\chapter{The critical profile far above the critical temperature}\label{sec:outer-hot}

In this section and the next, we prove \cref{thm:intro-outer}, which describes the critical profile $F_\sigma$ as $\sigma\to-\infty$ and as $\sigma\to+\infty$. This section treats the limit $\sigma=-S\to-\infty$. The main result is \cref{oc-full-hot}. It states that for every realization of the reflected GOE Airy point process $(\chi_i)$, the limit of $N^{2/3}(2-\lambda_i)$, outside a null set, we have $F_{-S}(\tau^{\mathrm h}_S+u/S^2)\to\Psi(u)$ locally uniformly in $u$, where
\begin{equation}\label{eq:outh-center}
\tau^{\mathrm h}_S=\frac{\log S^3-\frac32\log\log S^3-\log(16\sqrt\pi)}{4S^2},
\end{equation}
and that the inverse functions converge correspondingly.

The proof combines the convergence theorem of the critical window, which holds along deterministic sequences of half-precisions (\cref{cd-profile,cd-quantiles,cd-mean}), with four finite-dimensional estimates at high temperature whose constants do not depend on the dimension: the comparison of the time-$t$ law with the Bingham comparison law (\cref{mgf-global-bridge}), the likelihood ratio estimate for Gaussian laws conditioned on a common sphere (\cref{gp-pinned-lan-quantitative-third}), the decay of the relative errors of the covariance evolution (\cref{hi3-information-envelopes}), and the formula for the information (\cref{mc-exact-info}). We also use the properties of the GOE Airy point process in \cref{sp-airy-counts,sp-raw-gap-resolvent}, the convergence criterion for positive measures in \cref{pr-real-measure-transfer}, and \cref{cot-profile,hp-monotone-profile-transfer}. The finite approximations of the limiting scaled half-precisions constructed in \cref{oc-approximants} are used again in \cref{sec:outer-cold}.

\section{The order of limits}\label{sec:outh-limits}

The proof first approximates $F_{-S}$ by finite-dimensional diffusions at each fixed $S$, taking the dimension $N$ to infinity. It then lets $S\to\infty$.

This order of limits is different from the one in case (3) of \cref{thm:intro-main}, and neither statement implies the other. If we set $\delta_N=SN^{-1/3}$ in case (3), then $M_N=N\delta_N^3=S^3$, and the center and window of case (3), divided by $N^{2/3}$, converge to $\tau^{\mathrm h}_S$ and $S^{-2}$. However, the theorem behind case (3) assumes $M_N\to\infty$, and it gives no information when $M_N$ stays fixed while $N\to\infty$. Conversely, the critical theorem holds at fixed $\sigma$ and gives no information as $\sigma\to-\infty$.

The link between the two regimes is that several estimates of \cref{part:tools,part:hot} hold at every finite $N$, with constants that depend only on the constant in two-sided bounds on the spacings $p_i-p_1$ of the precisions, and not on $N$ or on $M=Nd^{3/2}$, as soon as $M$ exceeds a fixed threshold. We apply them to finite sequences of precisions that approximate the limiting scaled half-precisions of the critical window and have a fixed large value of $M$, let $N\to\infty$ with $M$ fixed, and let $M\to\infty$ only afterwards. All comparisons with Gaussian laws conditioned on the sphere are made at finite $N$, for the diffusion started from $\nu_N$; in particular, the limiting Gibbs law in the critical window is never compared with an unconditioned Gaussian product law.

\section{Notation and inputs}\label{sec:outh-inputs}

\paragraph{The limiting equation.}
Recall from \cref{sec:spec-critical} the GOE Airy point process $\chi_1<\chi_2<\cdots$, its gaps $\Delta_i=\chi_i-\chi_1$, with $\Delta_1=0$, the regularized inverse-gap statistic $\Xi$, the function $\mathscr S$ of \eqref{sp-airy-root-function}, its roots $\varrho_\sigma$, and the limiting scaled half-precisions $R^\sigma_i=(\Delta_i+\varrho_\sigma)/2$ of \eqref{eq:spec-limiting-half-precisions}. In this section and the next, we regard $\Xi$ and $(\Delta_i)_{i\ge2}$ as deterministic. We fix $\Xi\in\R$ and $0<\Delta_2<\Delta_3<\cdots$, set $\Delta_1=0$ and $t_i=(3\pi i/2)^{2/3}$, and assume that
\begin{gather}
\frac{\Delta_i}{t_i}\longrightarrow1\qquad(i\to\infty),\label{oc-count}\\
\sum_{i\ge1}\frac1{(x+\Delta_i)^2}\sim\frac1{2\sqrt x}\qquad(x\to\infty),\label{oc-trace}\\
\varrho_{-S}=S^2+O(S^{3/4})\qquad(S\to\infty).\label{oc-root}
\end{gather}
For these deterministic data, we define
\begin{equation}\label{eq:outh-saddle}
\mathscr S(x)=\Xi+\frac1x+\sum_{i\ge2}\Big(\frac1{x+\Delta_i}-\frac1{\Delta_i}\Big)\qquad(x>0).
\end{equation}
This is the first expression in \eqref{sp-airy-root-function}; its equality there with a regularized limit of partial sums is used only for the GOE Airy data, and not below. We show that \eqref{eq:outh-saddle} is meaningful under \eqref{oc-count}. Since $\Delta_i/(i-1)^{2/3}$ is positive for $i\ge2$ and converges to $(3\pi/2)^{2/3}$ by \eqref{oc-count}, there exist $0<c_\Delta\le1\le C_\Delta$, depending only on $(\Delta_i)$, such that
\begin{equation}\label{eq:outh-two-sided}
c_\Delta(i-1)^{2/3}\le\Delta_i\le C_\Delta(i-1)^{2/3}\qquad(i\ge2).
\end{equation}
In particular, $\sum_{i\ge2}\Delta_i^{-2}<\infty$. For $x$ in a compact subset of $(0,\infty)$, the bounds $|(x+\Delta_i)^{-1}-\Delta_i^{-1}|\le x\Delta_i^{-2}$ and $|\partial_x(x+\Delta_i)^{-1}|\le\Delta_i^{-2}$ show that the series in \eqref{eq:outh-saddle} and its termwise derivative converge uniformly there. We conclude that $\mathscr S$ is continuously differentiable on $(0,\infty)$, with
\begin{equation}\label{eq:outh-derivative}
-\mathscr S'(x)=\sum_{i\ge1}\frac1{(x+\Delta_i)^2}>0,\qquad
\mathscr S(x)-\mathscr S(y)=\sum_{i\ge1}\Big(\frac1{x+\Delta_i}-\frac1{y+\Delta_i}\Big)\quad(x,y>0).
\end{equation}
As $x\downarrow0$, the term $1/x$ tends to infinity while the rest of the series stays bounded, and thus $\mathscr S(x)\to+\infty$. By \eqref{oc-trace}, the integral of $-\mathscr S'$ over $(1,\infty)$ diverges, and therefore $\mathscr S(x)\to-\infty$ as $x\to\infty$. Then $\mathscr S$ is a decreasing bijection of $(0,\infty)$ onto $\R$, and $\varrho_\sigma=\mathscr S^{-1}(\sigma)$ and $R^\sigma_i=(\Delta_i+\varrho_\sigma)/2$ are defined for every $\sigma\in\R$.

By \cref{sp-goe-rigidity,sp-raw-gap-resolvent,sp-airy-counts}, the GOE Airy point process satisfies \eqref{oc-count}--\eqref{oc-root} almost surely. Indeed, the points are distinct and $\Xi$ is finite almost surely; \eqref{oc-count} follows from $|\chi_k-t_k|\le C\,t_k^{3/8}$ in the first part of \cref{sp-airy-counts}; \eqref{oc-trace} is the case $k=2$ of \eqref{sp-airy-traces}, since $C_2=1/2$; and \eqref{oc-root} is the pathwise outer-root bound of \cref{sp-outer-airy-saddle}. Every statement of this section and the next that is made under \eqref{oc-count}--\eqref{oc-root} therefore holds almost surely for the GOE Airy point process.

