\chapter{Approach to the critical temperature from below}\label{sec:cold-meso}\label{sec-mesoscopic-cold} In this section, we prove case (4) of \cref{thm:intro-main}: when $c_N=1+\epsilon_N$ with $\epsilon_N\to0$ and $N\epsilon_N^3\to\infty$, the distance $D_N$ has the Gaussian--Cauchy profile $\cV$, with the random center $\log(2\theta_NNa_2)/(4a_2)$ and the random window $a_2^{-1}$. As elsewhere in the paper, the dynamical statements are deterministic. In \cref{ccf-thm:full-profile}, we prove the profile for every sequence of scaled eigenvalue gaps that satisfy the bounds of \cref{def:coldm-admissible}, provided that the top fraction $\theta$ tends to zero and that $m=\theta Na_2$ tends to infinity. We deduce the statement for the GOE in \cref{thm:coldm-goe}, and the limits of the mixing times in \cref{ccf-cor:accuracy-cutoff}. We also show in \cref{ccf-cor:uniform-envelope} that the convergence is uniform over all vectors in $\mathcal A_N$ with $\theta\le1/M$ and $m\ge M$, in the iterated limit in which $N\to\infty$ first and $M\to\infty$ afterwards; this is used in \cref{sec:outer-cold}. The proof follows the strategy of \cref{sec:cold-fixed} for a fixed temperature $c>1$. Now every estimate must be uniform in the top fraction, which tends to zero, and the errors must be expressed through the parameter $m$ alone, which may diverge arbitrarily slowly. We use the profile lemmas of \cref{sec:model} (\cref{cot-profile,hp-monotone-profile-transfer}), the spectral geometry at low temperature (\cref{sp-condensed-geometry}), and the unnormalized process (\cref{cd-lift}). We also use the score bounds, Gaussian smoothing, and conditional relative entropy estimates of \cref{sec:scores}, the density bounds of \cref{sec:density} (\cref{thm:dens-main}), two functional inequalities of \cref{sec:inequalities} (\cref{cg-many-soft-pi,cg-three-soft-lsi}), and the comparison of the Gibbs measure with a Gaussian product law in \cref{mesocold-prop:stationary-graph}. \Cref{sec:coldm-setting} states the hypotheses and the main results, and describes the proof. \Cref{sec:coldm-means,sec:coldm-moments} estimate the means and all fixed moments of the unnormalized process. \Cref{sec:coldm-path} shows that the radial part of the unnormalized process is determined, along the time interval that we study, by its first coordinate at an earlier time. \Cref{sec:coldm-residence} gives a Gaussian approximation of the coordinates at an early time $\tau$ and controls the domain visited by the path. \Cref{sec:coldm-weak} identifies the limit of finitely many coordinates and proves the lower bound on the distance. \Cref{sec:coldm-renyi} bounds the density of the time-$t$ law at time $\theta^{-2}$, \cref{sec:coldm-entropy} bounds the conditional relative entropy of the remaining coordinates, and \cref{sec:coldm-proofs} proves the main results. \section{Setting and main results}\label{sec:coldm-setting}\label{mesocold-sec:geometry} \paragraph{The dynamics.} Throughout this section, $N\ge3$ and $a=(a_1,\dots,a_N)$ is a deterministic vector with \[ 0=a_10$. We say that $a$ is admissible with constants $(c_*,C_*,\varepsilon_*)$ if \begin{equation}\label{mc-eq:cp1} c_*(i-1)^{2/3}N^{-2/3}\le a_i\le C_*(i-1)^{2/3}N^{-2/3}\quad(2\le i\le N),\qquad \frac{a_3}{a_2}\ge1+\varepsilon_* . \end{equation} We write $\mathcal A_N=\mathcal A_N(c_*,C_*,\varepsilon_*)$ for the set of admissible vectors of length $N$. \end{definition} By \cref{sp-condensed-geometry}, the eigenvalue gaps of the model satisfy \eqref{mc-eq:cp1} at every fixed confidence. Note that the upper bound in \eqref{mc-eq:indexed-geometry} with $i=N$ implies that $a_N\le C_*$. \paragraph{Parameters.} We write \begin{equation}\label{mc-eq:scales} a=a_2,\qquad \theta=1-\frac1N\sum_{i=2}^N\frac1{2a_i},\qquad m=\theta Na . \end{equation} For the model, these are $a_2$, the top fraction $\theta_N$, and $m_N=\theta_NR_N$ of \eqref{eq:model-cold-parameters}. A centered Gaussian vector $(x_2,\dots,x_N)$ with independent coordinates of variances $(2a_i)^{-1}$ has mean squared norm $(1-\theta)N$; when $\theta>0$ is small, the Gibbs measure places the small fraction $\theta$ of $|x|^2=N$ on the first coordinate (\cref{mesocold-prop:stationary-graph}). We will see that $m$ is comparable to $(N\theta^3)^{1/3}$, and that it measures how far the system is from the critical window: for the model, $m_N$ is of order $(N\epsilon_N^3)^{1/3}$. Finally, for $m>0$ and $u\in\R$ we define \begin{equation}\label{mc-eq:cp2} t_m(u)=\frac{\log(2m)}{4a}+\frac ua , \end{equation} and we recall from \eqref{eq:model-cauchy-profiles} that $\cV(u)=\|\Law(G+e^{-2u}C)-\Law(G)\|_\TV$, where $G$ is a standard Gaussian random variable and $C$ is an independent standard Cauchy random variable. We use the following elementary consequences of the indexed bounds throughout the section. \begin{lemma}[Consequences of the indexed bounds]\label{lem:coldm-indexed} Suppose that $a$ satisfies \eqref{mc-eq:indexed-geometry}. There