\chapter{Introduction}\label{sec:intro} The spherical Sherrington--Kirkpatrick model is the simplest mean-field spin glass. Its configurations are points $x$ of the sphere $S_N=\{x\in\R^N:|x|^2=N\}$, and its energy is a quadratic form in $x$ whose coefficients form a random symmetric matrix, so that its equilibrium can be analyzed exactly through the spectrum of that matrix. The Langevin diffusion on $S_N$ that is reversible with respect to the Gibbs measure has been studied in physics for more than forty years, mostly through the evolution of observables such as the energy and two-time correlation functions in the limit of infinite system size. The goal of this paper is to describe how the law of this diffusion approaches equilibrium when the dimension $N$ is large and the disorder is fixed. We start the dynamics from the uniform distribution on $S_N$, independently of the disorder, and measure its distance to the Gibbs measure in total variation. We determine the asymptotic behavior of this distance as a function of time at fixed temperatures above and below the critical one, in the crossover to infinite temperature, along every mesoscopic approach to the critical temperature from either side, and in the critical window. In the regimes treated here outside the critical window we find cutoff, except in the part of the crossover to infinite temperature where $N\beta^2$ stays bounded; there the distance relaxes over bounded times. In each case we identify the center and the window of the transition and its limiting shape, which we call the mixing profile. In the critical window the mixing profile is random and determined by the edge of the spectrum of the disorder, and there is no cutoff. We also show that the critical mixing profiles match those of the two mesoscopic regimes far from the critical temperature. \section{The model and its Langevin dynamics}\label{sec:intro-background} \paragraph{Spherical models.} Berlin and Kac introduced the spherical model of a ferromagnet in \cite{BerlinKac1952}, in which the Ising spins of a lattice model are replaced by real variables subject to a single constraint on the sum of their squares. Lewis and Wannier considered the mean spherical model, in which this constraint holds only in expectation \cite[(1)--(2), p.~683]{LewisWannier1952}, and Stanley identified the spherical model with the limit of classical vector spin models as the number of spin components tends to infinity \cite{Stanley1968}. Sherrington and Kirkpatrick introduced the mean-field Ising spin glass with independent Gaussian couplings \cite{SherringtonKirkpatrick1975}. Kosterlitz, Thouless, and Jones introduced its spherical version, whose couplings form a random symmetric matrix, and solved its thermodynamics through the spectrum of that matrix \cite{KosterlitzThoulessJones1976}. We use the following normalization. Let $W_N$ be an $N\times N$ matrix from the Gaussian orthogonal ensemble (GOE), that is, a real symmetric matrix whose entries on and above the diagonal are independent with $W_{ij}\sim\cN(0,(1+\ind_{\{i=j\}})/N)$ for $i\le j$. Let $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_N$ be its eigenvalues, and let $\nu_N$ be the uniform probability measure on $S_N$. For an inverse temperature $\beta\ge0$ we define the Hamiltonian and the Gibbs measure by \begin{equation}\label{eq:intro-gibbs} H_N(x)=\frac{x^\T W_Nx}{\sqrt2},\qquad \pi_{N,c}(dx)=\frac1{Z_{N,c}}e^{\beta H_N(x)}\,\nu_N(dx)=\frac1{Z_{N,c}}e^{(c/2)x^\T W_Nx}\,\nu_N(dx), \end{equation} where $Z_{N,c}$ is the normalizing constant and $c=\sqrt2\,\beta$. Throughout the paper we parameterize the temperature by the rescaled inverse temperature $c$, so that increasing $c$ lowers the temperature. The critical value is $c=1$, that is, $\beta_c=1/\sqrt2$. Since its density with respect to $\nu_N$ is the exponential of a quadratic form, $\pi_{N,c}$ is a Bingham distribution \cite{Bingham1974}. Because $-W_N$ has the same law as $W_N$, negative inverse temperatures lead to the same results with the roles of the top and the bottom of the spectrum exchanged, and it suffices to consider $\beta\ge0$. We work in an orthonormal eigenbasis of $W_N$ and write $x_1,\dots,x_N$ for the coordinates in this basis, the eigencoordinates. The Gibbs measure has the following Gaussian representation. Let $z>c\lambda_1$ be the unique solution of \begin{equation}\label{eq:intro-multiplier} \frac1N\sum_{i=1}^N\frac1{z-c\lambda_i}=1, \end{equation} and set $p_i=z-c\lambda_i$ and $q_i=1/p_i$. Since $-\sum_ip_ix_i^2/2=-zN/2+(c/2)\sum_i\lambda_ix_i^2$ on $S_N$, the density of a centered Gaussian vector with independent coordinates of variances $q_1,\dots,q_N$ is proportional to the Gibbs density on $S_N$. The gradient of $|x|^2$ has constant length on $S_N$, and it follows that $\pi_{N,c}$ is the law of this Gaussian vector conditioned on $|x|^2=N$, where the choice of $z$ in \eqref{eq:intro-multiplier} makes the expected squared norm of the unconditioned vector equal to $N$. The numbers $z$, $p_i$, and $q_i$ are the Lagrange multiplier, the precisions, and the variances of \cref{def:model-multiplier}. Equation \eqref{eq:intro-multiplier} is the saddle-point equation in the solution of Kosterlitz, Thouless, and Jones \cite{KosterlitzThoulessJones1976}; see \cite[Lemma~1.3]{BaikLee2016} for a modern account of their contour-integral formula for $Z_{N,c}$. The representation of Bingham distributions by conditioned Gaussian vectors is classical in directional statistics \cite{KumeWood2005}, and for spherical models it appears in \cite[Proposition~2.7]{AruKorzhenkova2024} and, in the critical regime of the present model, in \cite[Section~1.2]{DuHuang2026}. As $N\to\infty$, the empirical spectral measure of $W_N$ converges to the semicircle law $\rho_{\mathrm{sc}}(\lambda)\,d\lambda=(2\pi)^{-1}\sqrt{4-\lambda^2}\,d\lambda$ on $[-2,2]$ \cite{Wigner1955,Wigner1958}; for convergence in probability in the present normalization, see \cite[Theorem~2.1.1, p.~7]{AndersonGuionnetZeitouni2010}. Also, $\lambda_1$ converges to $2$ by the edge estimates recalled in \cref{sp-goe-rigidity}. For $c<1$ we have $z\to1+c^2$ and the smallest precision $d=p_1=z-c\lambda_1$ converges to $(1-c)^2$, so that all variances $q_i$ remain bounded. For $c>1$ we have $d\to0$, and the first eigencoordinate carries a macroscopic fraction of the squared norm under $\pi_{N,c}$, in the sense that $x_1^2/N\to1-1/c$ in probability. This is the condensation of the spherical model onto the top eigenvector \cite{KosterlitzThoulessJones1976}, \cite[Section~III]{CugliandoloDean1995}. At low temperature the Gibbs measure is therefore concentrated on two antipodal regions in which $x_1$ is close to $\sqrt{(1-1/c)N}$ or to $-\sqrt{(1-1/c)N}$. The free energy has a phase transition at $c=1$ \cite{KosterlitzThoulessJones1976}; we refer to \cite{CrisantiSommers1992} for the replica analysis of general spherical $p$-spin models and to \cite{Talagrand2006} for a proof of the free-energy formula for spherical models with even interactions, which include the present one. \paragraph{Langevin dynamics.