\chapter{The critical profile far below the critical temperature}\label{sec:outer-cold}

This section treats the limit $\sigma=S\to+\infty$ of the critical mixing profile, which is the second half of \cref{thm:intro-outer}. The main result is \cref{cot-transfer}. 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 $F_S(\tau^{\mathrm c}_S+2u/\Delta_2)\to\cV(u)$ locally uniformly in $u$, where
\begin{equation}\label{eq:outc-center}
\tau^{\mathrm c}_S=\frac{\log\big((S-\Xi)\Delta_2\big)}{2\Delta_2},
\end{equation}
and that the inverse functions converge correspondingly.

We use the notation and the conditions \eqref{oc-count}--\eqref{oc-root} of \cref{sec:outh-inputs}, the finite approximations of \cref{oc-approximants}, and the critical theorem in the form \eqref{oc-critical-convergence}. The other inputs are the uniform version of the low-temperature profile theorem (\cref{ccf-cor:uniform-envelope}), the identity for the top fraction in \cref{sp-raw-gap-resolvent}, the properties of $\cV$ in \cref{cot-profile}, and \cref{hp-monotone-profile-transfer}.

The order of limits is the same as in \cref{sec:outer-hot}: first $N\to\infty$ at fixed $S$, then $S\to\infty$. If we set $\epsilon_N=SN^{-1/3}$ in case (4) of \cref{thm:intro-main}, then $N\epsilon_N^3=S^3$ stays bounded, and the profile parameter $m_N=\theta_NNa_2$ converges in distribution to the finite limit $(S-\Xi)\Delta_2/2$. The theorem behind case (4), on the other hand, requires $m_N\to\infty$. What makes the passage possible is that the low-temperature theorem holds uniformly over the vectors of scaled eigenvalue gaps that satisfy the scaled-gap bounds and the separation condition of \cref{def:coldm-admissible} with fixed constants, in the iterated sense $\lim_{L\to\infty}\limsup_{N\to\infty}\sup$ of \eqref{cot-uniform-input} below. This statement gives no bound at a fixed finite value of $m$ and fixed $N$, and none is needed.

\section{Finite approximations at a positive shift}\label{sec:outc-approx}

In the form \eqref{eq:model-raw} of the generator, the low-temperature theorems are stated in terms of the scaled eigenvalue gaps $a_i=(p_i-p_1)/2$ and the top fraction $\theta_N$ of \eqref{eq:model-cold-parameters}. We compute these quantities for the finite approximations of \cref{oc-approximants} with shift $\xi=\varrho_S$.

\begin{lemma}[Scaled eigenvalue gaps and top fraction of the approximations]\label{cot-deficit}
Assume \eqref{oc-count} and \eqref{oc-trace}. Fix $S\in\R$, write $Q_S=S-\Xi$, and let $p_{i,N}$, $N\ge N_0(\varrho_S)$, be the sequences of \cref{oc-approximants} with $\xi=\varrho_S$. Let
\[
a_{i,N}=\frac{p_{i,N}-p_{1,N}}2,\qquad
\theta_N=1-\frac1N\sum_{i=2}^N\frac1{2a_{i,N}},\qquad
m_N=\theta_NNa_{2,N} .
\]
\begin{enumerate}
\item The diffusion generated by $\Delta_{S_N}-\nabla_{S_N}(\sum_ia_{i,N}x_i^2)\cdot\nabla_{S_N}$ is the diffusion with half-precisions $p_{i,N}/2$ of \eqref{oc-critical-convergence}. Moreover, $0=a_{1,N}<a_{2,N}<\cdots<a_{N,N}$, $N^{2/3}a_{2,N}=\Delta_2/2$, $a_{3,N}/a_{2,N}=\Delta_3/\Delta_2$, and
\begin{equation}\label{eq:outc-gap-class}
\frac{c_\Delta}2\Big(\frac{i-1}N\Big)^{2/3}\le a_{i,N}\le\frac{5C_\Delta}2\Big(\frac{i-1}N\Big)^{2/3}\qquad(2\le i\le N),
\end{equation}
with $c_\Delta,C_\Delta$ as in \eqref{eq:outh-two-sided}, which do not depend on $S$.
\item As $N\to\infty$,
\begin{equation}\label{cot-deficit-limit}
N^{1/3}\theta_N\longrightarrow Q_S .
\end{equation}
\item If $Q_S>0$, then, as $N\to\infty$,
\[
m_N\longrightarrow\frac{Q_S\Delta_2}2,\qquad N\theta_N^3\longrightarrow Q_S^3,\qquad
\frac{\log(2m_N)}{4N^{2/3}a_{2,N}}\longrightarrow\tau^{\mathrm c}_S .
\]
\item As $S\to\infty$, we have $\varrho_S\to0$, and $0\le Q_S^{-1}-\varrho_S\le Q_S^{-3}\sum_{i\ge2}\Delta_i^{-2}$ whenever $Q_S>0$.
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\xi=\varrho_S$. For the first part, on $S_N$ we have $\sum_ip_{i,N}x_i^2/2=\sum_ia_{i,N}x_i^2+p_{1,N}N/2$, and the constant does not change the generator, by \eqref{eq:model-shift}. The remaining claims follow from the first two parts of \cref{oc-approximants}. Indeed, since the indices $2$ and $3$ are at most $n$, we have $N^{2/3}a_{2,N}=\Delta_2/2$ and $N^{2/3}a_{3,N}=\Delta_3/2$; and $\Delta_i/2\le N^{2/3}a_{i,N}\le5\Delta_i/2$ for all $2\le i\le N$, which gives \eqref{eq:outc-gap-class} by \eqref{eq:outh-two-sided}.