exists $C\ge1$, depending only on $c_*$ and $C_*$, such that the following hold. \begin{enumerate} \item We have $c_*N^{-2/3}\le a\le C_*N^{-2/3}$, and $a_k\le(C_*/c_*)(k-1)^{2/3}a$ for $2\le k\le N$. \item If $\theta>0$, then \[ \frac{m^3}{C_*^3}\le N\theta^3\le\frac{m^3}{c_*^3},\qquad \frac{m^2}{C_*^3}\le\frac{\theta^2}a\le\frac{m^2}{c_*^3},\qquad m\le C_*N^{1/3}. \] \item For every $x>0$, \[ \min\Big\{N-1,\Big\lfloor N\Big(\frac x{C_*}\Big)^{3/2}\Big\rfloor\Big\}\le\#\{i\ge2:a_i\le x\}\le N\Big(\frac x{c_*}\Big)^{3/2} . \] \item For all $t\ge0$, $h>0$, $s>0$, and integers $1\le k\le N-1$, \begin{gather*} \sum_{i=2}^Ne^{-4a_it}+\sum_{i=2}^N\frac1{(1+a_it)^2}\le CN(1+t)^{-3/2},\qquad \sum_{i=2}^N\min\{t,a_i^{-1}\}^2\le CN\sqrt t,\\ \sum_{i=2}^N\frac{\min\{h,a_i^{-1}\}}{a_i}\le CN\sqrt h,\qquad \sum_{i=2}^N\frac{e^{-a_is}}{a_i}\le\frac{CN}{\sqrt s},\\ \sum_{i\ge k+1}\Big(\frac a{a_i}\Big)^2\le Ck^{-1/3},\qquad \sum_{i=2}^{k+1}\frac1{a_i}\le CN^{2/3}k^{1/3} . \end{gather*} \end{enumerate} \end{lemma} \begin{proof} The first part is \eqref{mc-eq:indexed-geometry} with $i=2$ and $i=k$. For the second part, we have $\theta=m/(Na)$, so that $N\theta^3=m^3/(N^2a^3)$ and $\theta^2/a=m^2/(N^2a^3)$, and the first part implies that $c_*^3\le N^2a^3\le C_*^3$. Also $\theta\le1$, and $m\le Na\le C_*N^{1/3}$. For the third part, if $a_i\le x$ then $c_*(i-1)^{2/3}N^{-2/3}\le x$, that is, $i-1\le N(x/c_*)^{3/2}$; conversely, every $i\ge2$ with $i-1\le N(x/C_*)^{3/2}$ satisfies $a_i\le x$. For the fourth part, the first two bounds at $t=0$ follow from $2(N-1)\le2N$ and $\sum_{i=2}^N\min\{0,a_i^{-1}\}^2=0$. In proving those bounds below, assume $t>0$. We write $j=i-1$ and compare the sums with integrals of decreasing functions of $j$. For the first sum, using the lower bound in \eqref{mc-eq:indexed-geometry}, we find \[ \sum_{i\ge2}e^{-4a_it}\le\int_0^\infty e^{-4c_*tN^{-2/3}y^{2/3}}\,dy=\frac{N}{(4c_*t)^{3/2}}\int_0^\infty e^{-x^{2/3}}\,dx , \] and the sum is also at most $N$; the sum of $(1+a_it)^{-2}$ is bounded in the same way by $N(c_*t)^{-3/2}\int_0^\infty(1+x^{2/3})^{-2}\,dx$, which is finite. For the second sum, the indices with $a_i\le1/t$ number at most $N(c_*t)^{-3/2}$ and contribute at most $t^2$ each, while \[ \sum_{i:a_i>1/t}a_i^{-2}\le\frac{N^{4/3}}{c_*^2}\sum_{j>N(C_*t)^{-3/2}}j^{-4/3}\le CN\sqrt t \] when $N(C_*t)^{-3/2}\ge1$; when $N(C_*t)^{-3/2}<1$, the full sum $\sum_{i\ge2}a_i^{-2}\le4c_*^{-2}N^{4/3}$ is at most $CN\sqrt t$. The third sum is the same computation with the summand $\min\{h,a_i^{-1}\}a_i^{-1}$, which equals $h/a_i$ when $a_i\le1/h$ and $a_i^{-2}$ otherwise; for the first group we use $\sum_{j\le x}j^{-2/3}\le3x^{1/3}$. For the fourth sum, we have \[ \sum_{i\ge2}\frac{e^{-a_is}}{a_i}\le\frac{N^{2/3}}{c_*}\int_0^\infty y^{-2/3}e^{-c_*sN^{-2/3}y^{2/3}}\,dy=\frac{3\sqrt\pi}{2c_*}\,\frac{N^{2/3}}{(c_*sN^{-2/3})^{1/2}}=\frac{CN}{\sqrt s}, \] where we substituted $x=y^{2/3}$. For the last two sums, we use $a/a_i\le(C_*/c_*)j^{-2/3}$, $a_i^{-1}\le c_*^{-1}N^{2/3}j^{-2/3}$, $\sum_{j\ge k}j^{-4/3}\le4k^{-1/3}$, and $\sum_{j\le k}j^{-2/3}\le3k^{1/3}$. \end{proof} \subsection{Main results}\label{sec:coldm-results} The first theorem is the deterministic form of the main result. We recall that $\cV$ is continuous and strictly decreasing from $1$ to $0$ (\cref{cot-profile}). \begin{theorem}[The profile below the critical temperature]\label{ccf-thm:full-profile} Fix $00$. For each $n\ge1$, let $a^{(n)}\in\mathcal A_{N_n}(c_*,C_*,\varepsilon_*)$, and let $\theta_n$, $m_n$, $D_n$, and $t_{m_n}(u)$ be the quantities defined above for $a^{(n)}$. Suppose that $\theta_n>0$ for all $n$, that $\theta_n\to0$, and that $m_n\to\infty$. Then, for every compact interval $I\subset\R$, \begin{equation}\label{mc-eq:cp3} \sup_{u\in I}\big|D_n(t_{m_n}(u))-\cV(u)\big|\longrightarrow0\qquad(n\to\infty). \end{equation} \end{theorem} The hypotheses force $N_n\to\infty$, because $m_n\le C_*N_n^{1/3}$ by \cref{lem:coldm-indexed}, and the times $t_{m_n}(u)$, $u\in I$, are positive for all large $n$; for the finitely many other $n$, the convention $D(t)=D(0)$ for $t<0$ applies. The dimensions $N_n$ need not be distinct or increasing; this matters for the next corollary, which is proved by applying \cref{ccf-thm:full-profile} along a diagonal sequence. \begin{corollary}[Uniformity in the gaps]\label{ccf-cor:uniform-envelope} Fix $00$, and a nonempty compact interval $I\subset\R$. For $N\ge3$ and $M\ge1$ with $\log(2M)+4\min I>0$, let \[ \mathcal E_N(M)=\sup\Big\{\sup_{u\in I}\big|D(t_m(u))-\cV(u)\big|:a\in\mathcal A_N(c_*,C_*,\varepsilon_*),\ 0<\theta\le\frac1M,\ m\ge M\Big\}, \] where $D$, $\theta$, and $m$ are computed from $a$, and the supremum of the empty set is zero. The condition on $M$ ensures that $t_m(u)>0$ for $u\in I$ and $m\ge M$. Then \begin{equation}\label{mc-eq:cp14} \lim_{M\to\infty}\limsup_{N\to\infty}\mathcal E_N(M)=0 . \end{equation} Moreover, the following holds. Let $S_0>0$, and for $S\ge S_0$ and $N\ge3$ let $a^{(N,S)}\in\mathcal A_N(c_*,C_*,\varepsilon_*)$, with parameters $\theta_{N,S}>0$ and $m_{N,S}$. Suppose that for each fixed $S$ we have $\theta_{N,S}\to0$ and $m_{N,S}\to m_S\in(0,\infty)$ as $N\to\infty$, and that $m_S\to\infty$ as $S\to\infty$. Then \begin{equation}\label{mc-eq:cp15} \lim_{S\to\infty}\limsup_{N\to\infty}\sup_{u\in I}\big|D_{N,S}(t_{m_{N,S}}(u))-\cV(u)\big|=0, \end{equation} where $D_{N,S}$ and $t_{m_{N,S}}(u)$ are computed from $a^{(N,S)}$, with $D_{N,S}(t)=D_{N,S}(0)$ for $t<0$. For all sufficiently large fixed $S$, the times $t_{m_{N,S}}(u)$, $u\in I$, are positive for all large $N$. \end{corollary} The point of \cref{ccf-cor:uniform-envelope} is the order of the limits. At a fixed value of $m$ the error need not be small; the corollary states that the error is small in the limit $N\to\infty$, uniformly over $\mathcal A_N$, once $m$ is large and $\theta$ is small. This is the form in which the result enters the iterated limit of \cref{sec:outer-cold}, where at each fixed outer parameter $S$ the vectors $a^{(N,S)}$ have a finite limit $m_S$ of $m_{N,S}$. We now turn to the model. Recall from \eqref{eq:model-cold-parameters} that $a_i=c_N(\lambda_1-\lambda_i)/2$, $\theta_N=1-N^{-1}\sum_{i\ge2}(2a_i)^{-1}$, and $m_N=\theta_NNa_2$. \begin{theorem}[The profile for the GOE]\label{thm:coldm-goe} Let $c_N=1+\epsilon_N$, where $\epsilon_N>0$, $\epsilon_N\to0$, and $N\epsilon_N^3\to\infty$. Almost surely the eigenvalues are distinct, so that $a_2>0$. Let \[ T_N=\begin{cases} \dfrac{\log(2m_N)}{4a_2},&\theta_N>0\text{ and }2m_N>1,\\ 1,&\text{otherwise}, \end{cases} \] and let $D_N(t)=D_N(0)$ for $t<0$. Then $T_N>0$, $\PP(\theta_N>0,\ 2m_N>1)\to1$, $a_2T_N\to\infty$ in probability, and for every compact interval $I\subset\R$, \[ \sup_{u\in I}\big|D_N(T_N+u/a_2)-\cV(u)\big|\longrightarrow0\quad\text{in probability}. \] In particular, $D_N$ has cutoff profile $\cV$ with center $T_N$ and window $a_2^{-1}$ in the sense of \cref{def:model-cutoff}. \end{theorem} \begin{corollary}[Mixing times]\label{ccf-cor:accuracy-cutoff} The following hold. \begin{enumerate} \item In the setting of \cref{ccf-thm:full-profile}, let $T_n=\log(2m_n)/(4a^{(n)}_2)$, which is positive for all large $n$, and let $t_n(\eps)=\inf\{t\ge0:D_n(t)\le\eps\}$. Then, for every compact interval $I'\subset(0,1)$, \[ \sup_{\eps\in I'}\Big|a^{(n)}_2\big(t_n(\eps)-T_n\big)-\cV^{-1}(\eps)\Big|\longrightarrow0 , \] and for every fixed $\eta\in(0,1)$, we have $D_n((1-\eta)T_n)\to1$ and $D_n((1+\eta)T_n)\to0$. Moreover, $(a^{(n)}_2T_n)^{-1}=4/\log(2m_n)\to0$, and for $0<\eps_1<\eps_2<1$, \[ a^{(n)}_2\big(t_n(\eps_1)-t_n(\eps_2)\big)\longrightarrow\cV^{-1}(\eps_1)-\cV^{-1}(\eps_2)>0 . \] \item In the setting of \cref{thm:coldm-goe}, let $t_N(\eps)$ be the mixing time \eqref{model-distance} and $T_N$ as in \cref{thm:coldm-goe}. Then the convergence assertions of the first part hold in probability: $a_2(t_N(\eps)-T_N)\to\cV^{-1}(\eps)$ uniformly for $\eps$ in compact subsets of $(0,1)$, $D_N((1-\eta)T_N)\to1$ and $D_N((1+\eta)T_N)\to0$ for every fixed $\eta\in(0,1)$, $(a_2T_N)^{-1}\to0$, and $a_2(t_N(\eps_1)-t_N(\eps_2))\to\cV^{-1}(\eps_1)-\cV^{-1}(\eps_2)$ for $0<\eps_1<\eps_2<1$, all in probability. On the event $\{\theta_N>0,\ 2m_N>1\}$, whose probability tends to one, we have $(a_2T_N)^{-1}=4/\log(2m_N)$. In particular, there is cutoff. \end{enumerate} \end{corollary} \subsection{Outline of the proof}\label{sec:coldm-outline} We explain the mechanism and then list the steps of the proof. Let $X$ be the diffusion started from $\nu_N$, and write $X=\sqrt N\,Y/|Y|$, where $Y$ is the unnormalized process of \cref{sec:process}: \[ dY_i=-2a_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i,\qquad Y(0)\in S_N\text{ uniform}. \] The second moment $h_i(t)=\E Y_i(t)^2$ solves a linear equation whose kernel is $k(t)=(2/N)\sum_{i\ge2}e^{-4a_it}$, a defective probability density with total mass $1-\theta$. Since $a_1=0$, the first coordinate $Y_1$ is a martingale, and its mean square increases to order $\theta^{-1}$. The mean squared norm $\E|Y(t)|^2$ decreases, like $N(1+t)^{-1/2}$ up to the time $\theta^{-2}$ and like $N\theta^{-2}t^{-3/2}$ afterwards, until a time of order $a^{-1}$. After that time the mean squared norm is of order $\theta^{-2}$, while the mean square of the first coordinate is of order $\theta^{-1}$, so that $X_1^2=NY_1^2/|Y|^2$ is of order $\theta N$. In the first part of the proof (\cref{sec:coldm-means,sec:coldm-moments}), we estimate these means, and all fixed moments, uniformly in $\theta$. Let $\tau=m^{\mathfrak a}\theta^{-2}$ for a fixed $\mathfrak a\in(0,1)$, and let $H_i=\sqrt\theta\,Y_i(\tau)$. We show in \cref{sec:coldm-residence} that, for each fixed $J$, the vector $(H_1,\dots,H_J)$ is close to a standard Gaussian vector. After a time $T$ of order $a^{-1}\log m$, the process $\theta^2|Y(t)|^2$ stays close to $H_1^2$ along the whole