} The Langevin dynamics of the model is the diffusion on $S_N$ with generator \begin{equation}\label{eq:intro-generator} L=\Delta_{S_N}+\beta\,\nabla_{S_N}H_N\cdot\nabla_{S_N}, \end{equation} where $\Delta_{S_N}$ and $\nabla_{S_N}$ are the Laplace--Beltrami operator and the gradient on $S_N$. Its noise coefficient is $\sqrt2$, and it is reversible with respect to $\pi_{N,c}$. We write $(P_t)_{t\ge0}$ for its semigroup, so that $\nu_NP_t$ is the law at time $t$ of the diffusion started from the uniform distribution. In the eigenbasis the drift is linear apart from the term that keeps the process on the sphere. However, the noise is projected onto the tangent space of $S_N$, and for finite $N$ the eigencoordinates are coupled and are not independent Ornstein--Uhlenbeck processes. This dynamics has a long history in physics. Shukla and Singh studied relaxation in the spherical spin-glass model \cite{ShuklaSingh1981}. Ciuchi and de Pasquale studied nonlinear relaxation and ergodicity breaking in random anisotropy spin glasses with many spin components \cite{CiuchiDePasquale1988}. Cugliandolo and Dean solved the dynamics completely in the thermodynamic limit, for quenches from high temperature and for more general temperature protocols \cite{CugliandoloDean1995}. They diagonalized the coupling matrix and reduced the spherical constraint to a scalar Volterra equation for a time-dependent Lagrange multiplier \cite[(II.2)--(II.7)]{CugliandoloDean1995}, and from its solution they computed correlation and response functions and showed that the dynamics ages at low temperature. Zippold, K\"uhn, and Horner refined the time scales that separate stationary relaxation from aging and compared the model with spherical ferromagnets \cite[Sections~III--IV]{ZippoldKuhnHorner2000}. Cugliandolo and Kurchan developed a microscopic aging analysis for spherical $p>2$ spin glasses after taking the thermodynamic limit \cite[Introduction]{CugliandoloKurchan1993}, and the quadratic model behaves like a coarsening system \cite[Section~IV.C]{CugliandoloKurchanPeliti1997}. Caltagirone, Parisi, and Rizzo computed the off-equilibrium correlations at the critical temperature \cite[Section~III]{CaltagironeParisiRizzo2013}. On the mathematical side, Ben Arous, Dembo, and Guionnet proved aging for the Langevin dynamics of the spherical Sherrington--Kirkpatrick model with a soft spherical constraint \cite{BenArousDemboGuionnet2001}. The same authors derived rigorously the Cugliandolo--Kurchan equations for the correlation and response functions of spherical spin glasses with a soft spherical constraint \cite[Theorem~1.2]{BenArousDemboGuionnet2006}. Dembo, Guionnet, and Mazza obtained the limit of these equations under the hard spherical constraint \cite[Proposition~1.1]{DemboGuionnetMazza2007}, and proved that at sufficiently high temperature the solutions converge to a stationary pair that satisfies the fluctuation-dissipation relation and decays exponentially \cite[Theorem~1.3]{DemboGuionnetMazza2007}. These analyses take the thermodynamic limit first and then study the resulting correlation and response functions, including their long-time behavior. Finite-size effects enter through the edge of the spectrum. Rodgers and Moore identified the first-gap barrier between the two antipodal low-temperature regions \cite[Section~2, (25)--(26)]{RodgersMoore1989}; see also \cite[Section~4]{FyodorovPerretSchehr2015}. Fyodorov, Perret, and Schehr showed that at zero temperature the relaxation crosses over at times of order $N^{2/3}$ from the algebraic decay of the thermodynamic limit to an exponential decay governed by the random gap $\lambda_1-\lambda_2$, in a regime where observables are no longer self-averaging \cite[Section~3.1]{FyodorovPerretSchehr2015}. Barbier, de Freitas Pimenta, Cugliandolo, and Stariolo studied these finite-size effects further, including the escape from initial conditions concentrated near saddle points \cite{BarbierDeFreitasPimentaCugliandoloStariolo2021}. For gradient descent on the sphere started from the uniform distribution, which is the dynamics at zero temperature, Yu showed that the time needed to reach a fixed overlap with an extreme eigenvector lies between orders $N^{2/3}$ and $N^{2/3}\log N$ with probability at least $1-\delta$, for each fixed $\delta>0$ \cite[Section~3, Theorem~3.3]{Yu2024}. That result concerns a hitting time, while our results concern the law of the diffusion at positive temperature in total variation. In an approximation that neglects finite-size fluctuations of the spectral density, Cugliandolo, Dean, and Yoshino found a relaxation time of order $N(\beta-\beta_c)$ for the linear susceptibility at low temperature \cite[Sections~1.1 and~4.3]{CugliandoloDeanYoshino2007}. \paragraph{The question.} In this paper we study the law of the diffusion itself at finite $N$. We start the diffusion from $\nu_N$, independently of $W_N$, and measure its distance to equilibrium by \begin{equation}\label{eq:intro-distance} D_N(t)=\|\nu_NP_t-\pi_{N,c}\|_{\TV},\qquad\text{where }\|\mu-\eta\|_{\TV}=\sup_A|\mu(A)-\eta(A)| \end{equation} and the supremum is over Borel sets $A\subset S_N$. We ask how $D_N(t)$ behaves as a function of $t$ when $N$ is large, and how the answer depends on the temperature. We comment on three features of this formulation. First, total variation controls the law of the entire configuration, so that once $D_N(t)$ is small the law of every measurable observable is close to its equilibrium law, including observables that depend on all $N$ coordinates; it is also the standard distance in the theory of mixing times \cite{LevinPeres2017}. Second, the uniform distribution is the Gibbs measure at $\beta=0$, and the dynamics started from $\nu_N$ describes a quench from infinite temperature. In the eigenbasis a uniform point has coordinates $\sqrt N\,g_i/|g|$, where $g_1,\dots,g_N$ are independent standard Gaussian variables. Several of the computations cited above instead model the quench by giving every eigenmode the same deterministic initial amplitude \cite[(III.1)]{CugliandoloDean1995}, \cite[(6)]{FyodorovPerretSchehr2015}, \cite[(12)]{BarbierDeFreitasPimentaCugliandoloStariolo2021}. For a fixed matrix this is a different initial condition, and \cref{sec:intro-discussion} explains why the difference matters at low temperature. Third, the law $\nu_NP_t$ is averaged over the initial point and the Brownian motion but not over the disorder, so that $D_N(t)$ is a random variable that depends on $W_N$ only, and our results are statements in probability or in distribution with respect to $W_N$. \section{Cutoff and limit profiles}\label{sec:intro-cutoff} Diaconis and Shahshahani \cite{DiaconisShahshahani1981} and Aldous \cite{Aldous1983} found the cutoff phenomenon in their analysis of card shuffling and random walks on groups, and Aldous and Diaconis named it \cite{AldousDiaconis1986}; we refer to \cite{Diaconis1996} and \cite[Chapter~18]{LevinPeres2017} for surveys. Given a sequence of Markov processes, let $t_N(\varepsilon)$ be the first time at which the distance to equilibrium is at most $\varepsilon$. Ratios of mixing times are set to $1$ when their denominator is zero. In the deterministic setting, there is cutoff if $t_N(1-\varepsilon)>0$ for all sufficiently large $N$ and $t_N(\varepsilon)/t_N(1-\varepsilon)\to1$ for every $\varepsilon\in(0,1/2)$, which means that the distance drops from close to one to close to zero over a window of time that is much shorter than the mixing time. A finer question asks for a limit profile, which consists of a center $T_N$, a window $w_N$ with $w_N/T_N\to0$, and a function $P$ such that the distance at time $T_N+w_Nu$ converges to $P(u)$ for every $u\in\R$. Limit profiles are known for few processes. Diaconis, Graham, and Morrison studied the random walk on the hypercube $\{0,1\}^n$ that at each step stays put or flips one of the $n$ coordinates, each with probability $1/(n+1)$. They proved in \cite[Theorem~1]{DiaconisGrahamMorrison1990} that its distance to the uniform distribution at time $(n\log n)/4+un$ converges to \begin{equation}\label{eq:intro-Psi} \Psi(u)=2\Phi\bigl(e^{-2u}/2\bigr)-1, \end{equation} where $\Phi$ is the standard normal distribution function. The function $\Psi(u)$ is the total-variation distance between $\cN(e^{-2u},1)$ and $\cN(0,1)$, and it is also the