\chapter{The model, conventions, and the limiting profiles}\label{sec:model}\label{sec-model}
\aliaslabel{ht-general-matrix-model}{eq:model-general}
\aliaslabel{ht-canonical-equation}{eq:model-multiplier}

This section fixes the model and the conventions used throughout the paper.
In \Cref{sec:model-model}, we define the spherical Langevin dynamics and its Gibbs measure, and record three exact reductions: diagonalization of the disorder, invariance under adding a multiple of the identity to the quadratic form, and invariance under reflections of eigencoordinates (\Cref{lem:model-symmetry}). We also name the symmetry sectors of $L^2$ (\Cref{def:model-sectors}).
\Cref{sec:model-equilibrium} introduces the Lagrange multiplier and the precisions of the Gaussian representation of the Gibbs measure, the information of one variance sequence relative to another (\Cref{def:model-information}), and the parameters attached to each temperature regime.
\Cref{sec:model-distances} defines the total variation distance, the mixing times, and what we mean by a mixing profile, by a cutoff profile, and by cutoff (\Cref{def:model-profiles}), and recalls standard facts about total variation and relative entropy (\Cref{lem:model-tv}).
\Cref{sec:model-quantifiers} states once how probabilities over the disorder and the constants in our estimates are quantified.
In \Cref{sec:model-profiles}, we prove the properties of the three cutoff profiles $\Psi$, $\cV$, and $\cV_\theta$ (\Cref{cot-profile}), and show that pointwise convergence of a nonincreasing distance to a profile implies locally uniform convergence and convergence of the mixing times (\Cref{hp-monotone-profile-transfer}).

The section uses no earlier results. Its definitions are used everywhere. \Cref{cot-profile,hp-monotone-profile-transfer} are used in each section that identifies a profile, namely \Cref{sec:hot-fixed,sec:hot-meso,sec:infinite,sec:cold-fixed,sec:cold-meso,sec:outer-hot,sec:outer-cold}.

\section{The model}\label{sec:model-model}

\paragraph{Disorder.}
Fix $N\ge2$. Let $W_N=(W_{ij})_{i,j=1}^N$ be a matrix from the Gaussian orthogonal ensemble (GOE): a real symmetric random matrix whose entries $W_{ij}$, $1\le i\le j\le N$, are independent centered Gaussian random variables with $\E W_{ij}^2=1/N$ for $i<j$ and $\E W_{ii}^2=2/N$. We denote its eigenvalues by $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_N$. We write $\lambda_{i,N}$ when the dependence on $N$ matters. Almost surely the eigenvalues are distinct. Indeed, the discriminant of the characteristic polynomial of a symmetric matrix is a polynomial in its entries which does not vanish at a diagonal matrix with distinct diagonal entries. Its zero set has Lebesgue measure zero, while the law of $W_N$ has a density with respect to Lebesgue measure on the space of real symmetric matrices. We write
\[
\rho_{\mathrm{sc}}(x)=\frac{\sqrt{\max\{4-x^2,0\}}}{2\pi}
\]
for the semicircle density.

\paragraph{The dynamics.}
Let $S_N=\{x\in\R^N:|x|^2=N\}$, where $|\cdot|$ is the Euclidean norm, and let $\nu_N$ be the uniform probability measure on $S_N$. We write $\nabla_{S_N}$ and $\Delta_{S_N}$ for the gradient and the Laplace--Beltrami operator of $S_N$ with the metric induced from $\R^N$, and $\ip{\cdot}{\cdot}$ for the Euclidean inner product. For an inverse temperature $\beta\ge0$, we set $c=\sqrt2\,\beta$ and
\begin{equation}\label{model-original}
H_N(x)=\frac{x^\T W_Nx}{\sqrt2},\qquad
L=\Delta_{S_N}+\beta\nabla_{S_N}H_N\cdot\nabla_{S_N},\qquad
\pi_{N,c}(dx)=\frac{e^{\beta H_N(x)}\,\nu_N(dx)}{\int_{S_N}e^{\beta H_N}\,d\nu_N}.
\end{equation}
The inverse temperature multiplies $H_N$ and is not part of it: the Gibbs weight is $e^{+\beta H_N}$, so the physical energy is $-H_N$.
Note that $\beta H_N(x)=(c/2)\,x^\T W_Nx$. We call $c=\sqrt2\,\beta$ the rescaled inverse temperature, and use it in place of $\beta$ throughout. Increasing $c$ lowers the temperature. The critical value is $c=1$, that is, $\beta_c=1/\sqrt2$, and high and low temperature mean $c<1$ and $c>1$, respectively. The measure $\pi_{N,c}$ is the Gibbs measure of the quadratic spherical spin glass (the $2$-spin spherical Sherrington--Kirkpatrick model) introduced by Kosterlitz, Thouless, and Jones \cite{KosterlitzThoulessJones1976}. As a probability measure on a sphere with density proportional to the exponential of a quadratic form, it is a Bingham distribution \cite{Bingham1974}. We write $P_t=e^{tL}$ for the Markov semigroup generated by $L$, and, for a probability measure $\mu$ on $S_N$, we write $\mu P_t$ for the law at time $t$ of the diffusion with initial law $\mu$. We call $\mu P_t$ the time-$t$ law.

\paragraph{Normalization.}
The GOE variances $1/N$ off the diagonal and $2/N$ on the diagonal agree with those of the completion matrix for the Sherrington--Kirkpatrick model. In the convention of the Sherrington--Kirkpatrick literature in which the Gibbs weight is $e^{(\beta/2)x^\T Wx}$, our weight $e^{(c/2)x^\T W_Nx}$ has the same form. Thus $c$ plays the role of the SK inverse temperature, and $c=1$ corresponds to $\beta_c^{\mathrm{SK}}=1$. Here the reference measure $\nu_N$ is the uniform probability measure on $S_N$, rather than counting measure on the spin space.

In ambient coordinates, the diffusion generated by $L$ is the solution of
\begin{equation}\label{eq:model-sde}
dX_t=\sqrt2\,\Pi(X_t)\,dB_t+\Big(c\,\Pi(X_t)W_NX_t-\frac{N-1}{N}\,X_t\Big)\,dt,
\qquad \Pi(x)=I-\frac{xx^\T}{N},
\end{equation}
where $X_0\in S_N$ and $B$ is a standard Brownian motion in $\R^N$, independent of $(W_N,X_0)$. For the model, $X_0$ has law $\nu_N$ independently of $W_N$. The noise coefficient is $\sqrt2$, so that the generator contains $\Delta_{S_N}$ and not $\Delta_{S_N}/2$. The coefficients of \eqref{eq:model-sde} are polynomial, hence locally Lipschitz. To see that the solution stays on $S_N$, let $U_t=1-|X_t|^2/N$. Since $x^\T\Pi(x)=(1-|x|^2/N)\,x^\T$ and $\Tr(\Pi(x)\Pi(x)^\T)=N-2|x|^2/N+|x|^4/N^2$, Itô's formula gives
\[
dU_t=-\frac{2U_t}N\Big(\sqrt2\,X_t^\T dB_t+\big(c\,X_t^\T W_NX_t+N-1+U_t\big)\,dt\Big),
\]
up to the exit time of $X$ from any ball. This is a linear equation for $U$ with locally bounded adapted coefficients, and its unique solution with $U_0=0$ is $U\equiv0$. Hence $|X_t|^2=N$ for all $t\ge0$, and in particular the solution exists for all times. For a function $f$ that is smooth near $S_N$, the identities $\beta\nabla_{S_N}H_N(x)=c\,\Pi(x)W_Nx$ and $\Delta_{S_N}f(x)=\Tr(\Pi(x)\nabla^2f(x))-(N-1)N^{-1}\ip{x}{\nabla f(x)}$ for $x\in S_N$ show that the generator of \eqref{eq:model-sde} is $L$.