For the second part, let $P_{i,N}=N^{2/3}p_{i,N}$ and $B_{i,N}=P_{i,N}-\xi=2N^{2/3}a_{i,N}$. The trace condition $\sum_ip_{i,N}^{-1}=N$ reads $\sum_iP_{i,N}^{-1}=N^{1/3}$, and $P_{1,N}=\xi$. Hence
\[
N^{1/3}\theta_N=N^{1/3}-\sum_{i=2}^N\frac1{B_{i,N}}
=\frac1\xi-\sum_{i=2}^N\Big(\frac1{B_{i,N}}-\frac1{P_{i,N}}\Big)
=\frac1\xi-\sum_{i=2}^N\frac{\xi}{B_{i,N}(B_{i,N}+\xi)} .
\]
For each fixed $i\ge2$, we have $B_{i,N}=\Delta_i$ once $i\le n$. By the first part, $B_{i,N}\ge\Delta_i$, and the summands are bounded by $\xi\Delta_i^{-2}$, which is summable. By dominated convergence,
\[
N^{1/3}\theta_N\longrightarrow\frac1\xi-\sum_{i\ge2}\frac\xi{\Delta_i(\Delta_i+\xi)}=\mathscr S(\xi)-\Xi=S-\Xi,
\]
where we used \eqref{eq:outh-saddle} and $(\xi+\Delta_i)^{-1}-\Delta_i^{-1}=-\xi/(\Delta_i(\Delta_i+\xi))$.

The third part follows from $m_N=N^{1/3}\theta_N\cdot N^{2/3}a_{2,N}$, $N\theta_N^3=(N^{1/3}\theta_N)^3$, and
\[
\frac{\log(2m_N)}{4N^{2/3}a_{2,N}}=\frac{\log(2m_N)}{2\Delta_2}\longrightarrow\frac{\log(Q_S\Delta_2)}{2\Delta_2}=\tau^{\mathrm c}_S .
\]

For the fourth part, $\varrho_S\to0$ as $S\to\infty$ because $\mathscr S$ is a decreasing bijection of $(0,\infty)$ onto $\R$. The computation in the second part shows that $Q_S=\xi^{-1}-\sum_{i\ge2}\xi/(\Delta_i(\Delta_i+\xi))$. Multiplying by $\xi$, we obtain
\[
Q_S\,\xi=1-\xi^2\sum_{i\ge2}\frac1{\Delta_i(\Delta_i+\xi)} .
\]
The right side is at most one. If $Q_S>0$, this gives $\xi\le Q_S^{-1}$ and $0\le Q_S^{-1}-\xi=Q_S^{-1}\xi^2\sum_{i\ge2}(\Delta_i(\Delta_i+\xi))^{-1}\le Q_S^{-3}\sum_{i\ge2}\Delta_i^{-2}$.
\end{proof}

The fourth part shows that the smallest scaled precision $\varrho_S$ is close to $1/(S-\Xi)$ for large $S$. The second part shows that the top fraction of the approximations is of order $N^{-1/3}$, with $N^{1/3}\theta_N\to S-\Xi$; this agrees with the top fraction of the model in the critical window, by \cref{sp-raw-gap-resolvent}.