profile in our results at high temperature. Bayer and Diaconis found the profile of the riffle shuffle \cite[Theorem~4]{BayerDiaconis1992}, and Lacoin found that of the simple exclusion process on the circle \cite[Theorem~2.1]{Lacoin2016}. Teyssier showed that the profile of random transpositions is a total-variation distance between two Poisson laws \cite[Theorem~1.1]{Teyssier2020}, and Nestoridi and Olesker-Taylor developed a general method for the profiles of reversible Markov chains \cite{NestoridiOleskerTaylor2022}. Cutoff is also known for many diffusions. Saloff-Coste \cite[Theorems~2.1 and~4.2]{SaloffCoste1994} and M\'eliot \cite[Theorem~6]{Meliot2014} studied Brownian motion on spheres and other compact symmetric spaces, and Lachaud proved cutoff for samples of independent Ornstein--Uhlenbeck processes \cite[Propositions~1--2]{Lachaud2005}. Boursier, Chafa\"i, and Labb\'e proved it for the Dyson Ornstein--Uhlenbeck process \cite{BoursierChafaiLabbe2023} and computed exact profiles for Ornstein--Uhlenbeck processes \cite[Lemma~3.4]{BoursierChafaiLabbe2023}. Barrera and Jara \cite{BarreraJara2016,BarreraJara2020} studied total-variation cutoff for small Gaussian perturbations of attracting dynamical systems, where the profile is obtained through a Gaussian approximation \cite[Theorem~2.2]{BarreraJara2020}. For small L\'evy perturbations, Barrera, H\"ogele, and Pardo treated Ornstein--Uhlenbeck systems in Wasserstein distance \cite[Section~3]{BarreraHogelePardo2021a} and nonlinear Langevin systems with layered stable noise in total variation \cite[Section~1]{BarreraHogelePardo2021b}. Chen and Saloff-Coste gave cutoff criteria for maximum $L^p$ distances with $10)\to1$ for every $\varepsilon\in(0,1/2)$. In every supremum of this kind we restrict to offsets $u$ for which the evaluation time $T_N+w_Nu$ is nonnegative, as in \cref{def:model-cutoff}; equivalently, we set $D_N(t)=D_N(0)$ for $t<0$. Since $w_N/T_N\to0$ in probability, either convention changes nothing in the limit. If $P$ is continuous and decreases strictly from $1$ to $0$, then \cref{hp-monotone-profile-transfer} implies that $(t_N(\varepsilon)-T_N)/w_N\to P^{-1}(\varepsilon)$ in probability for every $\varepsilon\in(0,1)$, and in particular that there is cutoff. More generally, a mixing profile is a limit of the functions $u\mapsto D_N(T_N+w_Nu)$, or of $s\mapsto D_N(b_Ns)$ for a time scale $b_N$. The critical mixing profiles below are random, and they are not cutoff profiles. Besides the function $\Psi$ in \eqref{eq:intro-Psi}, we need the profiles \begin{equation}\label{eq:intro-V} \begin{aligned} \cV(u)&=\bigl\|\Law(G+e^{-2u}C)-\Law(G)\bigr\|_{\TV},\\ \cV_\theta(u)&=\bigl\|\Law(G+\sqrt{2\theta}\,e^{-2u}C)-\Law(G)\bigr\|_{\TV}, \end{aligned} \end{equation} where $G$ is a standard Gaussian variable, $C$ is an independent standard Cauchy variable, and $\theta>0$. The law of $G+aC$ is the convolution of a Gaussian and a Cauchy law, and its density is a Voigt profile \cite{Voigt1912}, \cite[Section~7.19]{OlverLozierBoisvertClark2010}. Both $\Psi$ and $\cV$ are continuous and decrease strictly from $1$ at $-\infty$ to $0$ at $+\infty$ (\cref{cq-jp-strict-profile-monotonicity-and-exact-accuracy-offsets}), and $\cV_\theta(u)=\cV(u-\log(2\theta)/4)$. Solving $\Psi(u)=\varepsilon$, we find $\Psi^{-1}(\varepsilon)=-\log\bigl(2\Phi^{-1}((1+\varepsilon)/2)\bigr)/2$. At low temperature we use the scaled eigenvalue gaps $a_i$ and the top fraction $\theta_N$ of \cref{sec:model-equilibrium}, given by \begin{equation}\label{eq:intro-gaps} a_i=\frac c2(\lambda_1-\lambda_i),\qquad \theta_N=1-\frac1N\sum_{i=2}^N\frac1{2a_i}. \end{equation} Since $c\lambda_1|x|^2/2$ is constant on $S_N$, we have $\pi_{N,c}(dx)\propto\exp(-\sum_{i\ge2}a_ix_i^2)\,\nu_N(dx)$. A centered Gaussian vector $(x_2,\dots,x_N)$ with variances $(2a_i)^{-1}$ has expected squared norm $(1-\theta_N)N$, which is why $\theta_N$ approximates the squared projection $x_1^2/N$ of $x/\sqrt N$ onto the top eigenvector under $\pi_{N,c}$. For fixed $c>1$ we have $\theta_N\to1-1/c$ in probability, and for $c_N=1+\epsilon_N$ with $\epsilon_N\to0$ and $N\epsilon_N^3\to\infty$ we have $\theta_N/\epsilon_N\to1$ in probability (\cref{sec:spectral}). The convergence \eqref{eq:intro-airy} of the top eigenvalues of $W_N$ to the Airy point process, stated below, implies that $N^{2/3}a_2$ converges in distribution to a positive random variable. At high temperature we use a deterministic evolution of variances, defined for a fixed matrix $W_N$ and $c\ge0$. Let $v_1(t),\dots,v_N(t)$ solve \begin{equation}\label{eq:intro-covariance} v_i'(t)=2+2\bigl(c\lambda_i-z(t)\bigr)v_i(t),\qquad v_i(0)=1,\qquad z(t)=1+\frac1N\sum_{j=1}^Nc\lambda_jv_j(t). \end{equation} These are the variances at time $t$ of the linear system $dx_i=(c\lambda_i-z(t))x_i\,dt+\sqrt2\,dB_i$ with independent standard Gaussian initial values, in which the spherical constraint holds only in expectation because the choice of $z(t)$ keeps $\sum_iv_i(t)=N$. Cugliandolo and Dean solved this system in the thermodynamic limit through the scalar Volterra equation mentioned above \cite[(II.4)--(II.7)]{CugliandoloDean1995}. As $t\to\infty$, $z(t)$ converges to the Lagrange multiplier $z$ and $v_i(t)$ converges to $q_i$. The Gaussian law with variances $v_i(t)$ conditioned on $S_N$ is the Bingham comparison law $\mu_t$ at time $t$, defined in \cref{def:heat-covariance}. Its distance to $\pi_{N,c}$ is governed by the relative errors and by the information of $v(t)$ relative to $q$ (\cref{def:model-information}), \begin{equation}\label{eq:intro-information} e_i(t)=\frac{v_i(t)}{q_i}-1,\qquad \cI_N(t)=\frac12\sum_{i=1}^Ne_i(t)^2. \end{equation} \subsection*{Away from the critical window} Our first result gives the profiles in the regimes treated here outside the critical window. Each of its five cases has its own section, and \cref{tab:intro-regimes} below summarizes the results. \begin{theorem}[Cutoff profiles away from the critical window]\label{thm:intro-main} Suppose that the rescaled inverse temperatures $c_N$ satisfy one of the conditions \textup{(1)}--\textup{(5)} below, and let $P$, $T_N$, and $w_N$ be the profile, the center, and the window given there. Then $w_N/T_N\to0$ in probability, and for every compact set $K\subset\R$ and every $\varepsilon\in(0,1)$, \begin{equation}\label{eq:intro-main} \sup_{u\in K}\bigl|D_N(T_N+w_Nu)-P(u)\bigr|\to0,\qquad \frac{t_N(\varepsilon)-T_N}{w_N}\to P^{-1}(\varepsilon) \end{equation} in probability. In particular, there is cutoff. \begin{enumerate} \item[\textup{(1)}] \emph{Infinite-temperature crossover.} Let $c_N\to0$ and $J_N=Nc_N^2\to\infty$. Then $P=\Psi$, $w_N=1$, and $T_N=\tau_N$, where $\tau_N=\inf\{t\ge0:\cI_N(t)\le1\}$ if $\cI_N(0)>1$ and $\tau_N=1$ otherwise; with probability tending to one, $\cI_N(0)>1$ and $\tau_N$ is the only solution of $\cI_N(t)=1$ in $[0,\infty)$. Here $\cI_N$ is the information \eqref{eq:intro-information} at $c=c_N$. Further, let $h_N=(c_N/4)\log J_N$ and \begin{equation*} \mathcal B(h)=\frac12\int_{-2}^2e^{4h\lambda}\rho_{\mathrm{sc}}(\lambda)\,d\lambda=\frac{I_1(8h)}{8h}, \end{equation*} where $I_1$ is the modified Bessel function of the first kind, and the quotient at $h=0$ is defined by continuous extension, with $\mathcal B(0)=1/2$. If $(h_N)$ is bounded, then $\tau_N=\bigl(\log J_N+\log\mathcal B(h_N)\bigr)/4+o_{\PP}(1)$. If $h_N\to\infty$, then $\tau_N-T_N'\to0$ in probability, where $T_N'$ is the center in \textup{(2)} evaluated at $c=c_N$, and \eqref{eq:intro-main} also holds with $T_N=T_N'$ and $w_N=(1-c_N)^{-2}$. \item[\textup{(2)}] \emph{Fixed high temperature.