\paragraph{The critical theorem along deterministic sequences.}
We use the results of \cref{sec:crit-process,sec:crit-profile} for deterministic sequences of half-precisions, in the terminology of \cref{sec:cprof-setup}. Let $\mathcal P$ be the class \eqref{eq:cprof-class} of sequences $R=(R_i)_{i\ge1}$ with $0<R_1<R_2\le R_3\le\cdots$ and $\lim_iR_i/i^{2/3}\in(0,\infty)$. Let $\mathcal N$ be an infinite set of integers, and for $N\in\mathcal N$ let $0<r_{1,N}\le\cdots\le r_{N,N}$ satisfy the critical spectral conditions \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0,\delta)$ that do not depend on $N$. Let $R_{i,N}=N^{2/3}r_{i,N}$, and suppose that $(r_{\cdot,N})$ converges to $R\in\mathcal P$ in the sense of \eqref{eq:cprof-conv}, that is,
\begin{equation}\label{eq:outh-critical-hypotheses}
\sum_{i\ge1}\big(R_{i,N}^{-1}\ind_{\{i\le N\}}-R_i^{-1}\big)^2\longrightarrow0,\qquad
\lim_{i\to\infty}i^{-2/3}R_i\in(0,\infty).
\end{equation}
The second condition, an exact Weyl asymptotic for the limiting scaled half-precisions, is part of the hypotheses of \cref{cd-profile,cd-mean}; two-sided bounds $R_i\asymp i^{2/3}$ alone do not suffice. Let $P^{(N)}_t$ be the semigroup generated by $\Delta_{S_N}-\nabla_{S_N}(\sum_ir_{i,N}x_i^2)\cdot\nabla_{S_N}$, let $\pi^{(N)}$ be its invariant law, and let $D^{(N)}(t)=\|\nu_NP^{(N)}_t-\pi^{(N)}\|_\TV$ and $t^{(N)}(\eps)=\inf\{t\ge0:D^{(N)}(t)\le\eps\}$. Then \cref{cd-profile,cd-quantiles} state that, as $N\to\infty$ along $\mathcal N$,
\begin{equation}\label{oc-critical-convergence}
D^{(N)}(N^{2/3}s)\longrightarrow F_R(s),\qquad
N^{-2/3}t^{(N)}(\eps)\longrightarrow F_R^{-1}(\eps),
\end{equation}
locally uniformly in $s\in(0,\infty)$ and in $\eps\in(0,1)$, respectively. Here $F_R$ is the critical mixing profile \eqref{eq:cprof-profile}, which is continuous and strictly decreasing on $(0,\infty)$, with $F_R(0+)=1$ and $F_R(\infty)=0$, and $F_R^{-1}$ is its inverse. We set $F_R(0)=1$.

We also need the mean of the covariance evolution. With $p_{i,N}=2r_{i,N}$, let
\[
H_N=-2\diag(p_{1,N},\dots,p_{N,N})+\frac2N\mathbf 1\mathbf 1^\T,\qquad
h_N(t)=e^{tH_N}\mathbf 1,\qquad
\bar h_N(t)=\frac1N\sum_{i=1}^Nh_{N,i}(t),
\]
be the heat matrix, the heat vector, and its mean, as in \eqref{eq:heat-objects}, and let $\psi_N(s)=N^{1/3}\bar h_N(N^{2/3}s)$. By \cref{cd-mean}, every derivative of $\psi_N$ converges uniformly on compact subsets of $(0,\infty)$ to the corresponding derivative of a positive completely monotone function $\psi_R$, which satisfies
\begin{equation}\label{oc-mean-transform}
\int_0^\infty e^{-ps}\psi_R(s)\,ds=\Big(\frac p4\sum_{i\ge1}\frac1{R_i(R_i+p/4)}\Big)^{-1}\qquad(p>0).
\end{equation}
For $\sigma\in\R$, we write $F_\sigma$ for the critical mixing profile $F_{R^\sigma}$, with limiting scaled half-precisions $R^\sigma=(R^\sigma_i)_{i\ge1}$ computed from $\Xi$ and $(\Delta_i)$. Under \eqref{oc-count}, $R^\sigma$ belongs to $\mathcal P$, with $i^{-2/3}R^\sigma_i\to(3\pi/2)^{2/3}/2$. When $\Xi$ and $(\Delta_i)$ come from the GOE Airy point process, $F_\sigma$ is the critical mixing profile of \cref{thm:intro-critical}.

\paragraph{Finite sequences of precisions.}
Let $0<p_1\le p_2\le\cdots\le p_N$ satisfy the trace condition \eqref{eq:heat-trace}, $\sum_ip_i^{-1}=N$. As in \cref{sec:heat}, we write $d=p_1$, $q_i=1/p_i$, and $\vartheta=N/\sum_iq_i^2$ for the weight of the zero eigenvalue. We let $H$, $h(t)$, $\bar h(t)$, $v_i(t)$, $e_i(t)$, and $\cI_N(t)=\sum_ie_i(t)^2/2$ be the heat matrix, the heat vector and its mean, the variances, the relative errors, and the information of \eqref{eq:heat-objects} and \eqref{eq:heat-errors}. We also write $M=Nd^{3/2}$. The matrix $A=-\diag(p_1,\dots,p_N)$ has Lagrange multiplier $0$ and precisions $p_i$, and we write $P_t=P_t^A$, which is generated by $\Delta_{S_N}-\nabla_{S_N}(\sum_ip_ix_i^2/2)\cdot\nabla_{S_N}$, and $\pi=\pi_A$. By \cref{prop:cond-bingham}, $\pi=\mu_q$ is the Gaussian law with variances $q_i$ conditioned on $S_N$, and $\mu_{v(t)}$ is the Bingham comparison law of $A$. We recall that $\mu_{v(t)}$ need not coincide with the time-$t$ law and that $v_i(t)$ need not equal its second moments; the estimates below compare these objects. We say that $p$ satisfies the precision-spacing bounds with constant $K\ge1$ if
\begin{equation}\label{eq:outh-geometry}
K^{-1}\Big(\frac{i-1}N\Big)^{2/3}\le p_i-p_1\le K\Big(\frac{i-1}N\Big)^{2/3}\qquad(2\le i\le N).
\end{equation}
We use the following four results.
\begin{enumerate}
\item By \cref{mgf-global-bridge}, for all $K\ge1$, $d_0\in(0,1)$, and $C_T>0$, there exist $C_{\mathrm{cp}}\ge1$ and $M_0\ge4$, depending only on $K$, $d_0$, and $C_T$, such that if $p$ satisfies the precision-spacing bounds with constant $K$, $0<d\le d_0$, and $M\ge M_0$, then
\begin{equation}\label{oc-finite-bridge}
\|\nu_NP_t-\mu_{v(t)}\|_\TV\le\min\big\{1,C_{\mathrm{cp}}\,e^{-(d-2/N)t}\big\}\qquad(0\le t\le C_Td^{-1}\log M).
\end{equation}
These are the hypotheses \eqref{eq:hotm-geometry} of that theorem. The estimate holds for every $N$; in particular, it applies when $M$ stays fixed as $N\to\infty$.
\item By \cref{hi3-information-envelopes}, for every $K\ge1$ there exists $C_{\mathrm{re}}$, depending only on $K$, such that if $p$ satisfies the precision-spacing bounds with constant $K$, $d\le1$, and $M\ge1$, which are the hypotheses \eqref{hi3-indexed-geometry}, then
\begin{equation}\label{oc-finite-deficit}
|e_i(d^{-1}+\tau)|\le C_{\mathrm{re}}\frac d{p_i}(1+d\tau)e^{-2d\tau}\qquad(\tau\ge0,\ 1\le i\le N).
\end{equation}
\item By \cref{gp-pinned-lan-quantitative-third}, applied with $v=q$ and $\tilde v=v(t)$, which have the same sum $N$, there exist a universal constant $\eta_{\mathrm{lik}}>0$ and, for every $B<\infty$, a constant $C_B$ with the following property. Let $\eta_q=q_1/(2\sum_iq_i^2)^{1/2}$, which is the quantity $\eta_v$ of \eqref{gp-standardization} for $v=q$, since $q_1=\max_iq_i$. If $\max_i|e_i(t)|\le1/4$, $\eta_q\le\eta_{\mathrm{lik}}$, and $\cI_N(t)\le B$, then
\begin{equation}\label{oc-lan}
\Big|\|\mu_{v(t)}-\pi\|_\TV-\Big(2\Phi\big(\sqrt{\cI_N(t)}/2\big)-1\Big)\Big|\le C_B\Big(\eta_q+\max_i|e_i(t)|\Big).
\end{equation}
\item By \cref{mc-exact-info}, we have $\cI_N(t)=N\bar h''(2t)/(8\bar h(t)^2)$ for $t\ge0$, and there exists a finite positive measure $\rho$ on $[2d,\infty)$ such that
\begin{equation}\label{eq:outh-heat-measure}
\bar h(t)=\vartheta+\int_{[2d,\infty)}e^{-\lambda t}\rho(d\lambda)\qquad(t\ge0).
\end{equation}
\end{enumerate}