interval $[T,K_0a^{-1}\log m]$, with an error bounded by a negative power of $m$ in every fixed $L^p$ norm (\cref{sec:coldm-path}). At the time $t=t_m(u)$, the standardized second coordinate is then \[ \sqrt{2a}\,X_2(t)=\frac{\sqrt{2aN}\,\theta Y_2(t)}{\theta|Y(t)|}\approx\sqrt{2m}\,e^{-2at}\,\frac{H_2}{|H_1|}+\sqrt{4a}\int_T^te^{-2a(t-s)}\,dB_2(s) \approx e^{-2u}\frac{H_2}{|H_1|}+G_2 , \] where $G_2$ is a standard Gaussian variable independent of $(H_1,H_2)$. The ratio $H_2/|H_1|$ is approximately a ratio of independent standard Gaussian variables, that is, approximately standard Cauchy. The coordinates $3\le i\le J$ are approximately independent standard Gaussian variables, because the separation $a_3\ge(1+\varepsilon_*)a$ makes their inherited part $(2m)^{(1-a_i/a)/2}H_i/|H_1|$ negligible. Under the Gibbs measure, the same coordinates are approximately independent standard Gaussian variables (\cref{mesocold-prop:stationary-graph}). This identifies the limit of finitely many coordinates and gives the lower bound $\liminf D(t_m(u))\ge\cV(u)$ (\cref{sec:coldm-weak}). For the upper bound we must control all coordinates at once. We keep the first $J$ coordinates, where $J$ is fixed so large that $b=a_{J+1}\ge6a$, and we compare the remaining coordinates with independent Gaussian coordinates of variances $(2a_i)^{-1}$. \Cref{sec:coldm-renyi} shows that the density of the time-$t$ law with respect to the Gibbs measure has all moments of size $e^{O(N\theta^3)}=e^{O(m^3)}$ at time $\theta^{-2}$; this gives the initial conditional relative entropy, of order $m^3$. The conditional relative entropy of the remaining coordinates then decays at a rate of at least $4\cdot(7/8)\cdot b$ up to errors of order $m^{-1}$ (\cref{sec:coldm-entropy}). Over the interval from $T$ to $t_m(u)$, of length about $(1/4-1/100)a^{-1}\log m$, the entropy is therefore multiplied by at most $\exp(-4\cdot(7/8)\cdot(b/a)\cdot(24/100)\log m)\le m^{-5}$, since $b/a\ge6$, and this beats the initial value $m^3$. The entropy estimate is proved for the endpoint measure weighted by the indicator of an event at time $\tau$ and killed on leaving a domain, multiplied by an endpoint cutoff function and normalized; the mass removed by the event restriction, the killing, and the cutoff is controlled in \cref{sec:coldm-residence}. Finally, the finitely many retained coordinates converge in total variation, by the Gaussian smoothing argument of \cref{sec:scores}, and \cref{sec:coldm-proofs} combines the two estimates with the stationary comparison. The parameters of the proof of \cref{ccf-thm:full-profile} are chosen in the order of \cref{tab:coldm-parameters}. All constants depend only on the parameters above them in the table, and every estimate holds for $m\ge m_0$ and $\theta\le1/2$, with a threshold $m_0$ that depends on the same parameters. \begin{table}[ht] \centering \caption{Parameters in the proof of \cref{ccf-thm:full-profile}, in the order in which they are fixed.}\label{tab:coldm-parameters} \begin{tabular}{@{}l>{\raggedright\arraybackslash}p{0.47\textwidth}>{\raggedright\arraybackslash}p{0.3\textwidth}@{}} \toprule Parameter & Role & Value or dependence\\ \midrule $c_*,C_*,\varepsilon_*$ & constants of \cref{def:coldm-admissible} & given\\ $J$ & number of retained coordinates, with $a_{J+1}\ge6a$ & $c_*,C_*$\\ $\mathfrak a,\mathfrak b,\zeta,K_0$ & exponents: $\tau=m^{\mathfrak a}\theta^{-2}$, $T=\mathfrak ba^{-1}\log(em)$, cone width $\delta=m^{-\zeta}$; horizon $K_0a^{-1}\log(em)$ & $\mathfrak a=1/10$, $\mathfrak b=\zeta=1/100$, $K_0=1$\\ $\gamma$ & tolerance for the mass removed by the event restriction & fixed; $\gamma\to0$ last\\ $\ell_-,\ell_+$ & the event $\{\ell_-\le|H_1|\le\ell_+,\ |H_i|\le\ell_+\}$ at time $\tau$ & $\gamma,J$\\ $s,L_0,L_1,L$ & cutoff and killing levels & $\ell_\pm,J$\\ $p,B_0$ & moment order in the bound on the mass of the set $\mathcal C$ outside which the cutoff equals one; failure exponent & $p=200$, $B_0=20$\\ $m_0$ & threshold for $m$ & all of the above\\ \bottomrule \end{tabular} \end{table} The moment order $p$ in the table is the order used for the ratio $\xi$ in \cref{cpm-cor:killed-collars}. The proof of \cref{cpm-cor:actual-residence} uses a larger moment order $p_{\mathrm{res}}$ of the path estimate of \cref{cpm-thm:radial-path}, chosen with $2p_{\mathrm{res}}\vartheta\ge B_0$. For the exponents above, $\vartheta=1/800$, and $p_{\mathrm{res}}=8000$ suffices. All these orders are fixed before the threshold for $m$. The main difficulty compared with \cref{sec:cold-fixed} is that the time scales $\theta^{-2}$ and $a^{-1}$ and the mean square $\theta^{-1}$ of the first coordinate all depend on $\theta$, which tends to zero at an arbitrary rate. The renewal estimates of \cref{sec:coldm-means} are therefore proved with constants that do not depend on $\theta$, the moment bounds of \cref{sec:coldm-moments} are proved by a comparison that keeps the small deficit $\theta$, and every