} Let $c_N=c\in(0,1)$. Then $P=\Psi$, $w_N=1/g$ with $g=(1-c)^2$, and \begin{equation*} T_N=\frac{\log N-\frac32\log\log N+\log A_c}{4g},\qquad A_c=\frac{\sqrt c\,(1-c)^3}{4\sqrt\pi\,(1+c)^2}. \end{equation*} \item[\textup{(3)}] \emph{Approach to the critical temperature from above.} Let $c_N=1-\delta_N$, where $\delta_N\to0$ and $M_N=N\delta_N^3\to\infty$. Then $P=\Psi$, $w_N=\delta_N^{-2}$, and \begin{equation*} T_N=\frac{\log M_N-\frac32\log\log M_N+\log B_{c_N}}{4\delta_N^2},\qquad B_c=\frac{\sqrt c}{4\sqrt\pi\,(1+c)^2}. \end{equation*} \item[\textup{(4)}] \emph{Approach to the critical temperature from below.} Let $c_N=1+\epsilon_N$, where $\epsilon_N>0$, $\epsilon_N\to0$, and $N\epsilon_N^3\to\infty$. Then $P=\cV$, $w_N=a_2^{-1}$, and \begin{equation*} T_N=\begin{cases} \dfrac{\log(2\theta_NNa_2)}{4a_2},&\theta_N>0\text{ and }2\theta_NNa_2>1,\\[4pt] 1,&\text{otherwise}. \end{cases} \end{equation*} \item[\textup{(5)}] \emph{Fixed low temperature.} Let $c_N=c>1$. Then $P$, $w_N$, and $T_N$ are as in \textup{(4)}. Equivalently, with $\theta=1-1/c$, \eqref{eq:intro-main} holds with $\cV_\theta$ in place of $P$ and with $T_N''$ in place of $T_N$, where $T_N''=\log(Na_2)/(4a_2)$ if $Na_2>1$ and $T_N''=1$ otherwise. Further, $a_2t_N(\varepsilon)/\log N\to1/12$ in probability. \end{enumerate} For finitely many fixed temperatures in \textup{(2)} and \textup{(5)}, the conclusions hold simultaneously for the same matrices $W_N$. \end{theorem} The centers $\tau_N$ in case (1) and $T_N$, $T_N''$ in cases (4) and (5) are random. In case (1), $\tau_N$ is the first time at which $\cI_N(t)\le1$ on the event $\{\cI_N(0)>1\}$, and $\tau_N=1$ elsewhere (see \eqref{eq:inf-first-crossing}). On $\{\cI_N(0)>1\}$, we have $\cI_N(\tau_N)=1$, and this solution is unique with probability tending to one. In cases (4) and (5), the events on which the centers are set equal to $1$ have probability tending to zero, so that $T_N=\log(2\theta_NNa_2)/(4a_2)$ and $T_N''=\log(Na_2)/(4a_2)$ with probability tending to one. We prove case (1) in \cref{sec:infinite} (\cref{wf-goe-classification,wf-explicit-clocks}), case (2) in \cref{sec:hot-fixed} (\cref{hp-fixed-profile}), and case (3) in \cref{sec:hot-meso} (\cref{mgf-all-mesoscopic-profile}). Case (4) is proved in \cref{sec:cold-meso} (\cref{ccf-thm:full-profile,thm:coldm-goe,ccf-cor:accuracy-cutoff}), and case (5) in \cref{sec:cold-fixed} (\cref{cq-full-profile}). When $Nc_N^2$ stays bounded, the uniform distribution and the Gibbs measure are not asymptotically singular, and the distance relaxes over a bounded time. \begin{theorem}[Bounded-time relaxation near infinite temperature]\label{thm:intro-bounded} Suppose that $c_N\to0$ and $\sqrt N\,c_N\to a\in[0,\infty)$. Then \begin{equation*} \sup_{t\ge0}\Bigl|D_N(t)-\Bigl(2\Phi\Bigl(\frac{ae^{-2t}}{2\sqrt2}\Bigr)-1\Bigr)\Bigr|\to0 \end{equation*} in probability, and for every $\varepsilon\in(0,1)$, \begin{equation*} t_N(\varepsilon)\to\max\Bigl\{\frac12\log\frac{a}{2\sqrt2\,\Phi^{-1}((1+\varepsilon)/2)},\,0\Bigr\} \end{equation*} in probability, where the limit is zero if $a=0$. If $c_N=0$, then $\nu_N=\pi_{N,0}$ and $D_N(t)=0$ for all $t\ge0$. \end{theorem} \Cref{thm:intro-bounded} is part of \cref{wf-goe-classification}. Its limit is $2\Phi(\sqrt{I(t)}/2)-1$ with $I(t)=a^2e^{-4t}/2$, and $I(t)$ is the limit of the information $\cI_N(t)$ in \eqref{eq:intro-information}, because the initial information satisfies $\cI_N(0)=(Nc_N^2/2)(1+o_{\PP}(1))$ and decays at rate $4+O(c_N)$ (\cref{sec:infinite}). There is no cutoff in this regime. \begin{table}[ht] \centering \renewcommand{\arraystretch}{1.3} \begin{tabular}{@{}llll@{}} \toprule Rescaled inverse temperature & Center $T_N$ (leading order) & Window $w_N$ & Profile\\ \midrule $c_N\to0$, $Nc_N^2\to a^2<\infty$ & \multicolumn{2}{l}{bounded relaxation time, no cutoff} & \cref{thm:intro-bounded}\\ $c_N\to0$, $Nc_N^2\to\infty$ & $\log(Nc_N^2)/4$ & $1$ & $\Psi$\\ $c\in(0,1)$ fixed & $\log N/(4(1-c)^2)$ & $(1-c)^{-2}$ & $\Psi$\\ $c_N=1-\delta_N$, $N\delta_N^3\to\infty$ & $\log(N\delta_N^3)/(4\delta_N^2)$ & $\delta_N^{-2}$ & $\Psi$\\ $c_N=1+\sigma N^{-1/3}$ & order $N^{2/3}$, no cutoff & --- & $F_\sigma$ (random)\\ $c_N=1+\epsilon_N$, $N\epsilon_N^3\to\infty$ & $\log(N\epsilon_N^3)/(12a_2)$ & $a_2^{-1}$ & $\cV$\\ $c>1$ fixed & $\log N/(12a_2)$ & $a_2^{-1}$ & $\cV$\\ \bottomrule \end{tabular} \caption{The regimes of \cref{thm:intro-main,thm:intro-bounded,thm:intro-critical}, from high to low temperature. The centers are given to leading order; the exact centers are in \cref{thm:intro-main}. At low temperature $a_2=c(\lambda_1-\lambda_2)/2$ is of order $N^{-2/3}$, and the center and the window are random at leading order, since $N^{2/3}a_2$ has a nondegenerate limit in distribution.} \label{tab:intro-regimes} \end{table} \subsection*{The critical window} In the critical window $c_N=1+\sigma N^{-1/3}$ we express our results through the Airy point process. Let $\chi_1<\chi_2<\cdots$ be the reflected GOE Airy point process, given by the eigenvalues of the stochastic Airy operator $-d^2/dx^2+x+2b'(x)$ on $L^2(0,\infty)$ with Dirichlet boundary condition at $0$, where $b'$ is white noise \cite[Section~2 and Theorem~1.1]{RamirezRiderVirag2011}. This is one random sequence, and for every fixed $k$ we have \begin{equation}\label{eq:intro-airy} \bigl(N^{2/3}(2-\lambda_i)\bigr)_{i=1}^k\Longrightarrow(\chi_i)_{i=1}^k \end{equation} in distribution \cite[Theorem~1.1]{RamirezRiderVirag2011}. The points $\chi_i$ are almost surely distinct and tend to infinity \cite[Proposition~3.5]{RamirezRiderVirag2011}. We write $\mathrm{TW}_1$ for the law of $-\chi_1$. By \eqref{eq:intro-airy} with $k=1$, $N^{2/3}(\lambda_1-2)$ converges in distribution to $\mathrm{TW}_1$, the GOE Tracy--Widom distribution \cite[Section~I and (53) in Section~VII]{TracyWidom1996}. We set $\Delta_i=\chi_i-\chi_1$ for $i\ge2$. Landon and Sosoe proved that the limit \begin{equation*} \Xi=\lim_{k\to\infty}\Bigl(\sum_{i=2}^k\frac1{\Delta_i}-\frac2\pi\Bigl(\frac{3\pi k}2\Bigr)^{1/3}\Bigr) \end{equation*} exists almost surely and that $N^{1/3}\bigl(N^{-1}\sum_{i\ge2}(\lambda_1-\lambda_i)^{-1}-1\bigr)$ converges in distribution to $\Xi$ \cite[Theorem~6.1]{LandonSosoe2022}. This is the regularized inverse-gap statistic of \cref{sp-raw-gap-resolvent}. Du and Huang showed that the function \begin{equation}\label{eq:intro-saddle} \mathscr S(x)=\Xi+\frac1x+\sum_{i\ge2}\Bigl(\frac1{x+\Delta_i}-\frac1{\Delta_i}\Bigr),\qquad x>0, \end{equation} is almost surely strictly decreasing, with limits $+\infty$ at $0$ and $-\infty$ at $+\infty$ \cite[Proposition~1.4]{DuHuang2026}. For every $\sigma\in\R$ we let $\varrho_\sigma>0$ be the unique solution of $\mathscr S(\varrho_\sigma)=\sigma$, and we set \begin{equation*} R_1^\sigma=\frac{\varrho_\sigma}2,\qquad R_i^\sigma=\frac{\Delta_i+\varrho_\sigma}2\quad(i\ge2). \end{equation*} These numbers are the limiting scaled half-precisions. If $c_N=1+\sigma N^{-1/3}$, then the equation \eqref{eq:intro-multiplier} for the Lagrange multiplier becomes $\mathscr S(\varrho_\sigma)=\sigma$ in the limit, and $N^{2/3}p_i/2$ converges in distribution to $R_i^\sigma$, jointly for finitely many $i$ (\cref{cg-canonical-weyl}). We let $\rho_\sigma$ be the law of a sequence $(\xi_i)_{i\ge1}$ of independent centered Gaussian variables with variances $(2R_i^\sigma)^{-1}$, conditioned on \begin{equation*} \sum_{i\ge1}\Bigl(\xi_i^2-\frac1{2R_i^\sigma}\Bigr)=0. \end{equation*} The series converges almost surely because $\sum_i(R_i^\sigma)^{-2}<\infty$, and since the event that it vanishes has probability zero, we define $\rho_\sigma$ through densities (\cref{sec:conditioning,sec:crit-process}). Note that $\rho_\sigma$ describes Gibbs measures in the critical window, in the sense that the first $k$ eigencoordinates under $\pi_{N,c_N}$, multiplied by $N^{-1/3}$, converge to the corresponding marginal of $\rho_\sigma$ (see \cref{cd-pin} for a precise statement). Du and Huang introduced this conditioned Gaussian sequence and the root $\varrho_0$ at the critical temperature $\sigma=0$, in their study of the overlap distribution \cite[Sections~1.2 and~2.2]{DuHuang2026}. We define the critical mixing profile $F_\sigma$ in \cref{sec:crit-profile} through the limit of the dynamics constructed in \cref{sec:crit-process}, as follows. Let $c_N=1+\sigma N^{-1/3}$, and let $X(t)$ be the diffusion started from $\nu_N$. We show that $N^{-1/3}X_i(N^{2/3}s)$ converges in distribution to $U_i(s)/\sqrt{Z(s)}$, jointly for finitely many $i$, where the processes $U_i$ solve \begin{equation}\label{eq:intro-critical-process} dU_i(s)=-2R_i^\sigma U_i(s)\,ds+\sqrt{2Z(s)}\,dB_i(s),\qquad U_i(0)=0, \end{equation} for independent Brownian motions $B_1,B_2,\dots$ and $Z$ is a positive process that solves a stochastic Volterra equation driven by the same Brownian motions. We show that $Z$ is the limit of the rescaled squared norm of the unnormalized process of \cref{sec:intro-ideas}. It tends to infinity as $s\downarrow0$, and this singular behavior at time zero is how the uniform initial distribution enters the limit. Conditionally on the Airy point process, we define \begin{equation}\label{eq:intro-F} F_\sigma(s)=\bigl\|\Law\bigl((U_i(s)/\sqrt{Z(s)})_{i\ge1}\bigr)-\rho_\sigma\bigr\|_{\TV},\qquad s>0. \end{equation} Note that $F_\sigma$ is a random function that is determined by the Airy point process. \begin{theorem}[The critical window]\label{thm:intro-critical} Fix $\sigma_1,\dots,\sigma_m\in\R$ and let $c_{N,j}=1+\sigma_jN^{-1/3}$. Let $D_N^{(j)}$ and $t_N^{(j)}$ be the distance and the mixing time at $c=c_{N,j}$, for the same matrix $W_N$. Then \begin{equation*} \Bigl(s\mapsto D_N^{(j)}(N^{2/3}s),\ \varepsilon\mapsto N^{-2/3}t_N^{(j)}(\varepsilon)\Bigr)_{j=1}^m \Longrightarrow \Bigl(F_{\sigma_j},\,F_{\sigma_j}^{-1}\Bigr)_{j=1}^m \end{equation*} in distribution, where all limits are defined from the same Airy point process. The topologies are those of uniform convergence on compact subsets of $(0,\infty)$ and of $(0,1)$, respectively. Almost surely, for each $j$, the function $F_{\sigma_j}$ is continuous and strictly decreasing on $(0,\infty)$, $F_{\sigma_j}(0+)=1$, and $F_{\sigma_j}(s)\to0$ as $s\to\infty$. Moreover, there exist $c_1,C_1>0$, depending on the Airy point process and on $\sigma_j$, such that $F_{\sigma_j}(s)\ge c_1e^{-C_1s}$ for all $s>0$. For every $\varepsilon\in(0,1/2)$ and every $j$, with the finite-$N$ quotient defined to be $1$ on the event $\{t_N^{(j)}(1-\varepsilon)=0\}$, \begin{equation*} \frac{t_N^{(j)}(\varepsilon)}{t_N^{(j)}(1-\varepsilon)}\Longrightarrow\frac{F_{\sigma_j}^{-1}(\varepsilon)}{F_{\sigma_j}^{-1}(1-\varepsilon)}, \end{equation*} and the limit is almost surely greater than one. Finally, for every $K>0$ and $\eta>0$ there exist $C>0$ and events $\mathcal A_N$ with $\liminf_{N\to\infty}\PP(\mathcal A_N)\ge1-\eta$ with the following property. On $\mathcal A_N$, for every $\sigma\in[-K,K]$ and every $\varepsilon\in(0,1/2)$, the mixing time $t_N(\varepsilon)$ at $c=1+\sigma N^{-1/3}$ satisfies \begin{equation*} C^{-1}N^{2/3}\log(1/\varepsilon)\le t_N(\varepsilon)\le CN^{2/3}\log(1/\varepsilon). \end{equation*} \end{theorem} In particular, \cref{thm:intro-critical} implies that there is no cutoff in the critical window, since $t_N(\varepsilon)/t_N(1-\varepsilon)$ converges in distribution to a limit that is almost surely greater than one. The convergence is stated in distribution: the theorem is formulated through the GOE edge limit without specifying a joint coupling across matrix sizes with the limiting Airy point process. We prove \cref{thm:intro-critical} in \cref{sec:crit-process,sec:crit-profile}. We prove convergence for deterministic spectra satisfying the hypotheses of \cref{sec:spectral} in \cref{cd-profile} and deduce the statement for the GOE in \cref{cd-goe-profile}. We prove the lower bound on $F_\sigma$ in \cref{critical-finite-time-positivity} and the bounds on $t_N(\varepsilon)$ in \cref{cl3-density-clock}. These bounds allow accuracies that depend on $N$, and for instance they show that the mixing time at accuracy $e^{-bN}$ is of order $N^{5/3}$. Our last main result describes critical mixing profiles far from the center of the window, where they converge to the profiles of the mesoscopic regimes on the same side of the critical temperature. \begin{theorem}[Matching with the mesoscopic regimes]\label{thm:intro-outer} Almost surely, for every compact set $K\subset\R$, as $S\to\infty$, \begin{equation*} \sup_{u\in K}\Bigl|F_{-S}\Bigl(\tau_S^{\mathrm h}+\frac u{S^2}\Bigr)-\Psi(u)\Bigr|\to0,\qquad \tau_S^{\mathrm h}=\frac{\log S^3-\frac32\log\log S^3-\log(16\sqrt\pi)}{4S^2}, \end{equation*} and \begin{equation*} \sup_{u\in K}\Bigl|F_{S}\Bigl(\tau_S^{\mathrm c}+\frac{2u}{\Delta_2}\Bigr)-\cV(u)\Bigr|\to0,\qquad \tau_S^{\mathrm c}=\frac{\log((S-\Xi)\Delta_2)}{2\Delta_2}. \end{equation*} Further, $S^2\bigl(F_{-S}^{-1}(\varepsilon)-\tau_S^{\mathrm h}\bigr)\to\Psi^{-1}(\varepsilon)$ and $(\Delta_2/2)\bigl(F_S^{-1}(\varepsilon)-\tau_S^{\mathrm c}\bigr)\to\cV^{-1}(\varepsilon)$, locally uniformly in $\varepsilon\in(0,1)$. \end{theorem} We prove \cref{thm:intro-outer} in \cref{sec:outer-hot} (\cref{oc-full-hot}) and \cref{sec:outer-cold} (\cref{cot-transfer}). The proof approximates each critical profile by finite-dimensional diffusions, taking $N\to\infty$ at fixed $\sigma=\mp S$ before letting $S\to\infty$. \section{Discussion}\label{sec:intro-discussion} \paragraph{Three mechanisms.} The three profiles $\Psi$, $\cV$, and $F_\sigma$ come from three different mechanisms. At high temperature the law of the diffusion at time $t$ is close in total variation to the Bingham comparison law, a Gaussian law with variances $v_i(t)$ conditioned on $S_N$, while $\pi_{N,c}$ is a Gaussian law with variances $q_i$ conditioned on $S_N$. The log-likelihood ratio of the Bingham comparison law with respect to $\pi_{N,c}$ is $\sum_ie_ix_i^2/(2v_i)$ up to an additive constant, and when the relative errors $e_i$ are small and many coordinates contribute, it is approximately Gaussian under $\pi_{N,c}$ with variance $\cI_N$ and mean $-\cI_N/2$. This is Le Cam's Gaussian experiment \cite{LeCam1960}, \cite[Chapters~6--7]{vanDerVaart1998}, in which the total-variation distance is $2\Phi(\sqrt{\cI_N}/2)-1$. The information decays over time, and for large $t$ its decay is governed by the edge of the spectrum. Indeed, each relative error $e_i(t)$ decays roughly like $e^{-2p_it}$, precisions near the edge satisfy $p_i-d=c(\lambda_1-\lambda_i)$, and the semicircle density vanishes like a square root at the edge. A Laplace-type computation then suggests that \begin{equation*} \cI_N(t)\approx A_c\,N\,(4gt)^{-3/2}e^{-4gt} \end{equation*} at fixed $c\in(0,1)$, with $A_c$ and $g$ as in \cref{thm:intro-main}. The center $T_N$ in case (2) agrees, up to $o(g^{-1})$, with the time at which the right side equals one, so that $\cI_N(T_N+u/g)\to e^{-4u}$ and the profile is $2\Phi(e^{-2u}/2)-1=\Psi(u)$. In particular, the correction $-(3/2)\log\log N$ comes from the factor $t^{-3/2}$, which reflects the square-root vanishing of the semicircle density. Factors $t^{-3/2}$ of this kind, which come from Laplace transforms of the spectral measure near the edge, appear in the physics literature on this model \cite[Appendix~A]{CugliandoloDean1995}, \cite[(14), (17)--(18)]{ZippoldKuhnHorner2000}, \cite[(27)--(28)]{FyodorovPerretSchehr2015}. At low temperature the mechanism is different. We describe it through the unnormalized process of \cref{sec:intro-ideas}, whose coordinates solve linear