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

We use the low-temperature profile theorem in the following uniform form. Fix $0<c_*\le C_*$ and $\eps_*>0$, and let $\mathcal A_N=\mathcal A_N(c_*,C_*,\eps_*)$ be the set of vectors of \cref{def:coldm-admissible}, that is, of vectors satisfying \eqref{mc-eq:cp1}: the vectors $0=a_1<a_2\le\cdots\le a_N$ such that
\[
c_*\Big(\frac{i-1}N\Big)^{2/3}\le a_i\le C_*\Big(\frac{i-1}N\Big)^{2/3}\quad(2\le i\le N),\qquad\frac{a_3}{a_2}\ge1+\eps_* .
\]
For $a\in\mathcal A_N$, let $\theta(a)=1-N^{-1}\sum_{i\ge2}(2a_i)^{-1}$ and $m(a)=\theta(a)Na_2$. When $\theta(a)>0$, define
\[
t_a(u)=\frac{\log(2m(a))}{4a_2}+\frac u{a_2}\qquad(u\in\R),
\]
and let $D^a(t)$ be the total variation distance between $\nu_NP_t^a$ and the invariant law of $P^a_t$, where $P^a_t$ is generated by $\Delta_{S_N}-\nabla_{S_N}(\sum_ia_ix_i^2)\cdot\nabla_{S_N}$. For a nonempty compact interval $I\subset\R$ and $L$ large enough that $\log(2L)+4\min I>0$, so that all the times below are positive, let
\begin{equation}\label{cot-uniform-input}
E_I(L)=\limsup_{N\to\infty}\,\sup\Big\{\sup_{u\in I}\big|D^a\big(t_a(u)\big)-\cV(u)\big|:a\in\mathcal A_N,\ 0<\theta(a)\le\frac1L,\ m(a)\ge L\Big\},
\end{equation}
where the supremum over the empty set is zero. In the notation of \cref{ccf-cor:uniform-envelope}, $E_I(L)=\limsup_N\mathcal E_N(L)$, and \eqref{mc-eq:cp14} states that $E_I(L)\to0$ as $L\to\infty$. This is stronger than \cref{ccf-thm:full-profile} applied to a single sequence with $\theta\to0$ and $m\to\infty$: it bounds the distance, for all large $N$ and uniformly over $\mathcal A_N$, at every fixed large value of $m$.

\begin{theorem}[The critical profile far below the critical temperature]\label{cot-transfer}
Assume \eqref{oc-count}--\eqref{oc-root}; in particular, the following holds almost surely for the GOE Airy point process. For every compact $I\subset\R$,
\begin{equation}\label{cot-outer-equality}
\sup_{u\in I}\Big|F_S\Big(\tau^{\mathrm c}_S+\frac{2u}{\Delta_2}\Big)-\cV(u)\Big|\longrightarrow0\qquad(S\to\infty).
\end{equation}
For every compact $J\subset(0,1)$,
\begin{equation}\label{cot-quantile}
\sup_{\eps\in J}\Big|\frac{\Delta_2}2\big(F_S^{-1}(\eps)-\tau^{\mathrm c}_S\big)-\cV^{-1}(\eps)\Big|\longrightarrow0\qquad(S\to\infty).
\end{equation}
Moreover, $F_S^{-1}(\eps)/\log S\to1/(2\Delta_2)$ for every $\eps\in(0,1)$.
\end{theorem}

Here $S$ is large enough that $S>\Xi$ and that all the times in \eqref{cot-outer-equality} are positive. The center and the window depend on the realization through $\Xi$ and $\Delta_2$.

\begin{proof}
The assertions for empty $I$ or $J$ are immediate under the convention of \cref{sec:outer-hot}. For the profile assertion we may enlarge a nonempty compact $I$ to its compact interval hull, since a supremum over $I$ is no larger than the supremum over that hull. We therefore fix a nonempty compact interval $I$. For the quantile assertion we likewise enlarge a nonempty compact $J\subset(0,1)$ to its compact interval hull, which is still contained in $(0,1)$. By \cref{cot-deficit}, for every $S$ and every $N\ge N_0(\varrho_S)$, the sequence $(a_{i,N})_{i\le N}$ belongs to $\mathcal A_N(c_*,C_*,\eps_*)$ with
\[
c_*=\frac{c_\Delta}2,\qquad C_*=\frac{5C_\Delta}2,\qquad\eps_*=\frac{\Delta_3}{\Delta_2}-1>0 .
\]
These constants depend only on $(\Delta_i)$, and not on $S$; only the dimension $N_0(\varrho_S)$ from which the sequences are defined depends on $S$. Let $D^{(N)}$ be the distance for the corresponding diffusion, which is the same for the gaps $a_{i,N}$ and for the half-precisions $p_{i,N}/2$, by the first part of \cref{cot-deficit}. For $u\in\R$, let
\[
t_N(u)=\frac{\log(2m_N)}{4a_{2,N}}+\frac u{a_{2,N}}
\]
be the time $t_a(u)$ for $a=(a_{i,N})$.