\section{Finite approximations of the limiting scaled half-precisions}\label{sec:outh-approx}

Let $\xi>0$. The precisions $N^{-2/3}(\Delta_i+\xi)$, $1\le i\le N$, which correspond to the limiting scaled half-precisions $(\Delta_i+\xi)/2$, do not satisfy the trace condition $\sum_ip_i^{-1}=N$. Indeed, by \eqref{eq:outh-partial-trace} below, $N^{-1}\sum_{i\le N}N^{2/3}(\Delta_i+\xi)^{-1}\to(12/\pi^2)^{1/3}\ne1$. The reason is that the gaps $\Delta_i$ grow like $i^{2/3}$ for every $i$, while the eigenvalues of the GOE follow the semicircle law in the bulk of the spectrum. We keep the first half of these precisions and multiply the second half by a factor that restores the trace exactly. This changes no fixed coordinate. We check that the resulting sequences satisfy both the hypotheses of the critical theorem and the precision-spacing bounds \eqref{eq:outh-geometry}, with a constant that does not depend on the shift $\xi$.

\begin{lemma}[Finite approximations of the limiting scaled half-precisions]\label{oc-approximants}\label{lem:outh-approximants}
Assume \eqref{oc-count}. There exists $K\ge1$, depending only on $(\Delta_i)$, with the following property. For every $\xi>0$, there exists $N_0(\xi)$ such that for every $N\ge N_0(\xi)$, with $n=\lfloor N/2\rfloor$, there exist $\gamma_N\in(1,3]$ and a sequence
\begin{equation}\label{eq:outh-approximants}
p_{i,N}=\begin{cases}N^{-2/3}(\Delta_i+\xi),&1\le i\le n,\\ \gamma_NN^{-2/3}(\Delta_i+\xi),&n<i\le N,\end{cases}
\end{equation}
such that the following hold.
\begin{enumerate}
\item We have $\sum_ip_{i,N}^{-1}=N$ and $0<p_{1,N}<p_{2,N}<\cdots<p_{N,N}$. The smallest precision is $p_{1,N}=\xi N^{-2/3}$, and $M=Np_{1,N}^{3/2}=\xi^{3/2}$.
\item The sequence $(p_{i,N})$ satisfies the precision-spacing bounds \eqref{eq:outh-geometry} with constant $K$, and $\max_ip_{i,N}\le K$. Moreover, $N^{2/3}(p_{i,N}-p_{1,N})=\Delta_i$ for $2\le i\le n$, and $\Delta_i\le N^{2/3}(p_{i,N}-p_{1,N})\le5\Delta_i$ for $n<i\le N$.
\item The half-precisions $r_{i,N}=p_{i,N}/2$ satisfy the critical spectral conditions \eqref{cg-weyl-hypotheses} with constants $(c_0,C_0,\delta)$, where $\delta=\Delta_2/2$ and $c_0,C_0$ depend only on $(\Delta_i)$ and $\xi$. With $R_i=(\Delta_i+\xi)/2$, we have $R=(R_i)\in\mathcal P$ and $i^{-2/3}R_i\to(3\pi/2)^{2/3}/2$, and $(r_{\cdot,N})$ converges to $R$ in the sense of \eqref{eq:outh-critical-hypotheses}.
\item The conclusions \eqref{oc-critical-convergence} and \eqref{oc-mean-transform} hold for these sequences with $R=R^{\mathscr S(\xi)}$, and $F_R=F_{\mathscr S(\xi)}$.
\end{enumerate}
\end{lemma}

\begin{proof}
Fix $\xi>0$, and let $c_\Delta,C_\Delta$ be as in \eqref{eq:outh-two-sided}.

\paragraph{Step 1: partial traces.}
For $0<a\le1$, let $\Sigma_N(a)=N^{-1/3}\sum_{1\le i\le aN}(\Delta_i+\xi)^{-1}$. We claim that
\begin{equation}\label{eq:outh-partial-trace}
\Sigma_N(a)\longrightarrow\Big(\frac{12a}{\pi^2}\Big)^{1/3}\qquad(N\to\infty).
\end{equation}
Fix $\eps\in(0,1)$. By \eqref{oc-count}, there exists $J$ such that $(1-\eps)t_i\le\Delta_i+\xi\le(1+\eps)t_i$ for $i>J$. The terms with $i\le J$ contribute at most $JN^{-1/3}/\xi$ to $\Sigma_N(a)$. The function $i\mapsto i^{-2/3}$ is decreasing, and comparison with an integral gives $\sum_{i\le aN}i^{-2/3}=3(aN)^{1/3}+O(1)$, so
\[
N^{-1/3}\sum_{J<i\le aN}\frac1{t_i}\longrightarrow\Big(\frac{3\pi}2\Big)^{-2/3}3a^{1/3}=\Big(\frac{12a}{\pi^2}\Big)^{1/3}.
\]
Combining these facts, we find that the limit points of $\Sigma_N(a)$ lie between $(1+\eps)^{-1}(12a/\pi^2)^{1/3}$ and $(1-\eps)^{-1}(12a/\pi^2)^{1/3}$, and $\eps$ is arbitrary. This proves \eqref{eq:outh-partial-trace}.

Let $\Sigma^-_N=\Sigma_N(1/2)$, the sum over $i\le n$, and $\Sigma^+_N=\Sigma_N(1)-\Sigma_N(1/2)$, the sum over $n<i\le N$. Then \eqref{eq:outh-partial-trace} implies that
\[
\Sigma^-_N\longrightarrow\Big(\frac6{\pi^2}\Big)^{1/3}<1,\qquad
\Sigma^-_N+\Sigma^+_N\longrightarrow\Big(\frac{12}{\pi^2}\Big)^{1/3}>1,
\]
where we used $6<\pi^2<12$. Let $\gamma_N=\Sigma^+_N/(1-\Sigma^-_N)$, which is defined once $\Sigma^-_N<1$. Then $\gamma_N$ converges to
\[
\gamma_*=\frac{(12/\pi^2)^{1/3}-(6/\pi^2)^{1/3}}{1-(6/\pi^2)^{1/3}}\approx1.440,
\]
and $\gamma_*>1$ because $(12/\pi^2)^{1/3}>1$. We choose $N_0(\xi)$ such that for $N\ge N_0(\xi)$ we have $1<\gamma_N\le3$, $n\ge3$, $\Delta_n\ge\xi$, and $\xi N^{-2/3}\le1$. This is possible since $\Delta_n\to\infty$.

\paragraph{Step 2: the trace and the order.}
Let $N\ge N_0(\xi)$ and define $p_{i,N}$ by \eqref{eq:outh-approximants}. By the definitions of $\Sigma^\pm_N$ and $\gamma_N$,
\[
\sum_{i=1}^Np_{i,N}^{-1}=N\Big(\Sigma^-_N+\frac{\Sigma^+_N}{\gamma_N}\Big)=N .
\]
The sequence is strictly increasing on $\{1,\dots,n\}$ and on $\{n+1,\dots,N\}$ because $(\Delta_i)$ is, and $p_{n+1,N}=\gamma_NN^{-2/3}(\Delta_{n+1}+\xi)>p_{n,N}$ because $\gamma_N>1$. This proves the first part.