equations in which the first coordinate has no drift and coordinate $i$ has drift $-2a_iY_i$. During an initial period the first two coordinates acquire values that, after a common normalization, are asymptotically independent standard Gaussian variables $H_1$ and $H_2$; these values are built from the initial coordinates and the early noise. Afterwards the first eigencoordinate of the diffusion grows until it carries a fraction $\theta_N$ of the squared norm, while the ratio of the second eigencoordinate to the first decays at rate $2a_2$. At time $T_N+u/a_2$ the second eigencoordinate, divided by its equilibrium standard deviation $(2a_2)^{-1/2}$, is approximately $G+e^{-2u}H_2/|H_1|$, where $G$ is a fresh standard Gaussian variable independent of $(H_1,H_2)$. Since $H_2/|H_1|$ has a standard Cauchy law, this leads to the profile $\cV$. The center $T_N$ solves $\sqrt{2\theta_NNa_2}\,e^{-2a_2T_N}=1$, where $\sqrt{\theta_NN}$ is the size of the first eigencoordinate, and all other eigencoordinates relax faster and do not contribute. The appearance of the Cauchy law depends on the initial distribution, since we need $H_1$ and $H_2$ to be asymptotically independent centered Gaussian variables with the same variance. A ratio of two such variables is a classical construction of the Cauchy law \cite{Marsaglia1965}, and what is specific to this model is that the ratio survives in the total-variation limit, independently of late Gaussian noise. In the critical window the relevant precisions are of order $N^{-2/3}$ and the time scale is $N^{2/3}$. On this scale eigencoordinates are of order $N^{1/3}$ and initial values of individual coordinates are negligible, while the limit \eqref{eq:intro-critical-process} retains infinitely many random rates $R_i^\sigma$. The profile $F_\sigma$ is determined by these random rates. \paragraph{Matching across regimes.} The constants in \cref{thm:intro-main} are consistent with each other. Since $A_c=(1-c)^3B_c$, the center in case (2) equals \begin{equation*} \frac{\log\bigl(N(1-c)^3\bigr)-\frac32\log\log N+\log B_c}{4(1-c)^2}. \end{equation*} This has the form of the center in case (3) with $\delta=1-c$, except that $\log\log N$ replaces $\log\log M_N$. If we replace $c$ by $c_N=1-\delta_N$ then the difference between the two centers is $-(3/8)\delta_N^{-2}\log(\log N/\log M_N)$, which is of the order of the window $\delta_N^{-2}$ or larger unless $\log M_N/\log N\to1$. For instance, it equals $(3/8)\delta_N^{-2}\log(1-3\alpha)$ for $\delta_N=N^{-\alpha}$ with $0<\alpha<1/3$. Thus the formula for fixed temperature does not extend to temperatures approaching the critical one, and the logarithmic correction must be computed with $M_N$. Case (1) connects to case (2) in a similar way. When $h_N\to\infty$ the center $\tau_N$ is given by the formula of case (2) with $c=c_N$, and when $h_N$ is bounded the function $\mathcal B$ describes the transition from relaxation governed by the bulk of the spectrum to relaxation governed by its edge. Below the critical temperature, cases (4) and (5) have the same formula. \Cref{thm:intro-outer} matches the critical profiles with cases (3) and (4). If we set $\delta_N=SN^{-1/3}$ in case (3), then $M_N=S^3$ and $B_{c_N}\to1/(16\sqrt\pi)$, so that the center and the window of case (3), divided by $N^{2/3}$, converge to $\tau_S^{\mathrm h}$ and $S^{-2}$. If we set $\epsilon_N=SN^{-1/3}$ formally in case (4), then under a coupling realizing the convergence of the edge spectral data we have $N^{1/3}\theta_N\to S-\Xi$ and $N^{2/3}a_2\to\Delta_2/2$. On the event $\{(S-\Xi)\Delta_2>1\}$, the center and the window divided by $N^{2/3}$ then converge to $\tau_S^{\mathrm c}$ and $2/\Delta_2$. This event holds almost surely for all sufficiently large $S$. These substitutions are only formal, since $N\delta_N^3$ and $N\epsilon_N^3$ stay bounded. In case (4) itself we have $\epsilon_NN^{1/3}\to\infty$ and $\theta_N/\epsilon_N\to1$, and the statistic $\Xi$ appears only in the iterated limit, through $\tau_S^{\mathrm c}$. The mesoscopic theorems do not imply \cref{thm:intro-outer}, which concerns the order of limits in which $N\to\infty$ first, and its proof uses estimates that are uniform in the effective parameters $N\delta_N^3$ and $\theta_NNa_2$ before the limit is taken. \paragraph{Time scales and the width of the critical window.} The mixing time increases from order $\log N$ at fixed high temperature to order $\delta_N^{-2}\log(N\delta_N^3)$ in the approach from above and to order $N^{2/3}$ in the critical window, and it is of order $N^{2/3}\log(N\epsilon_N^3)$ in the approach from below and of order $N^{2/3}\log N$ at fixed low temperature. The window grows from order one to order $N^{2/3}$ and remains of order $N^{2/3}$ below the critical temperature, where the center and the window are random at leading order. For instance, in case (5), \eqref{eq:intro-airy} and $a_2t_N(\varepsilon)/\log N\to1/12$ imply that $t_N(\varepsilon)/(N^{2/3}\log N)$ converges in distribution to $1/(6c\Delta_2)$. The critical window has width $N^{-1/3}$ in $c$ for the following reason. At high temperature the smallest precision is $d\approx(1-c)^2$, and the number of precisions in $[d,2d]$ is of order $N(1-c)^3$, since this is the number of eigenvalues within distance $(1-c)^2$ of the edge. When $M_N=N\delta_N^3\to\infty$, many coordinates contribute comparably to $\cI_N$ and their sum produces the Gaussian profile. When $N\delta_N^3$ is bounded, the relevant precisions are comparable to the spacing $N^{-2/3}$ of the top eigenvalues, and the limit retains their randomness. Below the critical temperature the corresponding parameter is $\theta_NNa_2$, which is of order $(N\epsilon_N^3)^{1/3}$. The fluctuations of the free energy cross over on the wider scale $c=1+bN^{-1/3}\sqrt{\log N}$, as predicted by Baik and Lee (see \cite[Section~1.2]{JohnstoneKlochkovOnatskiPavlyshyn2024}). Landon proved the critical Gaussian limit, tightness in the crossover, and the Gaussian and Tracy--Widom outer limits \cite[Theorems~1.1--1.2]{Landon2022}. Johnstone, Klochkov, Onatski, and Pavlyshyn determined the full crossover law \cite[Theorem~1.1, (1.7)]{JohnstoneKlochkovOnatskiPavlyshyn2024}. Under their centering and normalization its GOE form is $Z+\sqrt{3/2}\,b_+\mathrm{TW}_1$, where $b_+=\max\{b,0\}$ and $Z$ is an independent standard Gaussian variable, so that the Gaussian fluctuations from the bulk of the spectrum and the Tracy--Widom fluctuations from the edge are of the same order there. The dynamical window is determined by the edge alone, and the logarithm does not appear. A temperature scale $N^{-1/3}$ also appears in the static analysis of Wang and Tang, which is based on a maximum-likelihood approximation of the spectrum \cite[Section~4.1.1]{WangTang2024}. \paragraph{Symmetry of the initial condition.} The generator \eqref{eq:intro-generator}, the Gibbs measure, and $\nu_N$ are all invariant under each reflection $x_i\mapsto-x_i$ of an eigencoordinate, and therefore so is $\nu_NP_t$. This symmetry is important at low temperature. There the potential $\sum_{i\ge2}a_ix_i^2$ vanishes at $\pm\sqrt N$ times the top eigenvector and is at least $Na_2$ on the equator $\{x_1=0\}$, where $Na_2$ is of order $N^{1/3}$. This is the antipodal barrier mechanism of Rodgers and Moore \cite[Section~2, (25)--(26)]{RodgersMoore1989}, discussed further in \cite[Section~4]{FyodorovPerretSchehr2015}. The sign of $x_1$ changes only when the process crosses the equator. Since $\nu_NP_t$ is symmetric under $x_1\mapsto-x_1$, that sign does not affect $D_N(t)$, and the distance is the same as for the process folded onto the hemisphere $\{x_1\ge0\}$. From a starting point in one hemisphere the law at time $t$ need not be symmetric, and \cref{thm:intro-main} concerns only the uniform initial law. Gheissari, Grant, and Yu compare symmetric and worst-case initial conditions for spiked matrix models \cite[Theorem~1.2]{GheissariGrantYu2026}, as we discuss in \cref{sec:intro-related}. For the Curie--Weiss model the censored dynamics of \cite{DingLubetzkyPeres2009b} removes the analogous bottleneck by changing the dynamics. \paragraph{Which temperature sequences are covered.