\paragraph{General quadratic potentials.}
Many tools in this paper hold for an arbitrary deterministic real symmetric matrix $A$ with eigenvalues $\alpha_1\ge\cdots\ge\alpha_N$. For such $A$, we write
\begin{equation}\label{eq:model-general}
L_A=\Delta_{S_N}+\nabla_{S_N}\Big(\frac12x^\T Ax\Big)\cdot\nabla_{S_N},\qquad
P_t^A=e^{tL_A},\qquad
\pi_A(dx)\propto e^{x^\T Ax/2}\,\nu_N(dx).
\end{equation}
The model \eqref{model-original} is the case $A=cW_N$, for which $\alpha_i=c\lambda_i$. Let $V(x)=x^\T Ax/2$. Then $L_Af=e^{-V}\operatorname{div}_{S_N}(e^{V}\nabla_{S_N}f)$, and the divergence theorem on the closed manifold $S_N$ gives, for smooth $f,g$,
\begin{equation}\label{eq:model-dirichlet}
-\int_{S_N}f\,L_Ag\,d\pi_A=\int_{S_N}\ip{\nabla_{S_N}f}{\nabla_{S_N}g}\,d\pi_A .
\end{equation}
In particular, $L_A$ is symmetric and nonpositive on $L^2(\pi_A)$, and $\pi_A$ is reversible and invariant for $P^A_t$. The Dirichlet form of $L_A$ is $f\mapsto\int|\nabla_{S_N}f|^2\,d\pi_A$, built from the tangential gradient. All Poincaré and logarithmic Sobolev inequalities in this paper refer to this Dirichlet form. For a probability measure $\pi$ on $S_N$ and a nonnegative measurable function $g$ with $\pi(g)<\infty$, let $\Ent_\pi(g)=\pi(g\log g)-\pi(g)\log\pi(g)\in[0,\infty]$, with $0\log0=0$. We say that $\pi$ satisfies a Poincaré inequality with constant $C_{\mathrm P}$, and a logarithmic Sobolev inequality with constant $C_{\mathrm{LS}}$, if for all smooth $f$,
\begin{equation}\label{eq:model-fi}
\Var_\pi(f)\le C_{\mathrm P}\,\pi\big(|\nabla_{S_N}f|^2\big),\qquad
\Ent_\pi(f^2)\le C_{\mathrm{LS}}\,\pi\big(|\nabla_{S_N}f|^2\big).
\end{equation}
The spectral gap of $L_A$ is
\[
\lambda_{\mathrm{gap}}(L_A)=\inf\Big\{\frac{\pi_A(|\nabla_{S_N}f|^2)}{\Var_{\pi_A}(f)}:f\text{ smooth and nonconstant}\Big\},
\]
the smallest nonzero eigenvalue of $-L_A$, by compact weighted elliptic spectral theory \cite[Section~2, p.~30]{Setti1998}. The optimal constant in the Poincaré inequality is $1/\lambda_{\mathrm{gap}}(L_A)$. Restricting the infimum to the symmetry sectors of \Cref{def:model-sectors} below, we obtain the spectral gaps of these sectors.

\paragraph{Eigencoordinates.}
Conditional on $W_N$, fix an orthonormal basis $u_1,\dots,u_N$ of eigenvectors of $W_N$, chosen measurably, with $W_Nu_i=\lambda_iu_i$, and set $x_i=\ip{x}{u_i}$. The map $x\mapsto(x_1,\dots,x_N)$ is an orthogonal transformation of $\R^N$. It maps $S_N$ to itself, preserves $\nu_N$, and commutes with $\nabla_{S_N}$ and $\Delta_{S_N}$. In these coordinates,
\begin{equation}\label{eq:model-eigen}
\beta H_N(x)=\frac c2\sum_{i=1}^N\lambda_ix_i^2 .
\end{equation}
The distance between the time-$t$ law and the Gibbs measure does not depend on the choice of eigenbasis, because $\nu_N$ is invariant under rotations and the initial point is independent of $W_N$. When we study the dynamics for a fixed realization of $W_N$, we may therefore assume that $W_N=\diag(\lambda_1,\dots,\lambda_N)$. We call $x_1,\dots,x_N$ the eigencoordinates. No property of the eigenvectors of $W_N$ is used.

\paragraph{Adding a multiple of the identity.}
For every $\tau\in\R$ and $x\in S_N$, we have $x^\T(A+\tau I)x/2=x^\T Ax/2+\tau N/2$. Since the added term is constant on $S_N$, we obtain
\begin{equation}\label{eq:model-shift}
L_{A+\tau I}=L_A,\qquad \pi_{A+\tau I}=\pi_A,\qquad P_t^{A+\tau I}=P_t^A .
\end{equation}
Two choices of $\tau$ are used for the model. Taking $A=cW_N$ and $\tau=-c\lambda_1$, we find the form used at low temperature:
\begin{equation}\label{eq:model-raw}
\begin{gathered}
L=\Delta_{S_N}-\nabla_{S_N}F\cdot\nabla_{S_N},\qquad
\pi_{N,c}(dx)\propto e^{-F(x)}\nu_N(dx),\\
F(x)=\sum_{i=2}^Na_ix_i^2,\qquad
a_i=\frac c2(\lambda_1-\lambda_i).
\end{gathered}
\end{equation}
Here $a_1=0\le a_2\le\cdots\le a_N$, and we call the $a_i$ the scaled eigenvalue gaps. Similarly, for any $z>c\lambda_1$, the choice $\tau=-z$ yields
\begin{equation}\label{eq:model-precision-form}
\begin{gathered}
L=\Delta_{S_N}-\nabla_{S_N}\Big(\sum_{i=1}^N r_ix_i^2\Big)\cdot\nabla_{S_N},\qquad
\pi_{N,c}(dx)\propto \exp\Big(-\sum_{i=1}^N r_ix_i^2\Big)\nu_N(dx),\\
r_i=\frac{z-c\lambda_i}{2}>0 .
\end{gathered}
\end{equation}
The particular choice of $z$ that is used for the model is made in \Cref{sec:model-equilibrium}.