probability of failure is bounded by an explicit negative power of $m$. \section{The unnormalized process and its mean}\label{sec:coldm-means}\label{cpm-sec:high-moments-path} In this section and in \cref{sec:coldm-moments,sec:coldm-path,sec:coldm-residence,sec:coldm-weak}, the vector $a$ satisfies the indexed bounds \eqref{mc-eq:indexed-geometry} with constants $(c_*,C_*)$, and $0<\theta\le1/2$. We fix $K_0\ge1$ and write \[ T_{\max}=\frac{K_0\log(em)}a \] for the end of the time interval that we study; the times $t_m(u)$ with $u$ in a compact set lie in $[0,T_{\max}]$ once $m$ is large. We also write \begin{equation}\label{eq:coldm-f} f_\theta(t)=(1+t)^{-1/2}\min\big\{1,\,[\theta^2(1+t)]^{-1}\big\}\qquad(t\ge0). \end{equation} Unless said otherwise, the constants $C,c_0>0$ depend only on $c_*$, $C_*$, and $K_0$, and the estimates hold for $m\ge m_0$, with a threshold $m_0$ that depends on the same parameters. Positive constants in lower bounds are written $c_1,c_2$, with the same dependence; the letter $c$ alone is the rescaled inverse temperature. In particular, no constant depends on $\theta$ or on $N$, and the thresholds involve no negative power of $\theta$ and no power of $N$. \subsection{The unnormalized process} Let $B=(B_1,\dots,B_N)$ be a standard Brownian motion in $\R^N$, let $g$ be a standard Gaussian vector in $\R^N$ independent of $B$, and let $X_0=\sqrt N\,g/|g|$, which has law $\nu_N$. Let $Y$ and $Y^G$ be the solutions of \begin{equation}\label{mc-eq:lift} dY_i=-2a_iY_i\,dt+\sqrt{2/N}\,|Y|\,dB_i\qquad(1\le i\le N) \end{equation} with $Y(0)=X_0$ and $Y^G(0)=g$, respectively. By \cref{cd-lift}, applied with the quadratic potential coefficients $\rho_i=a_i$, both solutions exist, are unique, and never vanish, and $X=\sqrt N\,Y/|Y|$ is the diffusion with generator $L$ of \eqref{mc-eq:model} started from $\nu_N$. Then $\rho_t$ is the law of $X(t)$. We call $Y$ the unnormalized process with the uniform start, and $Y^G$ the unnormalized process with the Gaussian start. Moments are easier to compute for the Gaussian start. The next lemma transfers upper bounds to the uniform start. \begin{lemma}[Gaussian and uniform starts]\label{lem:coldm-gaussian-start} Almost surely, $Y^G(t)=(|g|/\sqrt N)\,Y(t)$ for all $t\ge0$, and $|g|$ is independent of $(X_0,B)$, hence of the path $Y$. In particular, if $\Phi\ge0$ is a measurable function of the path such that $\Phi(\lambda y)=\lambda^{2p}\Phi(y)$ for all $\lambda>0$, for some real $p\ge1$, then $\E\Phi(Y)\le\E\Phi(Y^G)$, with equality if $p=1$. \end{lemma} \begin{proof} The drift in \eqref{mc-eq:lift} is linear, and the diffusion coefficient $|y|$ is positively homogeneous of degree one. By the third part of \cref{cd-lift}, applied with the random variable $\lambda=|g|/\sqrt N$, which is measurable at time zero, the process $\lambda Y$ is the unnormalized process with initial value $\lambda X_0=g$, driven by the same Brownian motion $B$. Pathwise uniqueness implies that $Y^G=(|g|/\sqrt N)Y$. The variables $|g|$ and $g/|g|$ are independent by the rotation invariance of $g$, and both are independent of $B$. We obtain $\E\Phi(Y^G)=\E\big[(|g|^2/N)^p\big]\,\E\Phi(Y)$, and $\E(|g|^2/N)^p\ge(\E|g|^2/N)^p=1$ by Jensen's inequality, with equality if $p=1$. \end{proof} \paragraph{The second moments.} Let $h_i(t)=\E Y_i(t)^2$, and let $\bar h(t)=N^{-1}\sum_ih_i(t)=N^{-1}\E|Y(t)|^2$ be their mean. The variable $\sup_{s\le t}|Y(s)|$ has finite moments of every order for each $t$, since the coefficients of \eqref{mc-eq:lift} are globally Lipschitz, and the stochastic integrals below are martingales. By Itô's formula, \begin{equation}\label{eq:coldm-square} d\big(Y_i^2\big)=\Big(-4a_iY_i^2+\frac2N|Y|^2\Big)dt+2\sqrt{\frac2N}\,|Y|\,Y_i\,dB_i . \end{equation} Recall that $\E X_{0,i}^2=1$. The second moments solve $h'=\mathsf Hh$ with $h(0)=\ind$, where \begin{equation}\label{eq:coldm-mean-matrix} \mathsf H=-4\diag(a_1,\dots,a_N)+\frac2N\ind\ind^\T . \end{equation} Using \cref{lem:coldm-gaussian-start} with $p=1$, we see that the Gaussian start has the same second moments. In coordinates, since $a_1=0$, \begin{equation}\label{eq:coldm-duhamel} h_1(t)=1+2\int_0^t\bar h(s)\,ds,\qquad h_i(t)=e^{-4a_it}+2\int_0^te^{-4a_i(t-s)}\bar h(s)\,ds\quad(i\ge2). \end{equation} In particular, $h_1$ is nondecreasing. The difference $h_1-h_i$ vanishes at zero and has derivative $4a_ih_i\ge0$. Therefore $h_i\le h_1$, and we have \begin{equation}\label{eq:coldm-ordering} 0\le h_1(t)-h_i(t)=\int_0^t4a_ih_i(s)\,ds\le4a_it\,h_1(t) . \end{equation} \paragraph{Renewal kernels.