\paragraph{Step 3: the precision-spacing bounds.}
For $2\le i\le n$, we have $N^{2/3}(p_{i,N}-p_{1,N})=\Delta_i$. For $n<i\le N$, we have
\[
N^{2/3}(p_{i,N}-p_{1,N})=\gamma_N\Delta_i+(\gamma_N-1)\xi\in\big[\Delta_i,(2\gamma_N-1)\Delta_i\big]\subset[\Delta_i,5\Delta_i],
\]
where we used $\gamma_N>1$, $\xi\le\Delta_n\le\Delta_i$, and $\gamma_N\le3$. Together with \eqref{eq:outh-two-sided}, this gives
\[
c_\Delta\Big(\frac{i-1}N\Big)^{2/3}\le p_{i,N}-p_{1,N}\le5C_\Delta\Big(\frac{i-1}N\Big)^{2/3}\qquad(2\le i\le N),
\]
and $\max_ip_{i,N}=p_{N,N}\le\xi N^{-2/3}+5C_\Delta\le1+5C_\Delta$. This shows that \eqref{eq:outh-geometry} and the bound $\max_ip_{i,N}\le K$ hold with $K=\max(c_\Delta^{-1},1+5C_\Delta)$, which depends only on $(\Delta_i)$. This proves the second part.

\paragraph{Step 4: the critical conditions.}
Let $r_{i,N}=p_{i,N}/2$. Then $\sum_i(2r_{i,N})^{-1}=N$, and $r_{2,N}-r_{1,N}=\Delta_2/(2N^{2/3})$ because $2\le n$. For $i=1$, we have $r_{1,N}=(\xi/2)(1/N)^{2/3}$. For $i\ge2$, we have $(i-1)^{2/3}\ge2^{-2/3}i^{2/3}$, and Step 3 gives
\[
\frac{c_\Delta}{2^{5/3}}\Big(\frac iN\Big)^{2/3}\le\frac{p_{i,N}-p_{1,N}}2\le r_{i,N}\le\frac{\xi N^{-2/3}+5C_\Delta((i-1)/N)^{2/3}}2\le\frac{\xi+5C_\Delta}2\Big(\frac iN\Big)^{2/3}.
\]
These bounds show that \eqref{cg-weyl-hypotheses} holds with $c_0=\min(\xi/2,2^{-5/3}c_\Delta)$, $C_0=(\xi+5C_\Delta)/2$, and $\delta=\Delta_2/2$.

Let $R_{i,N}=N^{2/3}r_{i,N}$ and $R_i=(\Delta_i+\xi)/2$. Then $R_{i,N}=R_i$ for $i\le n$, and $R_{i,N}=\gamma_NR_i>R_i$ for $n<i\le N$. It follows that $0\le R_i^{-1}-R_{i,N}^{-1}\ind_{\{i\le N\}}\le R_i^{-1}$ for every $i>n$. Since this difference vanishes for $i\le n$, we have
\[
\sum_{i\ge1}\big(R_{i,N}^{-1}\ind_{\{i\le N\}}-R_i^{-1}\big)^2\le\sum_{i>n}R_i^{-2}\le4\sum_{i>n}\Delta_i^{-2}\longrightarrow0 .
\]
Finally, $i^{-2/3}R_i=i^{-2/3}(\Delta_i+\xi)/2\to(3\pi/2)^{2/3}/2$ by \eqref{oc-count}, and $0<R_1<R_2\le R_3\le\cdots$, so that $R\in\mathcal P$. This proves the third part. Since $\varrho_{\mathscr S(\xi)}=\xi$, the limiting scaled half-precisions are $R=R^{\mathscr S(\xi)}$, and the fourth part follows from \eqref{oc-critical-convergence} and \eqref{oc-mean-transform}.
\end{proof}

The constant $K$ in the precision-spacing bounds does not depend on $\xi$, while $N_0(\xi)$ and the constants $c_0,C_0$ of the critical conditions do. Only $K$ enters the estimates \eqref{oc-finite-bridge} and \eqref{oc-finite-deficit}, and only the limit $N\to\infty$ at fixed $\xi$ uses the critical conditions.

\section{The information of the limiting covariance evolution}\label{sec:outh-info}

For $\alpha>0$, let $p_{i,N}$ be the sequences of \cref{oc-approximants} with $\xi=\alpha^2$, so that $d_N=p_{1,N}=\alpha^2N^{-2/3}$ and $M=\alpha^3$. Let $\bar h_N$, $\psi_N$, and $\cI_N$ be the quantities of \cref{sec:outh-inputs} for $p_{i,N}$, let $R_i=(\Delta_i+\alpha^2)/2$, and write $\psi=\psi_R$ for the limit in \eqref{oc-mean-transform}. We measure time in units of $1/d_N$: the time $N^{2/3}s$ corresponds to $w=\alpha^2s=d_NN^{2/3}s$. Accordingly, we define for $w>0$
\begin{equation}\label{eq:outh-scaled}
f_\alpha(w)=\frac1\alpha\,\psi\Big(\frac w{\alpha^2}\Big),\qquad
\omega_\alpha=\Big(\alpha\sum_{i\ge1}\frac1{(\Delta_i+\alpha^2)^2}\Big)^{-1},\qquad
j_\alpha(w)=f_\alpha(w)-\omega_\alpha .
\end{equation}
By \eqref{oc-positive-transient} below, for each fixed $\alpha>0$ and $w_0>0$, we have $0\le j_\alpha(w)\le e^{-2(w-w_0)}j_\alpha(w_0)$ for $w\ge w_0$, and hence $f_\alpha(w)\to\omega_\alpha$ as $w\to\infty$. \Cref{oc-derivative-heat} below gives a relative asymptotic for $j_\alpha(w)$ in the joint regime $\alpha\to\infty$, $w\to\infty$, $w=o(\alpha^2)$. Note that \eqref{oc-trace} implies that $\omega_\alpha\to2$ as $\alpha\to\infty$.

\begin{lemma}[The limiting information]\label{oc-information-limit}
Assume \eqref{oc-count}, and fix $\alpha>0$.
\begin{enumerate}
\item For every $s>0$, as $N\to\infty$,
\begin{equation}\label{oc-information-formula}
\cI_N(N^{2/3}s)\longrightarrow\cI_\alpha(s)=\frac{\psi''(2s)}{8\psi(s)^2}=\frac{\alpha^3j_\alpha''(2\alpha^2s)}{8f_\alpha(\alpha^2s)^2}.
\end{equation}
\item For every $z>0$,
\begin{equation}\label{oc-scaled-transform}
\int_0^\infty e^{-zw}f_\alpha(w)\,dw=\frac\alpha{2\big[\mathscr S(\alpha^2)-\mathscr S\big(\alpha^2(1+z/2)\big)\big]}.
\end{equation}
\item There exists a positive Borel measure $\mathfrak n_\alpha$ on $[0,\infty)$ such that $\int(3+2x)^{-1}\mathfrak n_\alpha(dx)<\infty$ and
\begin{equation}\label{oc-positive-transient}
j_\alpha(w)=e^{-2w}\int_{[0,\infty)}e^{-2wx}\,\mathfrak n_\alpha(dx)\qquad(w>0).
\end{equation}
\item For every real $z>-2$ with $z\ne0$,
\begin{equation}\label{oc-real-transient-resolvent}
\int_{[0,\infty)}\frac{\mathfrak n_\alpha(dx)}{z+2+2x}=\frac\alpha{2\big[\mathscr S(\alpha^2)-\mathscr S\big(\alpha^2(1+z/2)\big)\big]}-\frac{\omega_\alpha}z .
\end{equation}
The right side extends continuously to $z=0$, and the identity holds there as well.
\end{enumerate}
\end{lemma}

The measure $\mathfrak n_\alpha$ need not have finite total mass. The point of \eqref{oc-real-transient-resolvent} is that it holds for negative $z$, where the Laplace transform of $f_\alpha$ itself diverges because $f_\alpha$ tends to $\omega_\alpha>0$. In \cref{oc-derivative-heat} below, we use it at $z$ close to $-2$, and this is how the decay rate $e^{-2w}$ enters.

\begin{proof}
The first part follows from the formula for the information and \cref{cd-mean}, and the second is a computation with \eqref{oc-mean-transform}. In the third part, the fact that $\mathfrak n_\alpha$ lives on $[0,\infty)$ comes from the lower bound $2d_N$ on the nonzero frequencies of the covariance evolution at finite $N$. The fourth part follows from the second and the third by analytic continuation.