\paragraph{Reflections.}
For $1\le j\le N$, let $R_j$ be the reflection of the $j$th eigencoordinate, $R_j(x)=(x_1,\dots,-x_j,\dots,x_N)$, and let $\mathcal G$ be the group generated by $R_1,\dots,R_N$. It contains the antipodal map $x\mapsto-x$. For $J\subset\{1,\dots,N\}$, let $Q_J$ be the map that replaces $x_j$ by $|x_j|$ for all $j\in J$.

\begin{lemma}[Reflections of eigencoordinates]\label{lem:model-symmetry}
Let $A=\diag(\alpha_1,\dots,\alpha_N)$.
\begin{enumerate}
\item For every $g\in\mathcal G$, every bounded measurable $f$ on $S_N$, and every $t\ge0$, we have $P_t^A(f\circ g)=(P_t^Af)\circ g$. The measures $\nu_N$ and $\pi_A$ are invariant under $\mathcal G$, and so is $\nu_NP_t^A$ for every $t\ge0$.
\item Let $\mathcal H$ be a subgroup of $\mathcal G$, and let $\mu,\mu'$ be $\mathcal H$-invariant probability measures on $S_N$. Then
\[
\|\mu-\mu'\|_{\TV}=\sup\big\{|\mu(B)-\mu'(B)|:B\subset S_N\text{ Borel and }\mathcal H\text{-invariant}\big\}.
\]
In particular, if $\mathcal H$ is generated by $(R_j)_{j\in J}$, then $\|Q_{J\#}\mu-Q_{J\#}\mu'\|_{\TV}=\|\mu-\mu'\|_{\TV}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Each $g\in\mathcal G$ is a diagonal orthogonal matrix. It commutes with $A$, it satisfies $g\,\Pi(x)=\Pi(gx)\,g$, and $gB$ is a standard Brownian motion. If $X$ solves \eqref{eq:model-sde} with $c\,W_N$ replaced by $A$ and $X_0=x$, then multiplying the equation by $g$ shows that $gX$ solves the same equation with Brownian motion $gB$ and initial point $gx$. By uniqueness in law, we have $P_t^A(f\circ g)(x)=\E f(gX_t)=(P_t^Af)(gx)$. The measures $\nu_N$ and $\pi_A$ are invariant because $g$ is an isometry of $S_N$ and $x^\T Ax=(gx)^\T A(gx)$. For every bounded measurable $f$, we then have
\[
(\nu_NP^A_t)(f\circ g)=\nu_N\big((P^A_tf)\circ g\big)=\nu_N(P^A_tf)=(\nu_NP_t^A)(f),
\]
which proves the first part.

For the second part, the inequality $\ge$ is immediate. Let $\sigma=\mu-\mu'$, and let $P\subset S_N$ be a positive set of a Hahn decomposition of $\sigma$, so that $\sigma(P)=\|\mu-\mu'\|_{\TV}$ and $\sigma(D)\ge0$ for all Borel sets $D\subset P$. Each set $hP$, $h\in\mathcal H$, is also a positive set, because $\sigma$ is $\mathcal H$-invariant. The set $B=\bigcup_{h\in\mathcal H}hP$ is $\mathcal H$-invariant, and it is a positive set because every Borel subset of $B$ is a finite disjoint union of Borel subsets of the sets $hP$. We have $\sigma(B)=\sigma(P)+\sigma(B\setminus P)\ge\sigma(P)$ because $P\subset B$, and the equality follows. If $\mathcal H$ is generated by $(R_j)_{j\in J}$, every $\mathcal H$-invariant Borel set $B$ satisfies $B=Q_J^{-1}(B')$ with $B'=\{x\in B:x_j\ge0\text{ for }j\in J\}$. For such $B$, we have $|\mu(B)-\mu'(B)|=|Q_{J\#}\mu(B')-Q_{J\#}\mu'(B')|\le\|Q_{J\#}\mu-Q_{J\#}\mu'\|_{\TV}$. The reverse inequality holds because $Q_{J\#}\mu(B'')-Q_{J\#}\mu'(B'')=\mu(Q_J^{-1}B'')-\mu'(Q_J^{-1}B'')$ for all Borel sets $B''$.
\end{proof}

It follows from the first part of \Cref{lem:model-symmetry} that the density of $\nu_NP_t$ with respect to $\pi_{N,c}$ is invariant under every reflection of an eigencoordinate. It is therefore orthogonal in $L^2(\pi_{N,c})$ to every function that is odd in some eigencoordinate, and such eigenfunctions of $L$ do not affect the distance of the time-$t$ law from equilibrium. An analogous observation for symmetric initial laws, in a spherical quadratic model with a large spike, was made by Gheissari, Grant, and Yu \cite[Definition~1.1 and Theorem~1.2(2)]{GheissariGrantYu2026}.

We use three classes of symmetric functions, whose spectral gaps can differ.

\begin{definition}[Symmetry sectors]\label{def:model-sectors}
Let $A=\diag(\alpha_1,\dots,\alpha_N)$, and let $\mu$ be either $\nu_N$ or $\pi_A$. The \emph{coordinate-even sector}, the \emph{antipodal-even sector}, and the \emph{sector even in $x_1$} are the closed subspaces of $L^2(\mu)$ that consist of the functions $f$ with $f\circ R_j=f$ for all $j$, with $f(-x)=f(x)$, and with $f\circ R_1=f$, respectively. The spectral gap of $L_A$ in one of these sectors is the infimum in the definition of $\lambda_{\mathrm{gap}}(L_A)$, restricted to nonconstant smooth functions in that sector; both the variance and the Dirichlet energy are taken under $\pi_A$. We speak of the coordinate-even and the antipodal-even spectral gaps, and of the spectral gap of the sector even in $x_1$.
\end{definition}

The measures $\nu_N$ and $\pi_A$ have equivalent $L^2$ norms, since $d\pi_A/d\nu_N$ is continuous and strictly positive on the compact sphere, so these sectors have the same functions and null-set convention for either choice of $\mu$. The coordinate-even sector is contained in the other two. By the first part of \Cref{lem:model-symmetry}, the semigroup $P^A_t$ maps each sector into itself. A function in the sector even in $x_1$ is determined by its restriction to the hemisphere $\{x_1\ge0\}$, and its variance and Dirichlet energy under $\pi_A$ equal those of this restriction under $\pi_A(\cdot\mid x_1\ge0)$, with the Dirichlet form of the hemisphere that imposes no boundary condition at the equator $\{x_1=0\}$ (see the proof of \Cref{lem:ineq-reflection}). In this sense, the sector even in $x_1$ describes the dynamics folded onto the hemisphere. Similarly, the coordinate-even sector describes the dynamics folded onto the orthant $\{x_i\ge0\text{ for all }i\}$.

\section{Equilibrium parameters and temperature regimes}\label{sec:model-equilibrium}

\begin{definition}[Lagrange multiplier and precisions]\label{def:model-multiplier}
Let $A$ be a real symmetric matrix with eigenvalues $\alpha_1\ge\cdots\ge\alpha_N$. The function $z\mapsto N^{-1}\sum_{i}(z-\alpha_i)^{-1}$ is continuous and strictly decreasing on $(\alpha_1,\infty)$, and it tends to $+\infty$ as $z\downarrow\alpha_1$ and to $0$ as $z\to\infty$. By the intermediate value theorem, there exists a unique $z>\alpha_1$ such that
\begin{equation}\label{eq:model-multiplier}
\frac1N\sum_{i=1}^N\frac1{z-\alpha_i}=1 .
\end{equation}
We call $z$ the Lagrange multiplier of $A$. We define the precisions $p_i$, the variances $q_i$, the matrix $Q$, the smallest precision $d$, the largest precision $s$, the ratio $\kappa$, and the half-precisions $r_i$ by
\begin{equation}\label{ht-canonical-quantities}
p_i=z-\alpha_i>0,\quad q_i=\frac1{p_i},\quad Q=(zI-A)^{-1},\quad d=p_1,\quad s=p_N,\quad \kappa=\frac sd,\quad r_i=\frac{p_i}2 .
\end{equation}
By \eqref{eq:model-multiplier}, $\sum_iq_i=\Tr Q=N$. We call this identity the Gaussian covariance trace constraint, or briefly the trace condition.
\end{definition}