} For a vector $\kappa=(\kappa_i)_{i\ge2}$ of positive numbers, we define \begin{equation}\label{mc-eq:bath-heat} k_\kappa(t)=\frac2N\sum_{i=2}^N\kappa_ie^{-4a_it},\qquad U_\kappa=\sum_{n\ge1}k_\kappa^{*n},\qquad \theta_\kappa=1-\int_0^\infty k_\kappa(t)\,dt=1-\frac1N\sum_{i=2}^N\frac{\kappa_i}{2a_i}, \end{equation} where $k^{*n}$ is the $n$-fold convolution on $[0,\infty)$. We write $k$, $U$ for the case $\kappa_i=1$ for all $i$, for which $\theta_\kappa=\theta$. If $\theta_\kappa>0$, the series defining $U_\kappa$ converges in $L^1$, $\int_0^\infty U_\kappa=(1-\theta_\kappa)/\theta_\kappa$, and $U_\kappa$ is the unique locally bounded solution of the renewal equation $U_\kappa=k_\kappa+k_\kappa*U_\kappa$. Summing the second identity in \eqref{eq:coldm-duhamel} over $i\ge2$, we find $N\bar h-h_1=(N/2)k+k*(N\bar h)$, and convolving with $\delta_0+U$ yields \begin{equation}\label{eq:coldm-V-renewal} \bar h=\frac12\,U+\frac1N(\delta_0+U)*h_1 , \end{equation} where we used $k+U*k=U$. The next lemma collects the properties of these kernels. Its constants do not depend on $\theta$, and this is what makes all later estimates uniform in the top fraction. \begin{lemma}[Renewal estimates uniform in the top fraction]\label{lem:coldm-renewal} Let $K_1\ge1$, and suppose that $1/2\le\kappa_i\le K_1$ for all $i\ge2$ and that $\theta/2\le\theta_\kappa\le\min\{2\theta,2/3\}$. There exist $C\ge1$ and $c_0\in(0,1]$, depending only on $c_*$, $C_*$, and $K_1$, such that if $m\ge C$, the following hold. \begin{enumerate} \item Let $\bar u_i=\sqrt{\kappa_i}$ and $\mathsf H_\kappa=-4\diag(a_2,\dots,a_N)+(2/N)\bar u\bar u^\T$. Then $U_\kappa(t)=(2/N)\,\bar u^\T e^{t\mathsf H_\kappa}\bar u$, and all eigenvalues of the symmetric matrix $\mathsf H_\kappa$ are smaller than $-2a$. In particular, $U_\kappa(t)=\int e^{-zt}\,\mu_\kappa(dz)$ for a finite positive measure $\mu_\kappa$ supported in $(2a,\infty)$, and $U_\kappa$ is positive and nonincreasing. \item For $t\ge1$, $\int_t^\infty U_\kappa\le C\theta^{-2}t^{-1/2}$. \item We have $U_\kappa(t)\le Cf_\theta(t)$ for $0\le t\le c_0/a$, and $U_\kappa(t)\le C(N\theta^2)^{-1}e^{-2a(t-c_0/a)}$ for $t\ge c_0/a$. In particular, $U_\kappa(t)\le C\theta^{-2}(1+t)^{-3/2}$ for all $t\ge0$. \item If $\kappa_i=1$ for all $i$, then $U(t)\ge C^{-1}f_\theta(t)$ for $0\le t\le1/a$. \end{enumerate} \end{lemma} \begin{proof} We prove the four parts in order. Throughout the proof, $C$ denotes a constant depending only on $c_*$, $C_*$, and $K_1$. \paragraph{Step 1: spectral representation.} By the Duhamel formula for the rank-one perturbation, \[ e^{t\mathsf H_\kappa}=e^{-4tD}+\int_0^te^{-4(t-s)D}\,\frac2N\bar u\bar u^\T e^{s\mathsf H_\kappa}\,ds,\qquad D=\diag(a_2,\dots,a_N). \] Multiplying by $(2/N)\bar u^\T$ on the left and by $\bar u$ on the right, we find that $\Phi(t)=(2/N)\bar u^\T e^{t\mathsf H_\kappa}\bar u$ solves $\Phi=k_\kappa+k_\kappa*\Phi$, so $\Phi=U_\kappa$. For $v\in\R^{N-1}$, we have $(\bar u\cdot v)^2\le\sum_i\kappa_i(4a_i)^{-1}\cdot\sum_i4a_iv_i^2$ by the Cauchy--Schwarz inequality, and therefore \[ v^\T\mathsf H_\kappa v\le-4\sum_ia_iv_i^2\Big(1-\frac2N\sum_i\frac{\kappa_i}{4a_i}\Big)=-4\theta_\kappa\sum_ia_iv_i^2<0\qquad(v\ne0). \] This shows that $\mathsf H_\kappa$ is negative definite, and the spectral theorem yields $U_\kappa(t)=\int e^{-zt}\mu_\kappa(dz)$, where $\mu_\kappa$ puts mass $(2/N)\ip{\bar u}{\varphi}^2$ at $z$ for each unit eigenvector $\varphi$ of $-\mathsf H_\kappa$ with eigenvalue $z>0$. It remains to locate the eigenvalues. Let $G_\kappa(y)=(2/N)\sum_{i\ge2}\kappa_i(4a_i+y)^{-1}$, which is decreasing in $y>-4a$, with $G_\kappa(0)=1-\theta_\kappa$. If $-z$ is an eigenvalue of $\mathsf H_\kappa$ with eigenvector $\varphi$ and $z\notin\{4a_i\}$, then $(4D-z)\varphi=(2/N)\bar u(\bar u\cdot\varphi)$, which implies that $\bar u\cdot\varphi\ne0$ and $G_\kappa(-z)=1$. Using $4a_i-2a\ge2a_i$, we find \[ G_\kappa(-2a)-G_\kappa(0)=\frac2N\sum_{i\ge2}\frac{2a\kappa_i}{4a_i(4a_i-2a)}\le\frac{K_1a}{2N}\sum_{i\ge2}a_i^{-2}\le\frac C{Na}=\frac{C\theta}m, \] by \cref{lem:coldm-indexed}. Together with $\theta_\kappa\ge\theta/2$, this implies that $G_\kappa(-2a)\le1-\theta/2+C\theta/m<1$ for $m>2C$. If $0t)\le n\PP(X_1>t)+\frac nt\,\E\big[X_1\ind_{\{X_1\le t\}}\big]\le\frac{Cn}{\sqrt t} . \] Using $U_\kappa=\sum_n(1-\theta_\kappa)^np_\kappa^{*n}$, we find $\int_t^\infty U_\kappa=\sum_n(1-\theta_\kappa)^n\PP(S_n>t)\le Ct^{-1/2}\theta_\kappa^{-2}\le4Ct^{-1/2}\theta^{-2}$. \paragraph{Step 3: the upper bound before time $c_0/a$.} For $y>0$, the Laplace transform of $U_\kappa$ is $\widehat U_\kappa(y)=G_\kappa(y)/(1-G_\kappa(y))$, by the renewal equation. Let $C'=2^{2/3}C_*/c_*$, so that $y\ge C'a$ implies $N(y/C_*)^{3/2}\ge2$, and let $C'a\le y\le1$. By the third part of \cref{lem:coldm-indexed}, at least $\min\{N-1,N(y/C_*)^{3/2}/2\}$ indices $i\ge2$ satisfy $a_i\le y$. For these indices $4a_i+y\le5y$ and $a_i^{-1}\ge y^{-1}$, and since $\kappa_i\ge1/2$, we find \[ 1-G_\kappa(y)=\theta_\kappa+\frac2N\sum_{i\ge2}\frac{\kappa_iy}{4a_i(4a_i+y)}\ge\theta_\kappa+\frac1N\sum_{i\ge2:\,a_i\le y}\frac1{20a_i}\ge\theta_\kappa+c_2\sqrt y , \] where in the last step we used $y\le1$ when the minimum is $N-1$. Splitting the sum at $a_i=y$ and using the third and fourth parts of \cref{lem:coldm-indexed}, we also find \[ -G_\kappa'(y)=\frac2N\sum_i\frac{\kappa_i}{(4a_i+y)^2}\le\frac{2K_1}N\Big(\frac{\#\{i:a_i\le y\}}{y^2}+\sum_{i:a_i>y}\frac1{16a_i^2}\Big)\le\frac C{\sqrt y} . \] Since $U_\kappa$ is nonincreasing, for $t>0$, \[ -\widehat U_\kappa'(1/t)=\int_0^\infty se^{-s/t}U_\kappa(s)\,ds\ge U_\kappa(t)\int_0^tse^{-1}\,ds=\frac{t^2U_\kappa(t)}{2e} . \] On the other hand, $-\widehat U_\kappa'=-G_\kappa'/(1-G_\kappa)^2$. Let $c_0=1/C'$. For $1\le t\le c_0/a$, the two bounds at $y=1/t$ yield \[ U_\kappa(t)\le\frac{2eC\,t^{-3/2}}{(\theta_\kappa+c_2t^{-1/2})^2}\le C\min\big\{t^{-1/2},\theta^{-2}t^{-3/2}\big\}\le Cf_\theta(t) . \] For $t\le1$, the renewal equation and $k_\kappa\le C$ imply that $U_\kappa(t)\le C+C\int_0^tU_\kappa$, and by Gronwall's inequality $U_\kappa\le C$ on $[0,1]$, while $f_\theta(t)\ge2^{-1/2}$ there because $\theta\le1/2$. \paragraph{Step 4: the upper bound after time $c_0/a$.} Let $t_0=c_0/a$. From Step~1, we have $U_\kappa(t)\le e^{-2a(t-t_0)}U_\kappa(t_0)$ for $t\ge t_0$. Step~3 gives $U_\kappa(t_0)\le Cf_\theta(t_0)\le C\theta^{-2}(a/c_0)^{3/2}\le C(N\theta^2)^{-1}$, where we used $a^{3/2}\le C_*^{3/2}N^{-1}$. This is the second bound of the third part. For the last bound, $f_\theta(t)\le\theta^{-2}(1+t)^{-3/2}$ covers $t\le t_0$, and for $t\ge t_0$ we have $(1+t)^{3/2}e^{-2at}\le Ca^{-3/2}\le CN$. \paragraph{Step 5: the lower bound.} Let $\kappa_i=1$ for all $i$. Let $1\le t\le1/a$. We claim that at least $c_2Nt^{-3/2}$ indices $i\ge2$ satisfy $a_i\le1/t$. If $N(tC_*)^{-3/2}\ge1$, this follows from the third part of \cref{lem:coldm-indexed}. Otherwise, the index $i=2$ qualifies, because $a\le1/t$, and $Nt^{-3/2}0$, depending only on $c_*$ and $C_*$, such that $\PP(S_{n-1}>t/2)\le1/2$ for $n\le1+c_1\sqrt t$. Let $n_*=\min\{\lfloor1+c_1\sqrt t\rfloor,\lceil1/\theta\rceil\}$. Using $1-\theta\ge e^{-2\theta}$ and $\theta n_*\le2$, we obtain \[ U(t)\ge\sum_{n=1}^{n_*}(1-\theta)^np^{*n}(t)\ge c_2\,t^{-3/2}e^{-4}\sum_{n=1}^{n_*}n\ge c_2\,t^{-3/2}\min\{t,\theta^{-2}\}\ge c_2f_\theta(t) . \] For $t\le1$, we have $U\ge k\ge c_2\ge c_2f_\theta$. This completes the proof. \end{proof} \subsection{A weighted mean} The proof of the moment bounds in \cref{sec:coldm-moments} compares the moments with products of the solutions of a linear system in which the coupling coefficients of the first coordinates are increased. The next lemma bounds these solutions. Its constants do not depend on the size $C_1$ of the perturbation of the remaining coordinates, which is chosen afterwards in terms of these constants. \begin{lemma}[Heat estimates uniform under bounded perturbations]\label{cpm-lem:weighted-heat} Fix $C_0\ge1$ and $C_1\ge0$, let $n_m=\lfloor m\rfloor$, and let $\kappa_i=C_0$ for $1\le i\le n_m$ and $\kappa_i=1+C_1\theta/n_m$ for $i>n_m$. Let $\omega(t)=(\omega_1(t),\dots,\omega_N(t))$ solve \[ \omega_i'=-4a_i\omega_i+\frac2N\kappa_i\,\omega_\Sigma,\qquad\omega_\Sigma=\sum_{j=1}^N\omega_j,\qquad\omega_i(0)=1 . \] There exist $C\ge1$, depending only on $c_*$, $C_*$, $C_0$, and $K_0$, and $m_0$, depending in addition on $C_1$, such that the following hold for $m\ge m_0$ and $0\le t\le T_{\max}$. \begin{enumerate} \item We have $\omega_1(t)\le C/\theta$, $\omega_\Sigma(t)\le C[Nf_\theta(t)+\theta^{-2}]$, and $\int_0^{T_{\max}}\omega_\Sigma\le CN/\theta$. \item For $i\ge2$, \[ \omega_i(t)\le \begin{cases} C\Big[\dfrac{\sqrt{1+t}}{1+a_it}+\dfrac{\min\{t,a_i^{-1}\}}{N\theta^2}\Big],& t\le\theta^{-2},\\[3mm] C\Big[\dfrac{e^{-2a_it}}\theta+\dfrac{\min\{t,a_i^{-1}\}}{\theta^2t^{3/2}}+\dfrac{\min\{t,a_i^{-1}\}}{N\theta^2}\Big],& t\ge\theta^{-2}. \end{cases} \] \item The mean satisfies $N\bar h(t)\ge C^{-1}[Nf_\theta(t)+\theta^{-2}]$. \item We have \[ N\bar h(t)\le\omega_\Sigma(t)\le CN\bar h(t),\qquad \omega_i(t)\le Ch_i(t)\quad(1\le i\le N),\qquad \max_{i>n_m}\frac{\omega_i(t)}{\omega_\Sigma(t)}\le\frac{C\theta}{n_m}. \] \end{enumerate} The case $C_0=1$, $C_1=0$ is $\omega=h$, and then the first two parts hold for the second moments $h_i$ and $N\bar h=\omega_\Sigma$. \end{lemma} \begin{proof} The proof has four steps: we compare the perturbed kernel with the unperturbed one, bound the first coordinate and the total, bound the other coordinates, and finally bound $\bar h$ from below and compare. Throughout, $C$ denotes a constant depending only on $c_*$, $C_*$, $C_0$, and $K_0$. \paragraph{Step 1: the perturbed kernel.} Let $\kappa_b=(\kappa_i)_{i\ge2}$. Using \cref{lem:coldm-indexed}, we have $\sum_{2\le i\le n_m}a_i^{-1}\le CN^{2/3}n_m^{1/3}$ and $N^{-1/3}=\theta N^{2/3}a/m\le C_*\theta/m$. It follows that \[ 0\le\theta-\theta_{\kappa_b}=\frac1N\sum_{i\ge2}\frac{\kappa_i-1}{2a_i}\le\frac{C_0}N\sum_{2\le i\le n_m}\frac1{a_i}+\frac{C_1\theta}{n_m}\le\frac{C\theta\,n_m^{1/3}}m+\frac{C_1\theta}{n_m}, \] which is at most $\theta/2$ for $m\ge m_0(C_1)$. Also $1\le\kappa_i\le C_0+1$ once $C_1\theta/n_m\le1$. We may therefore use \cref{lem:coldm-renewal} for $\kappa_b$ with $K_1=C_0+1$, and its constants depend only on $c_*$, $C_*$, and $C_0$. This is the only place where $C_1$ enters, through the threshold $m_0$. \paragraph{Step 2: the first coordinate and the total.