\paragraph{Step 1: the information.}
Recall from \cref{mc-exact-info} that $\cI_N(t)=N\bar h_N''(2t)/(8\bar h_N(t)^2)$. Using $\psi_N''(s)=N^{1/3}N^{4/3}\bar h_N''(N^{2/3}s)$, we find
\[
\cI_N(N^{2/3}s)=\frac{N\cdot N^{-5/3}\psi_N''(2s)}{8N^{-2/3}\psi_N(s)^2}=\frac{\psi_N''(2s)}{8\psi_N(s)^2}.
\]
The sequences of \cref{oc-approximants} satisfy the hypotheses of \cref{cd-mean}, which gives the convergence of $\psi_N$ and $\psi_N''$; moreover $\psi>0$. This gives the first expression in \eqref{oc-information-formula}. The second follows from $\psi(s)=\alpha f_\alpha(\alpha^2s)$, $\psi''(2s)=\alpha^5f_\alpha''(2\alpha^2s)$, and $f_\alpha''=j_\alpha''$.

\paragraph{Step 2: the Laplace transform.}
For $R_i=(\Delta_i+\alpha^2)/2$ and $p>0$, we have
\[
\frac p4\cdot\frac1{R_i(R_i+p/4)}=\frac{p}{(\Delta_i+\alpha^2)(\Delta_i+\alpha^2+p/2)}=2\Big(\frac1{\Delta_i+\alpha^2}-\frac1{\Delta_i+\alpha^2+p/2}\Big).
\]
Summing over $i$ and using \eqref{eq:outh-derivative}, we find that the right side of \eqref{oc-mean-transform} equals $(2[\mathscr S(\alpha^2)-\mathscr S(\alpha^2+p/2)])^{-1}$. The substitution $w=\alpha^2s$ gives $\int_0^\infty e^{-zw}f_\alpha(w)\,dw=\alpha\int_0^\infty e^{-z\alpha^2s}\psi(s)\,ds$, and taking $p=\alpha^2z$ proves \eqref{oc-scaled-transform}.

\paragraph{Step 3: the positive measure.}
Let $N\ge N_0(\alpha^2)$, and let $\vartheta_N$ and $\rho_N$ be the weight of the zero eigenvalue and the measure in \eqref{eq:outh-heat-measure} for the sequence $p_{i,N}$; the measure is supported on $[2d_N,\infty)$. Let $\mathfrak n_{\alpha,N}$ be the image of $\alpha^{-1}N^{1/3}\rho_N$ under the map $\lambda\mapsto\lambda/(2d_N)-1$, which maps $[2d_N,\infty)$ to $[0,\infty)$, and define
\[
f_{\alpha,N}(w)=\frac1\alpha\psi_N\Big(\frac w{\alpha^2}\Big),\qquad
\omega_{\alpha,N}=\frac{N^{1/3}\vartheta_N}\alpha,\qquad
j_{\alpha,N}=f_{\alpha,N}-\omega_{\alpha,N}.
\]
Inserting $N^{2/3}w/\alpha^2=w/d_N$ into \eqref{eq:outh-heat-measure}, we have
\begin{equation}\label{eq:outh-finite-transient}
j_{\alpha,N}(w)=\int_{[0,\infty)}e^{-(2+2x)w}\,\mathfrak n_{\alpha,N}(dx)\qquad(w\ge0).
\end{equation}
With $P_{i,N}=N^{2/3}p_{i,N}=2R_{i,N}$ and $q_i=p_{i,N}^{-1}$, we have $\sum_iq_i^2=N^{4/3}\sum_iP_{i,N}^{-2}$, and therefore $\omega_{\alpha,N}=(\alpha\sum_iP_{i,N}^{-2})^{-1}$. By the $\ell^2$ convergence in the third part of \cref{oc-approximants}, we have $\sum_iP_{i,N}^{-2}\to\sum_i(\Delta_i+\alpha^2)^{-2}$ and $\omega_{\alpha,N}\to\omega_\alpha$. Also $f_{\alpha,N}(w)\to f_\alpha(w)$ for every $w>0$, by \cref{cd-mean}. We conclude that $j_{\alpha,N}(w)\to j_\alpha(w)$ for every $w>0$. Note that $j_\alpha$ is continuous on $(0,\infty)$, since $\psi$ is.

Fix $\tau>0$, and let $m_N(dx)=e^{-(2+2x)\tau}\mathfrak n_{\alpha,N}(dx)$, a finite measure on $[0,\infty)$. Using \eqref{eq:outh-finite-transient}, we find that for every $t\ge0$,
\begin{equation}\label{eq:outh-laplace-finite}
\int e^{-(2+2x)t}\,m_N(dx)=j_{\alpha,N}(\tau+t)\longrightarrow j_\alpha(\tau+t).
\end{equation}
Taking $t=0$, we see that the total masses $m_N([0,\infty))=j_{\alpha,N}(\tau)$ are bounded. The family $(m_N)$ is also tight. Indeed, for $L>0$ and $t>0$, we have $1-e^{-(2+2x)t}\ge1-e^{-2Lt}$ for $x\ge L$, so
\[
(1-e^{-2Lt})\,m_N([L,\infty))\le\int\big(1-e^{-(2+2x)t}\big)m_N(dx)=j_{\alpha,N}(\tau)-j_{\alpha,N}(\tau+t).
\]
Letting $N\to\infty$, then $L\to\infty$, and then $t\downarrow0$, and using the continuity of $j_\alpha$ at $\tau$, we find that $\lim_{L\to\infty}\limsup_Nm_N([L,\infty))=0$. Let $b_\tau=j_\alpha(\tau)$, the limit of the total masses. If $b_\tau=0$, then $m_N$ converges weakly to the zero measure. If $b_\tau>0$, then for all large $N$ the probability measures $m_N/m_N([0,\infty))$ are tight, and by Prokhorov's theorem \cite[Theorem~5.1]{Billingsley1999} a subsequence of them converges weakly to a probability measure; multiplying it by $b_\tau$, we find that a subsequence of $(m_N)$ converges weakly to a finite measure $m^{(\tau)}$ on $[0,\infty)$ of mass $b_\tau$.
The function $x\mapsto e^{-(2+2x)t}$ is bounded and continuous, and \eqref{eq:outh-laplace-finite} implies that $\int e^{-(2+2x)t}m^{(\tau)}(dx)=j_\alpha(\tau+t)$ for every $t\ge0$. This determines $m^{(\tau)}$. Indeed, the value at $t=0$ is its total mass. If the mass is positive, we divide by it and cancel the factor $e^{-2t}$; the image of the resulting probability measure under $x\mapsto2x$ has the Laplace transform $t\mapsto e^{2t}j_\alpha(\tau+t)/j_\alpha(\tau)$, and a probability measure on $[0,\infty)$ is determined by its Laplace transform \cite[Section~XIII.1, Theorem~1]{Feller1971}. The case of zero mass is immediate.
For $\tau<\tau'$, the measures $e^{-(2+2x)(\tau'-\tau)}m^{(\tau)}(dx)$ and $m^{(\tau')}$ have the same Laplace transform $t\mapsto j_\alpha(\tau'+t)$, and they are equal by the same argument. We define $\mathfrak n_\alpha(dx)=e^{(2+2x)\tau}m^{(\tau)}(dx)$, which does not depend on $\tau$. For $w>0$ and $\tau<w$, we have
\[
\int e^{-(2+2x)w}\,\mathfrak n_\alpha(dx)=\int e^{-(2+2x)(w-\tau)}\,m^{(\tau)}(dx)=j_\alpha(w),
\]
which is \eqref{oc-positive-transient}. Note that $j_\alpha\ge0$. Using Tonelli's theorem and \eqref{oc-scaled-transform} with $z=1$, we find
\[
\int\frac{\mathfrak n_\alpha(dx)}{3+2x}=\int_0^\infty e^{-w}j_\alpha(w)\,dw\le\int_0^\infty e^{-w}f_\alpha(w)\,dw<\infty .
\]