For the model, $A=cW_N$ and $c\ge0$, we write $z_N=z_N(c)$ and $d_N$, $s_N$ for the corresponding quantities, so that
\begin{equation}\label{sp-canonical-definition}
\begin{gathered}
\frac1N\sum_{i=1}^N\frac1{z_N-c\lambda_i}=1,\qquad
p_i=z_N-c\lambda_i,\qquad
q_i=\frac1{z_N-c\lambda_i},\\
d_N=z_N-c\lambda_1,\qquad s_N=z_N-c\lambda_N .
\end{gathered}
\end{equation}
At $c=0$, we have $z_N=p_i=q_i=d_N=s_N=1$. With $z=z_N$ in \eqref{eq:model-precision-form}, the half-precisions $r_i=p_i/2$ are the coefficients of the quadratic potential. These names are used throughout the paper: the $a_i$ are the scaled eigenvalue gaps, the $r_i=p_i/2$ are the half-precisions, and in the critical window the limits of $N^{2/3}r_i$ are the limiting scaled half-precisions $R_i$ of \Cref{sec:spec-critical}. The unnormalized process of \Cref{sec:process} is built from a potential $\sum_i\rho_ix_i^2$, where the quadratic potential coefficients $\rho_i$ are either $r_i$ or $a_i$. Its coordinate $Y_i$ then has drift rate $2\rho_i$, that is, $2r_i=p_i$ or $2a_i$.

\begin{remark}[Gaussian representation]\label{rem:model-gaussian}
The names in \Cref{def:model-multiplier} come from the following representation. For any $z'>\alpha_1$, let $Y$ be a centered Gaussian vector whose coordinates in an eigenbasis of $A$ are independent with variances $(z'-\alpha_i)^{-1}$. Its density is proportional to $\exp(-z'|y|^2/2+y^\T Ay/2)$. Here conditioning on $|Y|^2=N$ means the weakly continuous version given by the coarea formula: its density with respect to the surface measure of $S_N$ is proportional to the density of $Y$ restricted to $S_N$, since the coarea factor $|\nabla|y|^2|^{-1}=(2\sqrt N)^{-1}$ is constant there. The term $-z'|y|^2/2$ is also constant on $S_N$, and this conditional law is $\pi_A$; see also \Cref{sec:conditioning}. The choice $z'=z$ in \eqref{eq:model-multiplier} makes the unconditioned mean squared norm $\E|Y|^2=\sum_iq_i$ equal to $N$, and $z$ plays the role of a Lagrange multiplier for this constraint. This is the device used by Kosterlitz, Thouless, and Jones \cite{KosterlitzThoulessJones1976} to solve the statics of the model. At the critical temperature $c=1$, this representation is used by Du and Huang \cite[Section~1.2, (1.7)--(1.8)]{DuHuang2026}.
Two cautions apply. The coordinates of a sample from $\pi_A$ are not independent, and in general their variances are not exactly $q_i$. Also, conditioning $Y$ on $|Y|^2=N$ is different from normalizing it radially: the law of $\sqrt N\,Y/|Y|$ has density proportional to $(\sum_ix_i^2/q_i)^{-N/2}$ with respect to $\nu_N$, which is not $\pi_A$ unless all $q_i$ are equal. In the isotropic case both constructions produce $\nu_N$. In particular, if $g$ is a standard Gaussian vector in $\R^N$, then $\sqrt N\,g/|g|$ has law $\nu_N$.
\end{remark}

The distance between two Gaussian laws conditioned on $S_N$ is governed by the following quantity, which we use for the high-temperature profiles.

\begin{definition}[Information]\label{def:model-information}
Let $q=(q_i)_{i=1}^n$ and $v=(v_i)_{i=1}^n$ be sequences of positive numbers with $\sum_iq_i=\sum_iv_i$, that is, the diagonals of two diagonal covariance matrices with the same trace. We call
\begin{equation}\label{eq:model-information}
\cI(q,v)=\frac12\sum_{i=1}^n\Big(\frac{v_i}{q_i}-1\Big)^2
\end{equation}
the \emph{information of $v$ relative to $q$}. For a real symmetric matrix $A$ with variances $q_i$ as in \Cref{def:model-multiplier}, and for $t\ge0$, we set $\cI_N(t)=\cI(q,v(t))$, where $v(t)$ is the covariance evolution of $A$, defined in \Cref{def:heat-covariance}. For the model, $A=cW_N$.
\end{definition}

In terms of the relative errors $e_i=v_i/q_i-1$, we have $\cI(q,v)=\frac12\sum_ie_i^2$. The information is not symmetric in $q$ and $v$, and it is not a Fisher information; in this paper, Fisher information and relative Fisher information always refer to integrated squared logarithmic derivatives of densities (\Cref{sec:scores}). The name comes from Le Cam's theory: when the relative errors are small and no coordinate carries a macroscopic share of the variance, the log-likelihood ratio between the conditioned Gaussian laws with variances $v$ and $q$ is approximately Gaussian with variance $\cI(q,v)$ (\Cref{sec:likelihood}), as in a Gaussian shift experiment with information $\cI(q,v)$ (\Cref{rem:model-profiles}).

\paragraph{Temperature regimes.}
The main results concern the following regimes and parameters. Temperature sequences need not be monotone.
\begin{itemize}
\item \emph{Infinite-temperature crossover:} $c_N\to0$, with $J_N=Nc_N^2$.
\item \emph{Fixed high temperature:} $0<c<1$ fixed. The smallest precision satisfies $d_N\to(1-c)^2$ (\Cref{sp-canonical-saddle}).
\item \emph{Approach to the critical temperature from above:} $c_N=1-\delta_N$ with $\delta_N\in(0,1)$, $\delta_N\to0$, and $M_N=N\delta_N^3\to\infty$. We write $\ell_N=\log(2+M_N)$.
\item \emph{Critical window:} $c_N=1+\sigma N^{-1/3}$ with $\sigma\in\R$ fixed, and times $t=N^{2/3}s$ with $s>0$ fixed. The half-precisions $r_i=p_i/2$ are used, and the random limiting objects are defined in \Cref{sec:spec-critical}.
\item \emph{Approach to the critical temperature from below:} $c_N=1+\epsilon_N$ with $\epsilon_N>0$, $\epsilon_N\to0$, and $N\epsilon_N^3\to\infty$.
\item \emph{Fixed low temperature:} $c>1$ fixed.
\end{itemize}
In the two low-temperature regimes we use the form \eqref{eq:model-raw} and the parameters
\begin{equation}\label{eq:model-cold-parameters}
\theta_N=1-\frac1N\sum_{i=2}^N\frac1{2a_i},\qquad
R_N=Na_2,\qquad
m_N=\theta_NR_N .
\end{equation}
The quantity $N(1-\theta_N)$ is the mean squared norm of a centered Gaussian vector $(Z_i)_{i\ge2}$ with independent coordinates of variances $(2a_i)^{-1}$, and $N\theta_N$ is the deficit of this mean with respect to $N$. We call $\theta_N$ the top fraction. It is a Gaussian variance deficit, defined through an unconditioned Gaussian mean. For general positive $a_2,\dots,a_N$, it satisfies $\theta_N<1$ but may be negative, and at finite $N$ it need not equal $\pi_{N,c}(x_1^2)/N$. Only asymptotically does it describe the squared projection $x_1^2/N$ of $x/\sqrt N$ onto the top eigenvector under the Gibbs measure: in either low-temperature regime, \Cref{cor:coldf-top-fraction} shows that for every fixed $\gamma>0$,
\[
\pi_{N,c_N}\Big(\Big|\frac{x_1^2}{N\theta_N}-1\Big|>\gamma\Big)\longrightarrow0\quad\text{in probability},
\]
where the left side is set to $0$ on the event $\{\theta_N\le0\}$, whose probability tends to zero. At fixed $c>1$, we have $\theta_N\to\theta=1-1/c>0$ in probability, while along the approach from below $\theta_N/\epsilon_N\to1$, so that $\theta_N\to0$ (\Cref{sp-condensed-geometry}). On the events of that lemma, $R_N\asymp N^{1/3}$, and therefore $m_N^3\asymp N\theta_N^3$; along an approach from below, $N\theta_N^3$ plays the role of $M_N$. In the low-temperature estimates, we abbreviate $m_N=\theta_NR_N$ by $m$.