\paragraph{Step 1: the bound at fixed $S$.}
Fix $L$ with $\log(2L)+4\min I>0$, and let $S$ be such that $Q_S=S-\Xi>0$, $Q_S\Delta_2/2>L$, and $\tau^{\mathrm c}_S+2\min I/\Delta_2>0$. The second and third parts of \cref{cot-deficit} give $N^{1/3}\theta_N\to Q_S>0$ and $m_N\to Q_S\Delta_2/2>L$. Then for all large $N$ we have $0<\theta_N\le1/L$ and $m_N\ge L$, so that $(a_{i,N})$ belongs to the set in \eqref{cot-uniform-input}. Therefore
\begin{equation}\label{cot-Nfirst-bound}
\limsup_{N\to\infty}\sup_{u\in I}\big|D^{(N)}\big(t_N(u)\big)-\cV(u)\big|\le E_I(L).
\end{equation}
We note that at fixed $S$ the parameter $m_N$ converges to a finite limit. For this reason \cref{ccf-thm:full-profile} itself does not apply, while the uniform bound \eqref{cot-uniform-input} does.

\paragraph{Step 2: identification of the left side.}
Using the first and third parts of \cref{cot-deficit}, we find
\[
\frac{t_N(u)}{N^{2/3}}=\frac{\log(2m_N)}{2\Delta_2}+\frac{2u}{\Delta_2}\longrightarrow\tau^{\mathrm c}_S+\frac{2u}{\Delta_2},
\]
uniformly in $u\in I$. The limits lie in the compact interval $[\tau^{\mathrm c}_S+2\min I/\Delta_2,\tau^{\mathrm c}_S+2\max I/\Delta_2]\subset(0,\infty)$, and for large $N$ all the times $t_N(u)/N^{2/3}$, $u\in I$, lie in a fixed compact subset $\mathcal K_S$ of $(0,\infty)$. By \cref{oc-approximants}, \eqref{oc-critical-convergence} holds with $F_R=F_S$, since $\xi=\varrho_S$. Thus
\begin{multline*}
\sup_{u\in I}\Big|D^{(N)}\big(t_N(u)\big)-F_S\Big(\tau^{\mathrm c}_S+\frac{2u}{\Delta_2}\Big)\Big|
\le\sup_{s\in\mathcal K_S}\big|D^{(N)}(N^{2/3}s)-F_S(s)\big|\\
+\sup_{u\in I}\Big|F_S\Big(\frac{t_N(u)}{N^{2/3}}\Big)-F_S\Big(\tau^{\mathrm c}_S+\frac{2u}{\Delta_2}\Big)\Big|,
\end{multline*}
and both terms tend to zero, the second by the uniform continuity of $F_S$ on $\mathcal K_S$. Combining this with \eqref{cot-Nfirst-bound}, we obtain
\[
\sup_{u\in I}\Big|F_S\Big(\tau^{\mathrm c}_S+\frac{2u}{\Delta_2}\Big)-\cV(u)\Big|\le E_I(L)
\]
for every $S$ satisfying the three conditions of Step 1.

\paragraph{Step 3: the limits $S\to\infty$ and $L\to\infty$.}
Since $Q_S\to\infty$ and $\tau^{\mathrm c}_S\to\infty$ as $S\to\infty$, the three conditions of Step 1 hold for all large $S$. By Step 2, the left side of \eqref{cot-outer-equality} has $\limsup_{S\to\infty}$ at most $E_I(L)$, for every $L$ with $\log(2L)+4\min I>0$. Letting $L\to\infty$ and using \eqref{mc-eq:cp14}, we obtain \eqref{cot-outer-equality}. Equivalently, after discarding finitely many dimensions at each fixed $S$, Steps 1 and 3 verify the hypotheses of the second statement \eqref{mc-eq:cp15} of \cref{ccf-cor:uniform-envelope} for the family $a^{(N,S)}=(a_{i,N})$, with $m_S=Q_S\Delta_2/2$, and Step 2 identifies its left side.

Finally, we apply the deterministic version of \cref{hp-monotone-profile-transfer} along an arbitrary sequence $S_n\to\infty$, with $D_n=F_{S_n}$, $b_n=\tau^{\mathrm c}_{S_n}$, $w_n=2/\Delta_2$, and $V=\cV$. Its hypotheses hold: $F_S$ is nonincreasing with $F_S(\infty)=0$ by \eqref{oc-critical-convergence}, $\cV$ has the required properties by \cref{cot-profile}, $b_n/w_n=\log(Q_{S_n}\Delta_2)/4\to\infty$, and the pointwise convergence is \eqref{cot-outer-equality}. As $F_S$ is continuous and strictly decreasing, we have $F_S^{-1}(\eps)=\inf\{s\ge0:F_S(s)\le\eps\}$, and the second part of \cref{hp-monotone-profile-transfer} gives \eqref{cot-quantile}. The last claim follows from \eqref{cot-quantile} and $\tau^{\mathrm c}_S/\log S\to1/(2\Delta_2)$, which holds because $Q_S/S\to1$.
\end{proof}