\paragraph{Step 4: real resolvents.}
Let $G(z)=\int(z+2+2x)^{-1}\mathfrak n_\alpha(dx)$ for $z\in\mathbb C\setminus(-\infty,-2]$. For $z$ in a compact subset $\mathcal K$ of this domain, there exists $c_{\mathcal K}>0$ with $|z+2+2x|\ge c_{\mathcal K}(3+2x)$ for all $x\ge0$, since the ratio $|z+2+2x|/(3+2x)$ is positive and continuous in $(z,x)\in\mathcal K\times[0,\infty]$, with value $1$ at $x=\infty$. By Step 3 and dominated convergence, $G$ is holomorphic on $\mathbb C\setminus(-\infty,-2]$. For real $z>0$, Tonelli's theorem, \eqref{oc-positive-transient}, and \eqref{oc-scaled-transform} give
\[
G(z)=\int_0^\infty e^{-zw}j_\alpha(w)\,dw=\frac\alpha{2\mathcal D(z)}-\frac{\omega_\alpha}z,\qquad
\mathcal D(z)=\mathscr S(\alpha^2)-\mathscr S\big(\alpha^2(1+z/2)\big).
\]
The function $\mathscr S$ extends holomorphically to $\mathbb C\setminus(-\infty,0]$, since the difference series in \eqref{eq:outh-saddle} converges uniformly on compact subsets there by \eqref{eq:outh-two-sided}. Then $\mathcal D$ is holomorphic on $\mathbb C\setminus(-\infty,-2]$. On the real interval $(-2,\infty)$, $\mathcal D$ vanishes only at $z=0$, because $\mathscr S$ is strictly decreasing on $(0,\infty)$, and this zero is simple, with
\[
\mathcal D'(0)=-\frac{\alpha^2}2\mathscr S'(\alpha^2)=\frac{\alpha^2}2\sum_{i\ge1}\frac1{(\Delta_i+\alpha^2)^2}=\frac\alpha{2\omega_\alpha}>0 .
\]
It follows that $\alpha/(2\mathcal D(z))$ has a simple pole at $0$ with residue $\alpha/(2\mathcal D'(0))=\omega_\alpha$, and the right side of \eqref{oc-real-transient-resolvent} has a removable singularity at $0$. Each $z_0\in(-2,\infty)$ is the center of an open disc in $\mathbb C\setminus(-\infty,-2]$ on which this right side is holomorphic, and the union $\Omega$ of these discs is a connected open set containing $(-2,\infty)$. Both sides of \eqref{oc-real-transient-resolvent} are holomorphic on $\Omega$ and agree on $(0,\infty)$, so they agree on $\Omega$ by the identity theorem. This proves \eqref{oc-real-transient-resolvent}, including the value at $z=0$.
\end{proof}

The next lemma computes the decay of $j_\alpha$ and of its second derivative in the regime where $w\to\infty$ but $w/\alpha^2\to0$, which contains the times relevant for the profile. Its proof reads the asymptotics off the real squared resolvents of the measures $\mathfrak n_\alpha$, using \cref{pr-real-measure-transfer}, without differentiating an asymptotic formula.

\begin{lemma}[Decay of the transient part]\label{oc-derivative-heat}
Assume \eqref{oc-count} and \eqref{oc-trace}. Let $\alpha_n\to\infty$ and $w_n\to\infty$ with $w_n/\alpha_n^2\to0$. Then
\begin{align}
j_{\alpha_n}(w_n)&=\big(1+o(1)\big)\frac{e^{-2w_n}w_n^{-3/2}}{4\sqrt{2\pi}},\label{oc-j}\\
j_{\alpha_n}''(2w_n)&=\big(1+o(1)\big)\frac{e^{-4w_n}w_n^{-3/2}}{4\sqrt\pi}.\label{oc-j-second}
\end{align}
Moreover, for every compact $K\subset\R$, these relations hold uniformly in $u\in K$ for $w=w_\alpha^*+u$ as $\alpha\to\infty$, where
\begin{equation}\label{eq:outh-scaled-center}
w_\alpha^*=\frac14\Big(\log\alpha^3-\frac32\log\log\alpha^3-\log(16\sqrt\pi)\Big).
\end{equation}
\end{lemma}

\begin{proof}
We abbreviate $\alpha=\alpha_n$ and $w=w_n$. Let $\nu_n$ be $w^{3/2}$ times the image of $\mathfrak n_\alpha$ under $x\mapsto wx$, a positive measure on $[0,\infty)$. For $s>0$, let $z=-2+2s/w>-2$. Using $z+2+2x=2(s+wx)/w$ and differentiating \eqref{oc-real-transient-resolvent} in $z$, we obtain
\begin{equation}\label{eq:outh-squared-transform}
\int\frac{\nu_n(dy)}{(s+y)^2}=\frac4{\sqrt w}\int\frac{\mathfrak n_\alpha(dx)}{(z+2+2x)^2}=-\frac{\alpha^3\mathscr S'(es)}{\sqrt w\,[\mathscr S(\alpha^2)-\mathscr S(es)]^2}-\frac{4\omega_\alpha}{\sqrt w\,z^2},
\end{equation}
where $e=\alpha^2/w$, so that $\alpha^2(1+z/2)=es$. Here the left side of \eqref{oc-real-transient-resolvent} is differentiated under the integral, which is justified because it is holomorphic, and the second term on the right requires $z\ne0$, that is, $w\ne s$. We note that the left side of \eqref{eq:outh-squared-transform} is finite.

We now let $n\to\infty$ at fixed $s>0$. Then $e\to\infty$, and $z\to-2$. Using \eqref{eq:outh-derivative} and \eqref{oc-trace}, we find $-\mathscr S'(x)=(1+o(1))/(2\sqrt x)$ as $x\to\infty$. Then $-\mathscr S'(es)=(1+o(1))/(2\sqrt{es})$, and integrating $\mathscr S'$ over $[es,\alpha^2]$, we find
\[
\mathscr S(\alpha^2)-\mathscr S(es)=-\big(1+o(1)\big)\big(\alpha-\sqrt{es}\big)=-\big(1+o(1)\big)\alpha,
\]
where we used $\sqrt{es}/\alpha=\sqrt{s/w}\to0$. Also $\omega_\alpha\to2$ and $z^2\to4$. Inserting these into \eqref{eq:outh-squared-transform}, we find
\[
\int\frac{\nu_n(dy)}{(s+y)^2}=\big(1+o(1)\big)\frac{\alpha^3}{2\sqrt{es}\sqrt w\,\alpha^2}+O(w^{-1/2})\longrightarrow\frac1{2\sqrt s},
\]
since $\sqrt{es}\sqrt w=\alpha\sqrt s$. Let $\mu(dy)=\pi^{-1}\sqrt y\,dy$ on $[0,\infty)$. The substitution $y=sr^2$ gives
\[
\int_0^\infty\frac{\sqrt y\,dy}{\pi(s+y)^2}=\frac2{\pi\sqrt s}\int_0^\infty\frac{r^2\,dr}{(1+r^2)^2}=\frac2{\pi\sqrt s}\cdot\frac\pi4=\frac1{2\sqrt s}.
\]
We have shown that the squared Stieltjes transforms of $\nu_n$ converge to that of $\mu$ at every $s>0$; these are the hypotheses of \cref{pr-real-measure-transfer} for $\nu_n$ and $\mu$.