\section{Distances, mixing times, and profiles}\label{sec:model-distances}

For probability measures $\mu,\eta$ on a measurable space, the total variation distance is
\[
\|\mu-\eta\|_{\TV}=\sup_B|\mu(B)-\eta(B)|,
\]
the supremum over measurable sets. If $\mu\ll\eta$, the relative entropy is $\KL(\mu\Vert\eta)=\int\log(d\mu/d\eta)\,d\mu\in[0,\infty]$; otherwise $\KL(\mu\Vert\eta)=\infty$. For the model, we set
\begin{equation}\label{model-distance}
D_N(t)=\|\nu_NP_t-\pi_{N,c}\|_{\TV},\qquad
t_N(\eps)=\inf\{t\ge0:D_N(t)\le\eps\}\quad(0<\eps<1).
\end{equation}
We call $t_N(\eps)$ the mixing time at accuracy $\eps$. Ratios of mixing times are set to $1$ when their denominator is zero. Here the initial point has law $\nu_N$ and is independent of $W_N$, and both the initial point and the Brownian motion are integrated out, so that $D_N(t)$ and $t_N(\eps)$ are random only through $W_N$. When several temperatures are considered at once, they share the same matrix $W_N$. We never couple the diffusions at different temperatures.

The initial point is integrated out before the distance is taken: $D_N(t)$ is the distance of the averaged law $\nu_NP_t$ from equilibrium. A formulation in which an initial point is sampled once from the Gibbs measure and held fixed would instead concern the Gibbs mass of initial points $x$ for which $\|\delta_xP_t-\pi_{N,c}\|_{\TV}$ exceeds a threshold. Our results do not address that question; see \hyperref[sec:intro-discussion]{the discussion in the introduction}.

We collect standard facts about these distances.

\begin{lemma}\label{lem:model-tv}
Let $\mu,\eta$ be probability measures on a measurable space $E$.
\begin{enumerate}
\item If $\mu$ and $\eta$ have densities $f,g$ with respect to a $\sigma$-finite measure $\lambda$, then $\|\mu-\eta\|_{\TV}=\int(f-g)_+\,d\lambda$, which is half of $\int|f-g|\,d\lambda$. Moreover $\|\mu-\eta\|_{\TV}=\sup\{|\mu(\varphi)-\eta(\varphi)|:0\le\varphi\le1\}$.
\item If $K$ is a Markov kernel from $E$ to another measurable space, then $\|\mu K-\eta K\|_{\TV}\le\|\mu-\eta\|_{\TV}$ and $\KL(\mu K\Vert\eta K)\le \KL(\mu\Vert\eta)$. In particular, both inequalities hold for the push-forwards under a measurable map.
\item For the model, $t\mapsto D_N(t)$ is nonincreasing.
\item \textup{(Pinsker's inequality)} $\|\mu-\eta\|_{\TV}\le\sqrt{\KL(\mu\Vert\eta)/2}$.
\item If $\mu=\mu_1\otimes\mu_2$ and $\eta=\eta_1\otimes\eta_2$, then $\|\mu-\eta\|_{\TV}\le\|\mu_1-\eta_1\|_{\TV}+\|\mu_2-\eta_2\|_{\TV}$.
\item For $m\in\R$, $\|\cN(m,1)-\cN(0,1)\|_{\TV}=2\Phi(|m|/2)-1$, where $\Phi$ is the standard normal distribution function.
\end{enumerate}
\end{lemma}

\begin{proof}
For the first part, let $B=\{f>g\}$. For every measurable $B'$, we have $\mu(B')-\eta(B')\le\int_{B'\cap B}(f-g)\,d\lambda\le\mu(B)-\eta(B)$, and similarly $\eta(B')-\mu(B')\le\eta(B^c)-\mu(B^c)=\mu(B)-\eta(B)$. We conclude that $\|\mu-\eta\|_{\TV}=\int(f-g)_+\,d\lambda$, which is half of $\int|f-g|\,d\lambda$ because $\int(f-g)\,d\lambda=0$. The same argument bounds $\int\varphi(f-g)\,d\lambda$ for $0\le\varphi\le1$, which proves the last claim.
For the second part, observe that for $0\le\varphi\le1$, the function $K\varphi$ also takes values in $[0,1]$, and $(\mu K)(\varphi)-(\eta K)(\varphi)=\mu(K\varphi)-\eta(K\varphi)$. The claim follows from the last statement of the first part, applied to $\mu,\eta$ and $\lambda=\mu+\eta$. The inequality for relative entropy is the data processing inequality \cite[Theorem~9]{vanErvenHarremoes2014}, with order one.
The third part follows from the second. Indeed, $\pi_{N,c}P_s=\pi_{N,c}$, and the second part implies that $D_N(t+s)=\|(\nu_NP_t)P_s-\pi_{N,c}P_s\|_{\TV}\le D_N(t)$.
The fourth part is Pinsker's inequality \cite[Theorem~31]{vanErvenHarremoes2014}, whose variational distance is twice ours.
For the fifth part, by the triangle inequality, we have $\|\mu-\eta\|_{\TV}\le\|\mu_1\otimes\mu_2-\eta_1\otimes\mu_2\|_{\TV}+\|\eta_1\otimes\mu_2-\eta_1\otimes\eta_2\|_{\TV}$, and the second part, applied to the kernels $x\mapsto\delta_x\otimes\mu_2$ and $y\mapsto\eta_1\otimes\delta_y$, bounds each term.
For the sixth part, we may take $m>0$. The densities of $\cN(m,1)$ and $\cN(0,1)$ cross only at $m/2$. By the first part, the distance is then $\Phi(m/2)-\Phi(-m/2)=2\Phi(m/2)-1$.
\end{proof}

We now define what we mean by a mixing profile, by a cutoff profile, and by cutoff. The centers, windows, and time scales below depend on the realized spectrum, and they are random variables.

\begin{definition}[Mixing profiles, cutoff profiles, and cutoff]\label{def:model-cutoff}\label{def:model-profiles}
A \emph{mixing profile} of $D_N$ is a limit of the functions $u\mapsto D_N(T_N+w_Nu)$, for some centers $T_N\ge0$ and windows $w_N>0$, or of the functions $s\mapsto D_N(b_Ns)$, for some time scale $b_N>0$, in probability or in distribution, uniformly on compact subsets of the domain of the limit. A mixing profile may be random.