} \Cref{thm:intro-main,thm:intro-bounded,thm:intro-critical} cover fixed values $c\in[0,1)\cup(1,\infty)$, every sequence $c_N\to0$ along which $Nc_N^2$ converges in $[0,\infty]$, every sequence $c_N\to1$ that stays on one side of $1$ and satisfies $N^{1/3}|c_N-1|\to\infty$, and $c_N=1+\sigma N^{-1/3}$ with $\sigma$ fixed. Several cases are not covered. First, for fixed $c\notin\{0,1\}$ we state the results at the fixed temperature $c$ and not along sequences $c_N\to c$. The proofs suggest that the results extend to such sequences with centers evaluated at $c_N$, but a center evaluated at the limit $c$ instead is shifted by an amount that does not vanish unless $(c_N-c)\log N\to0$. Similarly, in the critical window we do not treat $c_N=1+\sigma_NN^{-1/3}$ with $\sigma_N\to\sigma$. Second, we do not cover the case $c_N\to\infty$, in which the temperature tends to zero, since our proofs at low temperature use a bounded range of the scaled eigenvalue gaps $a_i$ and a limiting top fraction in $(0,1)$ or tending to zero. Third, a sequence that oscillates between regimes is analyzed along subsequences. It need not have a single limiting profile, although subsequences with the same limiting shape can be combined using their respective centers and windows. \section{Related work}\label{sec:intro-related} \paragraph{Equilibrium.} The equilibrium theory of the model is well developed. Baik and Lee proved that the free energy has Gaussian fluctuations of order $N^{-1}$ at high temperature and Tracy--Widom fluctuations of order $N^{-2/3}$ at low temperature \cite[Theorems~1.1--1.2]{BaikLee2016}; see \cite{BaikLee2017,BaikLee2020,BaikLeeWu2018} for related models with a ferromagnetic interaction or a bipartite structure. At the critical temperature, Landon proved a central limit theorem for the free energy on the scale $N^{-1}\sqrt{\log N}$ \cite[Theorem~1.1]{Landon2022}, and Johnstone, Klochkov, Onatski, and Pavlyshyn identified its limit law throughout the transition window \cite[Theorem~1.1, (1.7)]{JohnstoneKlochkovOnatskiPavlyshyn2024}. Nguyen and Sosoe proved a central limit theorem for the overlap of two replicas near the critical temperature on the high-temperature side \cite[Theorem~1 and Section~5]{NguyenSosoe2019}. Landon and Sosoe identified the fluctuations of the moments of the overlap at low temperature and in the process constructed the statistic $\Xi$ \cite[Theorems~2.1--2.3 and~6.1]{LandonSosoe2022}. Du and Huang determined the limiting overlap distribution at the critical temperature \cite[Theorem~1.1]{DuHuang2026}, and our description of the Gibbs measure in the critical window follows theirs. Results with an external field are in \cite{BaikCollinsWoodfinLeDoussalWu2021,CollinsWoodfin2021,LandonSosoe2025}, variational descriptions of spherical models of Thouless--Anderson--Palmer type are in \cite{BeliusKistler2019,Subag2024}. Chen extended the spherical free-energy formula to general mixtures, including odd interactions \cite{Chen2013}, while Panchenko and Talagrand studied multiple spherical systems with constrained overlaps \cite[Theorem~1 and Section~3]{PanchenkoTalagrand2007}. The relation between exact and mean spherical constraints was studied by Kac and Thompson \cite{KacThompson1977}, Crisanti, Sarracino, and Zannetti \cite{CrisantiSarracinoZannetti2019}, and Aru and Korzhenkova \cite{AruKorzhenkova2024}, and Lukkarinen studied condensation in spherical models \cite{Lukkarinen2020}. Genovese and Tantari studied the Legendre duality between spherical and Gaussian spin-glass ensembles \cite{GenoveseTantari2015}. These works concern the Gibbs measure, which in our setting is the end point of the dynamics. \paragraph{Spherical spin glass dynamics and sampling.} Gheissari and Jagannath proved that the spectral gap of the Langevin dynamics of spherical $p$-spin glasses with $p\ge3$ is of order one at sufficiently high temperature and exponentially small at sufficiently low temperature \cite[Theorem~1]{GheissariJagannath2019}; the quadratic case is excluded \cite[Remark~5]{GheissariJagannath2019}. Ben Arous and Jagannath related spectral gaps to the free-energy landscape \cite{BenArousJagannath2018} and distinguished shattering from metastability \cite[Theorems~2.5--2.6]{BenArousJagannath2024}, and Ben Arous, Gheissari, and Jagannath bounded the evolution of observables along spherical dynamics \cite{BenArousGheissariJagannath2020a}. Huang, Mohanty, Rajaraman, and Wu proved that simulated annealing based on Langevin dynamics samples from spherical spin glasses at high temperature in polynomial time, with small error in total variation \cite[Theorem~7.2]{HuangMohantyRajaramanWu2025}. In \cite[Section~1.5]{HuangMohantyRajaramanWu2025} they discuss the different question of sampling with Langevin dynamics at a fixed temperature from a uniform start. Huang, Montanari, and Pham obtained sampling in total variation for spherical spin glasses by algorithmic stochastic localization \cite{HuangMontanariPham2024}, an algorithm that differs from Langevin dynamics; see also \cite{ElAlaouiMontanariSellke2022}. For models with discrete spins, functional inequalities and rapid mixing at high temperature were established in \cite{BauerschmidtBodineau2019,EldanKoehlerZeitouni2022,AnariLiuOveisGharan2020}. Ge, Lee, Lu, and Risteski gave an exact rejection sampler for Bingham distributions with expected runtime polynomial in the dimension and the spectral range, assuming an oracle for solving a univariate polynomial equation \cite[Theorem~1.1]{GeLeeLuRisteski2021}; it does not concern the Langevin dynamics. Other dynamics on the sphere have been analyzed as well; for instance, Kac's walk mixes in order $n\log n$ steps \cite{PillaiSmith2017} and has cutoff in total variation from a coordinate vector \cite[Theorem~1.1]{JainMizgerd2026}. The closest work is that of Gheissari, Grant, and Yu \cite{GheissariGrantYu2026}, who study the Langevin dynamics on the sphere for spiked matrix models, in which a rank-one perturbation is added to a GOE matrix. For a sufficiently large fixed spike, they prove that at low temperature the worst-case mixing time is exponential, while the dynamics mixes in logarithmic time from initial distributions that are invariant under the reflection of the top eigencoordinate, such as the uniform distribution \cite[Theorems~1.2--1.3]{GheissariGrantYu2026}. They also prove cutoff in a high-temperature regime \cite[Corollary~6.3]{GheissariGrantYu2026}. Their results do not cover the model without a spike, in which the gap between the two largest eigenvalues is of order $N^{-2/3}$ instead of order one. They also do not identify a profile at low temperature, and the question of cutoff from symmetric initializations at low temperature is left open there \cite[discussion following Theorem~1.2, p.