Using \eqref{oc-positive-transient}, differentiation under the integral, and the substitution $y=wx$, we have
\[
j_\alpha(w)=e^{-2w}w^{-3/2}\int e^{-2y}\,\nu_n(dy),\qquad
j_\alpha''(2w)=4e^{-4w}w^{-3/2}\int\Big(1+\frac yw\Big)^2e^{-4y}\,\nu_n(dy).
\]
The differentiation is justified for $w>0$ because $(2+2x)^2e^{-(2+2x)v}\le C_ve^{-(2+2x)v/2}$ and $\int e^{-(2+2x)v/2}\mathfrak n_\alpha(dx)=j_\alpha(v/2)<\infty$. The tests $\varphi(y)=e^{-2y}$, $\varphi(y)=e^{-4y}$, and $\varphi_n(y)=(1+y/w_n)^2e^{-4y}$ are continuous, $\varphi_n\to e^{-4y}$ uniformly on compact intervals, and $(1+y)^2|\varphi_n(y)|\le(1+y)^4e^{-4y}$ once $w_n\ge1$, which tends to zero as $y\to\infty$. By \cref{pr-real-measure-transfer}, we obtain
\begin{equation*}
\begin{aligned}
\int e^{-2y}\,\nu_n(dy)&\longrightarrow\frac1\pi\int_0^\infty\sqrt y\,e^{-2y}\,dy=\frac1{4\sqrt{2\pi}},\\
\int\varphi_n\,d\nu_n&\longrightarrow\frac1\pi\int_0^\infty\sqrt y\,e^{-4y}\,dy=\frac1{16\sqrt\pi},
\end{aligned}
\end{equation*}
where we used $\int_0^\infty\sqrt y\,e^{-ay}\,dy=\sqrt\pi\,a^{-3/2}/2$. This proves \eqref{oc-j} and \eqref{oc-j-second}.

For the last claim, $w=w^*_\alpha+u$ satisfies $w\to\infty$ and $w/\alpha^2\to0$ uniformly in $u\in K$. If the relations were not uniform in $u\in K$, there would exist $\eps>0$, $\alpha_n\to\infty$, and $u_n\in K$ such that one of the relative errors at $w_n=w^*_{\alpha_n}+u_n$ is at least $\eps$ for every $n$. This contradicts the first part applied to the sequences $\alpha_n$ and $w_n$.
\end{proof}

\section{The profile}\label{sec:outh-profile}

We now prove the hot half of \cref{thm:intro-outer}. Recall \eqref{eq:outh-center} and \eqref{eq:outh-scaled-center}. The function $\Psi$ and its inverse $u_\eps=\Psi^{-1}(\eps)$ are those of \eqref{hp-universal-profile}.

\begin{theorem}[The critical profile far above the critical temperature]\label{oc-full-hot}
Assume \eqref{oc-count}--\eqref{oc-root}; in particular, the following holds almost surely for the GOE Airy point process. For every compact $K\subset\R$,
\begin{equation}\label{oc-full-alpha}
\sup_{u\in K}\Big|F_{\mathscr S(\alpha^2)}\Big(\frac{w^*_\alpha+u}{\alpha^2}\Big)-\Psi(u)\Big|\longrightarrow0\qquad(\alpha\to\infty)
\end{equation}
and
\begin{equation}\label{oc-full-S}
\sup_{u\in K}\Big|F_{-S}\Big(\tau^{\mathrm h}_S+\frac u{S^2}\Big)-\Psi(u)\Big|\longrightarrow0\qquad(S\to\infty).
\end{equation}
For every compact $J\subset(0,1)$,
\begin{equation}\label{oc-full-quantile}
\sup_{\eps\in J}\Big|S^2\big(F_{-S}^{-1}(\eps)-\tau^{\mathrm h}_S\big)-u_\eps\Big|\longrightarrow0\qquad(S\to\infty).
\end{equation}
Moreover, $S^2F_{-S}^{-1}(\eps)/\log S\to3/4$ for every $\eps\in(0,1)$.
\end{theorem}

In this section and the next, a supremum of nonnegative errors over the empty set is defined to be zero. In \eqref{oc-full-alpha} and \eqref{oc-full-S}, the parameters $\alpha$ and $S$ are large enough that $\log\log\alpha^3$ and $\log\log S^3$ are defined and all the times are positive.

\begin{proof}
The assertions for empty $K$ or $J$ are immediate under this convention, so we consider nonempty compact sets. The proof has four steps: the asymptotics of the limiting information, the comparison at fixed $\alpha$ and finite $N$, the limit $N\to\infty$ followed by $\alpha\to\infty$, and the change of parameter from $\alpha$ to $S$.

\paragraph{Step 1: the limiting information at the center.}
Let $w=w^*_\alpha+u$ and $s=w/\alpha^2$, and write $\ell=\log\alpha^3$. \Cref{oc-derivative-heat} implies that $j_\alpha(w)\to0$, and since $\omega_\alpha\to2$, we have $f_\alpha(w)\to2$. Inserting this and \eqref{oc-j-second} into \eqref{oc-information-formula}, we obtain
\[
\cI_\alpha(s)=\big(1+o(1)\big)\frac{\alpha^3}{8\cdot4\cdot4\sqrt\pi}\,w^{-3/2}e^{-4w}=\big(1+o(1)\big)\frac{\alpha^3}{128\sqrt\pi}\,w^{-3/2}e^{-4w}.
\]
We have $e^{-4w}=16\sqrt\pi\,\ell^{3/2}\alpha^{-3}e^{-4u}$ and $w^{-3/2}=(1+o(1))8\ell^{-3/2}$ by \eqref{eq:outh-scaled-center}. Hence
\begin{equation}\label{eq:outh-info-limit}
\cI_\alpha\Big(\frac{w^*_\alpha+u}{\alpha^2}\Big)\longrightarrow e^{-4u}\qquad(\alpha\to\infty),
\end{equation}
uniformly in $u\in K$, by the uniformity in \cref{oc-derivative-heat}.

\paragraph{Step 2: the comparison at finite $N$.}
The constants are chosen in the order of \cref{t:outh-choices}. Let $K_{\mathrm{sh}}$ be the constant $K$ of \cref{oc-approximants} in the precision-spacing bounds, which depends only on $(\Delta_i)$. Let $C_{\mathrm{cp}}$ and $M_0$ be the constants of \eqref{oc-finite-bridge} for the constant $K_{\mathrm{sh}}$, $d_0=1/2$, and $C_T=1$, and let $C_{\mathrm{re}}$ be the constant of \eqref{oc-finite-deficit} for $K_{\mathrm{sh}}$. Let $B=1+2\max_{u\in K}e^{-4u}$, and let $C_B$ and $\eta_{\mathrm{lik}}$ be the constants of \eqref{oc-lan}. By \eqref{oc-trace},
\begin{equation}\label{eq:outh-radial-limit}
\frac{\alpha^{-2}}{\big(2\sum_{i\ge1}(\Delta_i+\alpha^2)^{-2}\big)^{1/2}}=\big(1+o(1)\big)\alpha^{-3/2}\qquad(\alpha\to\infty).
\end{equation}
Using \eqref{eq:outh-info-limit} and \eqref{eq:outh-radial-limit}, we choose $\alpha_0\ge1$ such that for all $\alpha\ge\alpha_0$ and $u\in K$, with $w=w^*_\alpha+u$:
\begin{equation}\label{eq:outh-alpha-choice}
\begin{gathered}
\alpha^3\ge M_0,\qquad 1\le w\le\log\alpha^3,\qquad C_{\mathrm{re}}e^2we^{-2w}\le\frac14,\qquad
\cI_\alpha\Big(\frac w{\alpha^2}\Big)\le\frac B2,\\
\frac{\alpha^{-2}}{\big(2\sum_{i\ge1}(\Delta_i+\alpha^2)^{-2}\big)^{1/2}}\le\min\Big(\frac{\eta_{\mathrm{lik}}}2,\frac2{\alpha^{3/2}}\Big).
\end{gathered}
\end{equation}

\begin{table}[ht]
\centering
\begin{tabular}{@{}lll@{}}
\toprule
Order & Choice & Depends on\\
\midrule
1 & $\Xi$, $(\Delta_i)$ satisfying \eqref{oc-count}--\eqref{oc-root}; constant $K_{\mathrm{sh}}$ in \eqref{eq:outh-geometry} & ---\\
2 & $d_0=1/2$, $C_T=1$; constants $C_{\mathrm{cp}}$, $M_0$ in \eqref{oc-finite-bridge} & $K_{\mathrm{sh}}$\\
3 & constant $C_{\mathrm{re}}$ in \eqref{oc-finite-deficit} & $K_{\mathrm{sh}}$\\
4 & compact $K$; $B$; constants $C_B$, $\eta_{\mathrm{lik}}$ in \eqref{oc-lan} & $K$\\
5 & $\alpha\ge\alpha_0$ satisfying \eqref{eq:outh-alpha-choice} & all of the above\\
6 & $u\in K$, and then $N\to\infty$ & $\alpha$\\
7 & $\alpha\to\infty$, and then $S\to\infty$ through $\alpha^2=\varrho_{-S}$ & ---\\
\bottomrule
\end{tabular}
\caption{The order of choices in the proof of \cref{oc-full-hot}. No constant in rows 1--5 depends on $N$, and the constants in rows 1--4 do not depend on $\alpha$.}\label{t:outh-choices}
\end{table}

Fix $\alpha\ge\alpha_0$ and $u\in K$, let $w=w^*_\alpha+u$ and $s=w/\alpha^2$, and let $p_{i,N}$ be the sequences of \cref{oc-approximants} with $\xi=\alpha^2$, for $N\ge N_0(\alpha^2)$ so large that $d_N=\alpha^2N^{-2/3}\le1/2$. Let $t=N^{2/3}s=w/d_N$. We bound the three errors in turn.