Let $P:\R\to(0,1)$ be continuous and strictly decreasing, with $P(u)\to1$ as $u\to-\infty$ and $P(u)\to0$ as $u\to\infty$. Let $T_N,w_N>0$ be random variables. We say that $D_N$ has \emph{cutoff profile $P$ with center $T_N$ and window $w_N$} if $w_N/T_N\to0$ in probability and, for every compact $K\subset\R$,
\[
\sup_{u\in K}\big|D_N(T_N+w_Nu)-P(u)\big|\longrightarrow0\quad\text{in probability},
\]
where the supremum is restricted to $u$ with $T_N+w_Nu\ge0$. We say that there is \emph{cutoff} if, for every $\eps\in(0,1/2)$, we have $t_N(1-\eps)>0$ with probability tending to one and $t_N(\eps)/t_N(1-\eps)\to1$ in probability.
\end{definition}

A cutoff profile is a mixing profile. The critical mixing profiles $F_\sigma$ of \Cref{sec:crit-profile} are limits in distribution of $s\mapsto D_N(N^{2/3}s)$, uniformly on compact subsets of $(0,\infty)$. They are random, and they are not cutoff profiles. This definition of cutoff refers to the fixed initial law $\nu_N$ and not to the worst initial point. For background on cutoff and profiles, see \cite[Chapter~18]{LevinPeres2017} and \cite{Diaconis1996}.
A cutoff profile implies cutoff by \Cref{cor:model-profile-cutoff} below.

\section{Disorder quantifiers and constants}\label{sec:model-quantifiers}

Our estimates for the dynamics are deterministic statements about a fixed matrix whose spectrum satisfies explicit hypotheses. The random matrix enters only through the probability that these hypotheses hold. We use the following convention throughout.

\paragraph{Eventual statements.}
A statement holds \emph{for all sufficiently large $N$}, or \emph{eventually}, if there exists $N_0$ such that it holds for all $N\ge N_0$. The threshold $N_0$ may depend on $c$ (or on the sequence $(c_N)$), on the confidence $\eta$, and on the fixed parameters named in the statement, including fixed time anchors. It never depends on $W_N$, on the initial point, or on a varying time over which uniformity is claimed.

\paragraph{Fixed-confidence events.}
A statement holds \emph{at every fixed confidence} if, for every $\eta\in(0,1)$, there exist constants depending only on $\eta$ and on the fixed parameters of the statement, and events $\mathcal G_N^\eta$ with $\liminf_{N\to\infty}\PP(\mathcal G_N^\eta)\ge1-\eta$, such that the statement holds on $\mathcal G_N^\eta$ with these constants for all sufficiently large $N$. The constants are chosen before $N\to\infty$, and the confidence $\eta$ is sent to zero last. Such estimates say nothing about exponentially small failure probabilities, and the events $\mathcal G_N^\eta$ are not a single family of events of probability tending to one with fixed constants.

\paragraph{Probability modes.}
Unless said otherwise, ``in probability'' refers to the law of $W_N$, and so do $X_N=O_{\PP}(a_N)$ (that is, $(X_N/a_N)$ is tight) and $X_N=o_{\PP}(a_N)$ (that is, $X_N/a_N\to0$ in probability). Probabilities over the initial point and the noise are written explicitly when they occur. For fixed $W_N$, they are deterministic quantities. Convergence in distribution, written $\Rightarrow$, is used only where a random limit appears, as in the critical window. ``With probability tending to one'' means with probability over $W_N$ tending to one as $N\to\infty$.

\paragraph{Power exponents.}
For a positive sequence $(a_N)$ and $r\in\R$, we write $a_N=N^{r+o(1)}$ if, for every fixed $\eta>0$, we have $N^{r-\eta}\le a_N\le N^{r+\eta}$ for all sufficiently large $N$, with a threshold that may depend on $\eta$. For nonnegative random variables $U_N$, we write $U_N=N^{r+o_{\PP}(1)}$ if $\PP\{N^{r-\eta}\le U_N\le N^{r+\eta}\}\to1$ for every fixed $\eta>0$.

\paragraph{Comparisons.}
We write $X\lesssim Y$ if $X\le CY$ for a positive constant $C$, and $X\asymp Y$ if $X\lesssim Y\lesssim X$, with constants as below. Subscripts indicate extra dependence, as in $\asymp_K$.

The next lemma records how a fixed-confidence statement is converted into convergence in probability. It is used in every application to the GOE without further comment.

\begin{lemma}[Convergence in probability from fixed-confidence estimates]\label{lem:model-localization}
Let $X_N$ be real random variables. Suppose that for every $\eta\in(0,1)$ there exist events $\mathcal G_N^\eta$ with $\liminf_N\PP(\mathcal G^\eta_N)\ge1-\eta$ such that $\PP(\mathcal G_N^\eta\cap\{|X_N|>\eps\})\to0$ for every $\eps>0$. Then $X_N\to0$ in probability. If instead, for every $\eta$, there exists a constant $C_\eta$ with $|X_N|\le C_\eta$ on $\mathcal G^\eta_N$ for all large $N$, then $X_N=O_{\PP}(1)$.
\end{lemma}

\begin{proof}
For every $\eps>0$ and $\eta\in(0,1)$, we have $\limsup_N\PP(|X_N|>\eps)\le\limsup_N\PP(\mathcal G_N^\eta\cap\{|X_N|>\eps\})+\limsup_N\PP((\mathcal G_N^\eta)^c)\le\eta$. Let $\eta\to0$. The second claim is proved in the same way.
\end{proof}

\paragraph{Constants.}
We write $C,C',C_0,c_0,c_*,\dots$ for positive constants whose dependence is stated each time. Their values may change from line to line. The letter $c$ alone is reserved for the rescaled inverse temperature $c=\sqrt2\,\beta$. Unless stated otherwise, constants do not depend on $N$ or on $W_N$; any dependence on the realized spectrum or on the Airy point process, as in \hyperref[thm:intro-critical]{the critical-window theorem}, is stated explicitly.

\section{The limiting profiles}\label{sec:model-profiles}

Let $\Phi$ and $\phi$ be the standard normal distribution function and density. Throughout, $G$, $G'$ denote standard Gaussian random variables and $C$ a standard Cauchy random variable, all independent. We define
\begin{equation}\label{hp-universal-profile}
\Psi(u)=2\Phi\Big(\frac{e^{-2u}}2\Big)-1,\qquad
u_\eps=-\frac12\log\Big[2\,\Phi^{-1}\Big(\frac{1+\eps}2\Big)\Big]\quad(0<\eps<1),
\end{equation}
and, for $u\in\R$ and $\theta>0$,
\begin{equation}\label{eq:model-cauchy-profiles}
\begin{aligned}
\cV(u)&=\big\|\Law(G+e^{-2u}C)-\Law(G)\big\|_{\TV},\\
\cV_\theta(u)&=\big\|\Law(G+\sqrt{2\theta}\,e^{-2u}C)-\Law(G)\big\|_{\TV},
\end{aligned}
\end{equation}
where $\Law(\cdot)$ denotes the law of a random variable. For $h>0$, let $K_h$ be the Ornstein--Uhlenbeck transition kernel
\begin{equation}\label{eq:model-ou-kernel}
K_h(x,\cdot)=\Law\big(e^{-2h}x+\sqrt{1-e^{-4h}}\,G'\big),
\end{equation}
which is the transition kernel at time $h$ of $dZ_h=-2Z_h\,dh+2\,dB_h$. Its density $y\mapsto(1-e^{-4h})^{-1/2}\phi\big((y-e^{-2h}x)(1-e^{-4h})^{-1/2}\big)$ is strictly positive, and $\cN(0,1)K_h=\cN(0,1)$.