~4]{GheissariGrantYu2026}. Compared to their results, \cref{thm:intro-main}(4)--(5) concern the model without a spike and identify the profile, and \cref{thm:intro-critical,thm:intro-outer} have no counterpart in \cite{GheissariGrantYu2026}. The physics works discussed in \cref{sec:intro-background} compute observables in the thermodynamic limit or at zero temperature, and several of them use a deterministic initial amplitude in each eigenmode. Our results concern the total-variation distance of the full law at positive temperature, from the uniform distribution and for fixed disorder. They are consistent with the physical picture, in which the relaxation rate $(1-c)^2$ at high temperature corresponds to the classical saddle point, and in which the role of the edge of the spectrum and of the gap $\lambda_1-\lambda_2$ at low temperature was identified in \cite{FyodorovPerretSchehr2015,BarbierDeFreitasPimentaCugliandoloStariolo2021}. We also note that the isolated (Hamiltonian) dynamics of the same model has additional conserved quantities that prevent relaxation to the Gibbs measure \cite{CugliandoloLozanoNessiPiccoTartaglia2018}. \paragraph{Tools.} We use several classical tools and credit them where they are used. These include the representation of uniform and Bingham laws by conditioned Gaussian vectors \cite{Bingham1974,KumeWood2005}, with the Gaussian approximation of spherical coordinates going back to Borel \cite[Section~6]{DiaconisFreedman1987}. They also include Le Cam's theory of Gaussian likelihood ratios \cite{LeCam1960,vanDerVaart1998}, Stein's method for normal approximation \cite{Stein1972}, the scalar equation of Cugliandolo and Dean for the Lagrange multiplier \cite{CugliandoloDean1995}, the Bakry--\'Emery calculus and logarithmic Sobolev inequalities \cite{BakryEmery1985,Gross1975,BakryGentilLedoux2014}, and stochastic calculus \cite{RevuzYor1999}. \section{Ideas of the proofs}\label{sec:intro-ideas} We describe the main ideas briefly here and refer to \cref{sec:overview} for a detailed overview. The proofs use one exact representation of the dynamics throughout. Let $g=(g_1,\dots,g_N)$ be a standard Gaussian vector, independent of independent Brownian motions $B_1,\dots,B_N$, and let $Y$ solve \begin{equation}\label{eq:intro-unnormalized} dY_i=-p_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i,\qquad Y(0)=g. \end{equation} Then $\sqrt N\,Y/|Y|$ is the Langevin diffusion with generator \eqref{eq:intro-generator} started from $\nu_N$ (\cref{sec:process}). Shifting the precisions $p_i$ by a common constant, for instance replacing them by $2a_i$, does not change $\sqrt NY/|Y|$. The process $Y$ is the unnormalized process of \cref{sec:process}, with quadratic potential coefficients $p_i/2$. Its coordinates solve linear equations that are coupled only through the factor $|Y|$ in the noise. At high temperature we compare $\nu_NP_t$ with the Bingham comparison law. We bound the total-variation distance between them by integrating the error that the comparison law makes in the forward equation of the diffusion (\cref{sec:comparison}), which requires functional inequalities on the sphere (\cref{sec:inequalities}) and moment bounds for the unnormalized process (\cref{sec:process}). The distance between the comparison law and $\pi_{N,c}$ is a distance between two Gaussian laws conditioned on the same sphere. We show that it equals $2\Phi(\sqrt{\cI_N}/2)-1$ up to an error that we control quantitatively (\cref{sec:conditioning,sec:likelihood}). We compute the information from the covariance evolution and the edge of the spectrum (\cref{sec:heat}). At low temperature we follow the mechanism described in \cref{sec:intro-discussion}. We analyze the first coordinates of the unnormalized process directly and control the remaining coordinates through conditional relative entropy and logarithmic Sobolev inequalities, and smoothing estimates then upgrade weak convergence of the first coordinates to convergence in total variation (\cref{sec:scores,sec:density}). In the critical window we construct the limit process \eqref{eq:intro-critical-process} and the limiting Gibbs law (\cref{cd-functional,cd-pin}). We also prove that at every time of order $N^{2/3}$ the law of the diffusion has a density with respect to the Gibbs measure that is bounded in $L^2$ uniformly in $N$ (\cref{cl3-actual-source-density}). With the smoothing of the finite-dimensional semigroup and a Poincar\'e inequality for the conditional law of the remaining coordinates, this bound controls the remaining coordinates at every fixed time (\cref{cd-averaged-tail}). Integration by parts under the Gibbs measure gives a uniform translation bound for the first-coordinate laws. Mollification upgrades their weak convergence to convergence in total variation at each fixed positive time (\cref{cd-head-current}). Combining this with the bound on the remaining coordinates gives pointwise convergence of the whole total-variation distance (\cref{cd-pointwise-profile}). Semigroup smoothing and weak convergence give continuity of the limiting profile; monotonicity then upgrades pointwise convergence to locally uniform convergence, and finite-dimensional spectral correlation bounds give strict decrease (\cref{cd-profile}). We prove the outer limits in \cref{thm:intro-outer} by applying finite-dimensional estimates that are uniform in the effective parameters $N\delta_N^3$ and $\theta_NNa_2$ to finite approximations of the Airy point process (\cref{sec:outer-hot,sec:outer-cold}). Two principles apply throughout the paper. First, the object of study is always the law of the original diffusion on $S_N$ started from $\nu_N$. We use Gaussian and Bingham laws through explicit total-variation comparisons with the finite-dimensional diffusion. In the critical window, the countable limiting laws define the limiting distance, whose convergence follows from finite-coordinate convergence in total variation, estimates for the remaining coordinates, and continuity of the limiting profile. Second, random-matrix inputs enter only through deterministic hypotheses on the spectrum, which hold with probability at least $1-\eta$ for every fixed $\eta>0$ with constants depending on $\eta$. The dynamical estimates are deterministic statements about a matrix that satisfies these hypotheses. We state the published random-matrix results that we use, in our normalization, in \cref{sec:spectral}. \section{Organization of the paper}\label{sec:intro-organization} For publication, the manuscript is divided into two volumes. The first, \emph{Part I: Foundations and tools}, contains Sections~1--12: the main statements, an overview of the proofs, the preliminaries, and the general tools. The second, \emph{Part II: Mixing profiles and critical dynamics}, contains Sections~13--21: the proofs in the temperature regimes, the construction and analysis of the critical dynamics, and the outer limits of the critical profiles. \Cref{sec:overview} gives an overview of the proofs. \Cref{part:prelim} contains the preliminaries: in \cref{sec:model} we define the model, fix conventions, and collect the properties of the profiles, and in \cref{sec:spectral} we state the random-matrix inputs and analyze the Lagrange multiplier. \Cref{part:tools} develops the tools, each in the generality needed later, namely Gaussian laws conditioned on a sphere (\cref{sec:conditioning}), their likelihood ratios (\cref{sec:likelihood}), the covariance evolution and its heat estimates (\cref{sec:heat}), functional inequalities on the sphere (\cref{sec:inequalities}), the comparison with Bingham laws (\cref{sec:comparison}), the unnormalized process (\cref{sec:process}), score bounds and conditional relative entropy (\cref{sec:scores}), and density bounds (\cref{sec:density}). \Cref{part:hot} proves \cref{thm:intro-main}(1)--(3) and \cref{thm:intro-bounded} (\cref{sec:hot-fixed,sec:hot-meso,sec:infinite}), and \cref{part:cold} proves \cref{thm:intro-main}(4)--(5) (\cref{sec:cold-fixed,sec:cold-meso}). \Cref{part:critical} constructs the critical limit process and proves \cref{thm:intro-critical,thm:intro-outer} (\cref{sec:crit-process,sec:crit-profile,sec:outer-hot,sec:outer-cold}).