First, $p_{i,N}$ satisfies the precision-spacing bounds with constant $K_{\mathrm{sh}}$, $d_N\le1/2$, and $M=\alpha^3\ge M_0$. Since $w\le\log\alpha^3$, we have $t\le d_N^{-1}\log M$, and \eqref{oc-finite-bridge} implies that
\[
\|\nu_NP^{(N)}_t-\mu_{v(t)}\|_\TV\le C_{\mathrm{cp}}e^{-(d_N-2/N)t}=C_{\mathrm{cp}}e^{-w}e^{2sN^{-1/3}}.
\]
Second, since $w\ge1$, we may write $t=d_N^{-1}+\tau$ with $d_N\tau=w-1\ge0$. Since $d_N\le p_{i,N}$, \eqref{oc-finite-deficit} gives
\[
\max_i|e_i(t)|\le C_{\mathrm{re}}\,w\,e^{-2(w-1)}=C_{\mathrm{re}}e^2we^{-2w}\le\frac14 .
\]
Third, with $q_i=p_{i,N}^{-1}$ and $P_{i,N}=N^{2/3}p_{i,N}$, we have
\[
\eta_q=\frac{q_1}{(2\sum_iq_i^2)^{1/2}}=\frac{\alpha^{-2}}{(2\sum_iP_{i,N}^{-2})^{1/2}},
\]
which converges as $N\to\infty$ to the left side of \eqref{eq:outh-radial-limit}, by the $\ell^2$ convergence in \cref{oc-approximants}. Also $\cI_N(t)\to\cI_\alpha(s)\le B/2$, by \eqref{oc-information-formula}. Then there exists $N_1(\alpha,u)$ such that for $N\ge N_1(\alpha,u)$ we have $\eta_q\le\eta_{\mathrm{lik}}$, $\eta_q\le3\alpha^{-3/2}$, and $\cI_N(t)\le B$. Then \eqref{oc-lan} applies, and, since $\pi^{(N)}=\mu_q$, the triangle inequality gives
\begin{equation}\label{eq:outh-finite-comparison}
\Big|D^{(N)}(t)-\Big(2\Phi\big(\sqrt{\cI_N(t)}/2\big)-1\Big)\Big|\le C_{\mathrm{cp}}e^{-w}e^{2sN^{-1/3}}+C_B\big(3\alpha^{-3/2}+C_{\mathrm{re}}e^2we^{-2w}\big)
\end{equation}
for $N\ge N_1(\alpha,u)$.

\paragraph{Step 3: the limits $N\to\infty$ and $\alpha\to\infty$.}
Let $N\to\infty$ in \eqref{eq:outh-finite-comparison}, at fixed $\alpha$ and $u$. By \eqref{oc-critical-convergence} and \cref{oc-approximants}, $D^{(N)}(N^{2/3}s)\to F_{\mathscr S(\alpha^2)}(s)$, and $\cI_N(t)\to\cI_\alpha(s)$. We obtain, for all $\alpha\ge\alpha_0$ and $u\in K$, with $w=w^*_\alpha+u$,
\begin{equation}\label{eq:outh-limit-comparison}
\Big|F_{\mathscr S(\alpha^2)}\Big(\frac w{\alpha^2}\Big)-\Big(2\Phi\big(\sqrt{\cI_\alpha(w/\alpha^2)}/2\big)-1\Big)\Big|\le C_{\mathrm{cp}}e^{-w}+C_B\big(3\alpha^{-3/2}+C_{\mathrm{re}}e^2we^{-2w}\big).
\end{equation}
The right side tends to zero as $\alpha\to\infty$, uniformly in $u\in K$, because $w\ge w^*_\alpha+\min K\to\infty$. By \eqref{eq:outh-info-limit} and the uniform continuity of $I\mapsto2\Phi(\sqrt I/2)-1$ on $[0,B]$, we have $2\Phi(\sqrt{\cI_\alpha(w/\alpha^2)}/2)-1\to2\Phi(e^{-2u}/2)-1=\Psi(u)$ uniformly in $u\in K$. This proves \eqref{oc-full-alpha}.

\paragraph{Step 4: the parameter $S$.}
Let $\alpha_S=\varrho_{-S}^{1/2}$, so that $\mathscr S(\alpha_S^2)=-S$ and $F_{\mathscr S(\alpha_S^2)}=F_{-S}$. We have $\alpha_S^2/S^2=1+O(S^{-5/4})$ by \eqref{oc-root}. In particular, $\alpha_S\to\infty$. Using the definitions \eqref{eq:outh-center} and \eqref{eq:outh-scaled-center}, we have
\[
\alpha_S^2\Big(\tau^{\mathrm h}_S+\frac u{S^2}\Big)-w^*_{\alpha_S}-u
=\Big(\frac{\alpha_S^2}{S^2}-1\Big)\big(S^2\tau^{\mathrm h}_S+u\big)
-\frac14\log\frac{\alpha_S^3}{S^3}+\frac38\log\frac{\log\alpha_S^3}{\log S^3}.
\]
The first term is $O(S^{-5/4}\log S)$ uniformly in $u\in K$, since $S^2\tau^{\mathrm h}_S=O(\log S)$; the second is $O(S^{-5/4})$; and the third is $O(S^{-5/4}/\log S)$. We conclude that $\tau^{\mathrm h}_S+u/S^2=(w^*_{\alpha_S}+u+\eps_S(u))/\alpha_S^2$ with $\sup_{u\in K}|\eps_S(u)|\to0$. Let $K'=K+[-1,1]$. For $S$ large, $|\eps_S|\le1$, and
\begin{multline*}
\sup_{u\in K}\Big|F_{-S}\Big(\tau^{\mathrm h}_S+\frac u{S^2}\Big)-\Psi(u)\Big|\\
\le\sup_{u'\in K'}\Big|F_{\mathscr S(\alpha_S^2)}\Big(\frac{w^*_{\alpha_S}+u'}{\alpha_S^2}\Big)-\Psi(u')\Big|+\sup_{u\in K}\big|\Psi\big(u+\eps_S(u)\big)-\Psi(u)\big| .
\end{multline*}
The first term tends to zero by \eqref{oc-full-alpha} with $K'$ in place of $K$, and the second by the uniform continuity of $\Psi$ on $K'$. This proves \eqref{oc-full-S}.

Finally, we apply the deterministic version of \cref{hp-monotone-profile-transfer} along an arbitrary sequence $S_n\to\infty$, with $D_n=F_{-S_n}$, $b_n=\tau^{\mathrm h}_{S_n}$, $w_n=S_n^{-2}$, and $V=\Psi$. Its hypotheses hold: $F_{-S}$ is nonincreasing with $F_{-S}(\infty)=0$ by \eqref{oc-critical-convergence}, $\Psi$ has the required properties by \cref{cot-profile}, $b_n/w_n=S_n^2\tau^{\mathrm h}_{S_n}\to\infty$, and the pointwise convergence is \eqref{oc-full-S}. As $F_{-S}$ is continuous and strictly decreasing, we have $F_{-S}^{-1}(\eps)=\inf\{s\ge0:F_{-S}(s)\le\eps\}$. The second part of \cref{hp-monotone-profile-transfer} gives \eqref{oc-full-quantile}. The last claim follows from \eqref{oc-full-quantile} and $S^2\tau^{\mathrm h}_S/\log S\to3/4$.
\end{proof}