\begin{lemma}[Properties of the limiting profiles]\label{cot-profile}\label{cq-jp-strict-profile-monotonicity-and-exact-accuracy-offsets}\label{lem:model-profiles}
\begin{enumerate}
\item We have $\Psi(u)=\|\cN(e^{-2u},1)-\cN(0,1)\|_{\TV}$. The function $\Psi$ is continuous and strictly decreasing on $\R$, with $\Psi(u)\to1$ as $u\to-\infty$ and $\Psi(u)\to0$ as $u\to\infty$, and $\Psi^{-1}(\eps)=u_\eps$ for $0<\eps<1$.
\item The function $\cV$ is continuous and strictly decreasing on $\R$, with $\cV(u)\to1$ as $u\to-\infty$ and $\cV(u)\to0$ as $u\to\infty$. Every $\eps\in(0,1)$ has a unique preimage $\cV^{-1}(\eps)$, which depends continuously on $\eps$. Moreover, for $z>0$ and $h>0$, we have $\Law(G+zC)K_h=\Law(G+e^{-2h}zC)$.
\item For every $\theta>0$ and $u\in\R$, we have $\cV_\theta(u)=\cV\big(u-\log(2\theta)/4\big)$. In particular, $\cV_\theta$ has the properties of the second part, and $\cV_\theta^{-1}(\eps)=\cV^{-1}(\eps)+\log(2\theta)/4$.
\end{enumerate}
\end{lemma}

\begin{proof}
The first part follows from the last part of \Cref{lem:model-tv} with $m=e^{-2u}$, since $m\mapsto2\Phi(m/2)-1$ is continuous and strictly increasing from $0$ to $1$ on $[0,\infty)$ and $u\mapsto e^{-2u}$ is a decreasing bijection of $\R$ onto $(0,\infty)$. Solving $2\Phi(e^{-2u}/2)-1=\eps$ for $u$ yields $u=u_\eps$.

For the second part, write $\mathsf T(z)=\|\Law(G+zC)-\Law(G)\|_{\TV}$ for $z\ge0$, so that $\cV(u)=\mathsf T(e^{-2u})$. The law of $G+zC$ has the density $f_z(y)=\E\,\phi(y-zC)$, which is continuous and strictly positive. If $z_n\to z\ge0$, then $f_{z_n}(y)\to f_z(y)$ for all $y$ by bounded convergence, with $f_0=\phi$, and Scheffé's lemma \cite[pp.~435--437]{Scheffe1947} implies that $f_{z_n}\to f_z$ in $L^1$; equivalently, apply dominated convergence to $\min\{f_{z_n},f_z\}$ and use that both densities integrate to one. By the first part of \Cref{lem:model-tv}, $\mathsf T$ is continuous on $[0,\infty)$ and $\mathsf T(0)=0$. This proves that $\cV$ is continuous and that $\cV(u)\to0$ as $u\to\infty$. For the other limit, fix $\ell>0$ and the interval $I=[-\ell,\ell]$. Conditioning on $G$ and using that the density of $zC$ is bounded by $1/(\pi z)$, we obtain $\PP(G+zC\in I)\le2\ell/(\pi z)$. Then
\[
\mathsf T(z)\ge\PP(G\in I)-\PP(G+zC\in I)\ge1-\PP(|G|>\ell)-\frac{2\ell}{\pi z}.
\]
Letting $z\to\infty$ and then $\ell\to\infty$, we obtain $\mathsf T(z)\to1$, that is, $\cV(u)\to1$ as $u\to-\infty$.

To prove strict monotonicity, we first note that $e^{-2h}G+\sqrt{1-e^{-4h}}\,G'$ is a standard Gaussian independent of $C$. Then
\[
\Law(G+zC)K_h=\Law\big(e^{-2h}G+\sqrt{1-e^{-4h}}\,G'+e^{-2h}zC\big)=\Law(G+e^{-2h}zC),
\]
and $\Law(G)K_h=\Law(G)$. Fix $z>0$ and $h>0$, and let $\sigma=\Law(G+zC)-\Law(G)$. We have $\sigma\ne0$, because the characteristic function of $G+zC$ is $e^{-\xi^2/2-z|\xi|}$. Let $\sigma=\sigma^+-\sigma^-$ be its Jordan decomposition. Since $\sigma(\R)=0$, both $\sigma^+$ and $\sigma^-$ are nonzero, with total mass $\mathsf T(z)$. The measures $\sigma^\pm K_h$ have densities $a^\pm(y)=\int k_h(x,y)\,\sigma^\pm(dx)$, where $k_h$ is the strictly positive density of $K_h(x,\cdot)$. In particular, $a^+(y)>0$ and $a^-(y)>0$ for all $y$. By the first part of \Cref{lem:model-tv},
\[
\mathsf T(e^{-2h}z)=\|\sigma K_h\|_{\TV}=\frac12\int|a^+-a^-|\,dy
=\frac12\int(a^++a^-)\,dy-\int\min(a^+,a^-)\,dy<\mathsf T(z).
\]
In the last step, we used $\int(a^++a^-)\,dy=2\,\mathsf T(z)$ and $\int\min(a^+,a^-)\,dy>0$. We conclude that $\mathsf T$ is strictly increasing on $(0,\infty)$, and that $\cV$ is strictly decreasing. A continuous strictly decreasing bijection of $\R$ onto $(0,1)$ has a continuous inverse, which proves the second part.

For the third part, note that $\sqrt{2\theta}\,e^{-2u}=\exp\big(-2(u-\log(2\theta)/4)\big)$, so that $\cV_\theta(u)=\cV(u-\log(2\theta)/4)$. The remaining claims follow from the second part.
\end{proof}

\begin{remark}[The Gaussian and Gaussian--Cauchy profiles]\label{rem:model-profiles}
The function $\Psi(u)=2\Phi(\sqrt I/2)-1$ with $I=e^{-4u}$ is the total variation distance in a Gaussian shift experiment with information $I$, the limit experiment in Le Cam's theory of local asymptotic normality \cite[Chapters~6--7]{vanDerVaart1998}. As a cutoff profile, $\Psi$ appears in the work of Diaconis, Graham, and Morrison on the random walk on the hypercube \cite[Theorem~1]{DiaconisGrahamMorrison1990}: at time $(n\log n)/4+un$ the distance tends to $\operatorname{erf}(e^{-2u}/\sqrt8)=\Psi(u)$. Gaussian profiles of this kind also arise for Ornstein--Uhlenbeck processes \cite[Lemma~3.4]{BoursierChafaiLabbe2023}.
The law of $G+zC$ is the convolution of a Gaussian and a Cauchy law, and its density is a Voigt profile \cite{Voigt1912}, \cite[\S7.19]{OlverLozierBoisvertClark2010}. We use the total variation representation \eqref{eq:model-cauchy-profiles} for $\cV$. In the low-temperature sections, the variable $C$ arises as the ratio of two independent centered Gaussian variables.
\end{remark}

The next lemma converts pointwise convergence of a nonincreasing distance to a profile into locally uniform convergence and convergence of the mixing times.

