & \cref{thm:intro-critical} & \ref{sec:spectral}, \ref{sec:conditioning}, \ref{sec:heat}, \ref{sec:inequalities}, \ref{sec:process}, \ref{sec:density}, \ref{sec:crit-process}\\ \ref{sec:outer-hot} & \cref{thm:intro-outer}, first part & \ref{sec:spectral}, \ref{sec:likelihood}, \ref{sec:heat}, \ref{sec:hot-meso}, \ref{sec:crit-process}, \ref{sec:crit-profile}\\ \ref{sec:outer-cold} & \cref{thm:intro-outer}, second part & \ref{sec:spectral}, \ref{sec:conditioning}, \ref{sec:cold-meso}, \ref{sec:crit-profile}, \ref{sec:outer-hot}\\ \bottomrule \end{tabular} \caption{Main results of each section and the earlier sections that it uses, besides \cref{sec:model}. Sections~\ref{sec:conditioning}--\ref{sec:density} form \cref{part:tools}.} \label{tab:ov-dependencies} \end{table}