\begin{lemma}[Monotone profiles and mixing times]\label{hp-monotone-profile-transfer}
Let $D_n:\Omega\times[0,\infty)\to[0,1]$ be jointly measurable, with nonincreasing sample functions $t\mapsto D_n(t)$ satisfying $\lim_{t\to\infty}D_n(t)=0$ almost surely, and let $b_n,w_n>0$ be random variables with $b_n/w_n\to\infty$ in probability. Let $V:\R\to(0,1)$ be continuous and strictly decreasing, with $V(u)\to1$ as $u\to-\infty$ and $V(u)\to0$ as $u\to\infty$. Suppose that for each fixed $u\in\R$ and $\eps>0$,
\begin{equation}\label{eq:model-pointwise}
\PP\big(b_n+w_nu\ge0,\ |D_n(b_n+w_nu)-V(u)|>\eps\big)\longrightarrow0 .
\end{equation}
Then the following hold, with all limits in probability.
\begin{enumerate}
\item For each compact interval $K\subset\R$, we have $\sup_{u\in K}|D_n(b_n+w_nu)-V(u)|\to0$, where the supremum is over $u\in K$ with $b_n+w_nu\ge0$; with probability tending to one, this is all of $K$.
\item Let $t_n(\eps)=\inf\{t\ge0:D_n(t)\le\eps\}$. For each compact interval $I\subset(0,1)$,
\[
\sup_{\eps\in I}\Big|\frac{t_n(\eps)-b_n}{w_n}-V^{-1}(\eps)\Big|\longrightarrow0 .
\]
\item For every sequence of random variables $\omega_n\to\infty$ in probability, $\inf\{D_n(t):0\le t\le b_n-w_n\omega_n\}\to1$, where the infimum over the empty set is $1$.
\end{enumerate}
If the hypotheses hold for deterministic $D_n$, $b_n$, $w_n$, with ordinary convergence in place of convergence in probability, then the conclusions hold with ordinary convergence.
\end{lemma}

\begin{proof}
For an empty $K$, the error supremum in the first part is zero. In its proof assume that $K$ is nonempty. For each fixed $u$, the event $\{b_n+w_nu\ge0\}$ has probability tending to one, because $b_n/w_n\to\infty$ in probability. We work on these events below. The restricted suprema in the first part are measurable. To see this, write $K=[a,b]$ and $\ell_n=\max\{a,-b_n/w_n\}$. If $\ell_n>b$, set the supremum over the empty interval to $0$. Otherwise, use a fixed countable dense subset of $K$ containing its endpoints, retain the points $u$ with $b_n+w_nu\ge0$, and also include the random left endpoint $\ell_n$. At every interior point, monotonicity places $D_n(b_n+w_nu)$ between its one-sided limits; continuity of $V$ and convexity of $x\mapsto|x-V(u)|$ therefore show that these values have the same supremum as the whole restricted interval. The countable supremum and the additional value $|D_n(b_n+w_n\ell_n)-V(\ell_n)|$ are measurable by joint measurability.

For the first part, fix $\delta>0$ and let $u_0<u_1<\cdots<u_k$ be a mesh with $K\subset[u_0,u_k]$ and $V(u_{j})-V(u_{j+1})\le\delta$ for all $j$, which is possible because $V$ is uniformly continuous on $[u_0,u_k]$. Let $e_n=\max_j|D_n(b_n+w_nu_j)-V(u_j)|$. Using \eqref{eq:model-pointwise} and a union bound over the finitely many mesh points, we find that $e_n\to0$ in probability. For $u\in[u_j,u_{j+1}]$, since $w_n>0$ and $D_n$ is nonincreasing, we have
\[
\begin{split}
V(u)-\delta-e_n&\le V(u_{j+1})-e_n\le D_n(b_n+w_nu_{j+1})\le D_n(b_n+w_nu)\\
&\le D_n(b_n+w_nu_j)\le V(u_j)+e_n\le V(u)+\delta+e_n .
\end{split}
\]
Taking the supremum over $u\in K$, we find $\sup_{u\in K}|D_n(b_n+w_nu)-V(u)|\le\delta+e_n$, and $\delta$ is arbitrary.

For the second part, the conclusion for $I=\emptyset$ is immediate, with the error supremum over the empty set equal to zero. Assume henceforth that $I$ is nonempty. Write $v=V^{-1}$, which is continuous on $(0,1)$. Fix $h>0$. The functions $\eps\mapsto V(v(\eps)-h)-\eps$ and $\eps\mapsto\eps-V(v(\eps)+h)$ are continuous and positive on the compact set $I$. Let $\gamma_h>0$ be a common lower bound for them on $I$. Let $K_h=[\min_Iv-h,\max_Iv+h]$. By the first part, with probability tending to one we have $\sup_{u\in K_h}|D_n(b_n+w_nu)-V(u)|<\gamma_h$, and then, simultaneously for all $\eps\in I$,
\[
D_n\big(b_n+w_n(v(\eps)-h)\big)>\eps,\qquad D_n\big(b_n+w_n(v(\eps)+h)\big)<\eps .
\]
The second inequality gives $t_n(\eps)\le b_n+w_n(v(\eps)+h)$. Because $D_n$ is nonincreasing, the first inequality implies that $D_n(t)>\eps$ for all $t\le b_n+w_n(v(\eps)-h)$, and then $t_n(\eps)\ge b_n+w_n(v(\eps)-h)$. We have shown that $|(t_n(\eps)-b_n)/w_n-v(\eps)|\le h$ for all $\eps\in I$, and $h$ is arbitrary. No continuity of $D_n$ is needed.

For the third part, fix $\eps>0$ and let $A>0$ be such that $V(-A)>1-\eps/2$. On the event $\{\omega_n\ge A\}$, which has probability tending to one, the infimum is either $1$ or equal to $D_n(b_n-w_n\omega_n)\ge D_n(b_n-w_nA)$, since $D_n$ is nonincreasing. By \eqref{eq:model-pointwise}, $D_n(b_n-w_nA)>1-\eps$ with probability tending to one.

The deterministic version follows from the same inequalities.
\end{proof}

\begin{corollary}[Cutoff profiles imply cutoff]\label{cor:model-profile-cutoff}
If $D_N$ has cutoff profile $P$ with center $T_N$ and window $w_N$ in the sense of \Cref{def:model-cutoff}, then $(t_N(\eps)-T_N)/w_N\to P^{-1}(\eps)$ in probability, uniformly for $\eps$ in compact subsets of $(0,1)$, and there is cutoff.
\end{corollary}

\begin{proof}
The definition of a cutoff profile includes the hypotheses of \Cref{hp-monotone-profile-transfer} with $b_n=T_N$ and $V=P$, since $D_N$ is nonincreasing by \Cref{lem:model-tv}. It is also jointly measurable in $(W_N,t)$, because it is measurable in $W_N$ for each $t$ and continuous in $t$: the density $P_th$ below is continuous in $t$ in $L^2(\pi_{N,c})$. Also $D_N(t)\to0$ as $t\to\infty$. Indeed, $\nu_N$ has a bounded density $h$ with respect to $\pi_{N,c}$, the density of $\nu_NP_t$ is $P_th$ by \eqref{eq:model-dirichlet}, and $\|P_th-1\|_{L^2(\pi_{N,c})}\to0$ because the spectrum of $-L$ on the compact connected manifold $S_N$ is discrete with kernel spanned by the constants. Then $2D_N(t)\le\|P_th-1\|_{L^2(\pi_{N,c})}$ by the Cauchy--Schwarz inequality. The first claim then follows from its second part. Fix $\eps\in(0,1/2)$ and write $t_N(\eps)=T_N\big(1+(w_N/T_N)(P^{-1}(\eps)+o_{\PP}(1))\big)$, and similarly for $1-\eps$. Since $w_N/T_N\to0$ in probability, we have $t_N(1-\eps)>0$ with probability tending to one and $t_N(\eps)/t_N(1-\eps)\to1$ in probability.
\end{proof}