} As in \eqref{eq:coldm-V-renewal}, summing the equations for $i\ge2$, we find \[ \omega_\Sigma=W+(\delta_0+U_{\kappa_b})*\omega_1,\qquad W=(\delta_0+U_{\kappa_b})*K_b,\qquad K_b(t)=\sum_{i\ge2}e^{-4a_it}, \] and $\omega_1=1+(2C_0/N)\int_0^t\omega_\Sigma$. Because $\kappa_i\ge1$, we have $K_b\le(N/2)k_{\kappa_b}$ and $W\le(N/2)(k_{\kappa_b}+U_{\kappa_b}*k_{\kappa_b})=(N/2)U_{\kappa_b}$. The function $\omega_1$ is nondecreasing, and $((\delta_0+U_{\kappa_b})*\omega_1)(t)\le\omega_1(t)(1+\int U_{\kappa_b})=\omega_1(t)/\theta_{\kappa_b}\le2\omega_1(t)/\theta$. We conclude that \begin{equation}\label{eq:coldm-total-bound} \omega_\Sigma(t)\le\frac N2U_{\kappa_b}(t)+\frac2\theta\omega_1(t), \end{equation} and, integrating, \[ \omega_1(t)\le1+\frac{C_0}{\theta_{\kappa_b}}+\frac{4C_0}{N\theta}\int_0^t\omega_1(s)\,ds\le\frac{3C_0}\theta+\frac{4C_0}{N\theta}\int_0^t\omega_1(s)\,ds . \] Gronwall's inequality implies that $\omega_1(t)\le3C_0\theta^{-1}e^{4C_0t/(N\theta)}$. For $t\le T_{\max}$, we have $t/(N\theta)\le K_0\log(em)/(aN\theta)=K_0\log(em)/m\le K_0$, and $\omega_1(t)\le C/\theta$. Inserting this and the third part of \cref{lem:coldm-renewal} into \eqref{eq:coldm-total-bound}, we find $\omega_\Sigma(t)\le C[Nf_\theta(t)+\theta^{-2}]$ for $t\le c_0/a$, and $\omega_\Sigma(t)\le C\theta^{-2}$ for $t\ge c_0/a$. This proves the first two bounds of the first part. For the third, we integrate \eqref{eq:coldm-total-bound}: $\int_0^{T_{\max}}\omega_\Sigma\le N/\theta+C\theta^{-2}T_{\max}$, and $\theta^{-2}T_{\max}=(N/\theta)K_0\log(em)/m\le CN/\theta$. \paragraph{Step 3: the other coordinates.} Let $i\ge2$. By variation of constants, \begin{equation}\label{eq:coldm-coordinate-duhamel} \omega_i(t)=e^{-4a_it}+\frac{2\kappa_i}N\int_0^te^{-4a_i(t-s)}\omega_\Sigma(s)\,ds . \end{equation} Let $t\le\theta^{-2}$. Then $f_\theta(s)\le(1+s)^{-1/2}$, and splitting the integral at $s=t/2$, we have \[ \int_0^te^{-4a_i(t-s)}(1+s)^{-1/2}\,ds\le2e^{-2a_it}\sqrt{1+t}+\frac{\sqrt2\min\{t,a_i^{-1}\}}{\sqrt{1+t}}\le\frac{C\sqrt{1+t}}{1+a_it}, \] where in the last step we used $a_i\le C_*$; the term with $\theta^{-2}$ contributes $\theta^{-2}\min\{t,(4a_i)^{-1}\}$. This proves the first case of the second part. Now let $t\ge\theta^{-2}$, and split \eqref{eq:coldm-coordinate-duhamel} at $s=t/2$. For $s\le t/2$, the exponential is at most $e^{-2a_it}$, and the first part gives $(2\kappa_i/N)\int_0^{T_{\max}}\omega_\Sigma\le C/\theta$. For $s\in[t/2,t]$, we have $f_\theta(s)\le\theta^{-2}(1+s)^{-3/2}\le3\theta^{-2}t^{-3/2}$, which implies that $\omega_\Sigma(s)\le C[N\theta^{-2}t^{-3/2}+\theta^{-2}]$, and $\int_{t/2}^te^{-4a_i(t-s)}ds\le\min\{t,a_i^{-1}\}$. Together with $e^{-4a_it}\le e^{-2a_it}\le\theta^{-1}e^{-2a_it}$, this proves the second case. \paragraph{Step 4: lower bound and comparisons.} By \eqref{eq:coldm-V-renewal}, we have $\bar h\ge U/2$, and the fourth part of \cref{lem:coldm-renewal} implies that $N\bar h(t)\ge c_1Nf_\theta(t)$ for $t\le1/a$. Recall from \eqref{eq:coldm-duhamel} that $h_1(s)=1+2\int_0^s\bar h\ge1+\int_0^sU$. Let $t\ge2$. Since $h_1$ is nondecreasing, \eqref{eq:coldm-V-renewal} implies that \[ N\bar h(t)\ge h_1(t)+\int_0^{t/2}U(s)h_1(t-s)\,ds\ge h_1(t/2)\Big(1+\int_0^{t/2}U\Big)\ge\Big(1+\int_0^{t/2}U\Big)^2 . \] The second part of \cref{lem:coldm-renewal} implies that $1+\int_0^{t/2}U=\theta^{-1}-\int_{t/2}^\infty U\ge\theta^{-1}-C\theta^{-2}t^{-1/2}$. We conclude that $N\bar h(t)\ge(4\theta^2)^{-1}$ for $t\ge C_2\theta^{-2}$, with $C_2$ depending only on $c_*,C_*$. We now combine these bounds. For $t\le C_2\theta^{-2}$, we have $f_\theta(t)\ge c_1\theta$, and the second part of \cref{lem:coldm-indexed} implies that $Nf_\theta(t)\ge c_1N\theta\ge c_1\theta^{-2}N\theta^3\ge c_1\theta^{-2}$. Then $N\bar h(t)\ge c_1Nf_\theta(t)\ge c_1[Nf_\theta(t)+\theta^{-2}]$. For $C_2\theta^{-2}\le t\le1/a$, both lower bounds apply; note that $C_2\theta^{-2}\le1/a$ for large $m$, because $\theta^2/a\ge m^2/C_*^3$. For $1/a\le t\le T_{\max}$, we have $Nf_\theta(t)\le N\theta^{-2}a^{3/2}\le C_*^{3/2}\theta^{-2}$, and $N\bar h(t)\ge(4\theta^2)^{-1}$. This proves the third part. For the fourth part, note that the matrix of the system for $\omega$ is $\mathsf H+(2/N)(\kappa-\ind)\ind^\T$, where $\mathsf H$ is the matrix of \eqref{eq:coldm-mean-matrix}. Since $\kappa_i\ge1$, this matrix dominates $\mathsf H$ entrywise. The difference $e=\omega-h$ solves $e'=\mathsf He+\phi$ with $e(0)=0$ and $\phi_i=(2/N)(\kappa_i-1)\omega_\Sigma\ge0$, and $e(t)=\int_0^te^{(t-s)\mathsf H}\phi(s)\,ds\ge0$, because $e^{t\mathsf H}$ has nonnegative entries. We conclude that $\omega\ge h$ componentwise, and in particular $\omega_\Sigma\ge N\bar h$. Combining the first and third parts, we have $\omega_\Sigma\le C[Nf_\theta+\theta^{-2}]\le CN\bar h$. Inserting $\omega_\Sigma\le CN\bar h$ into \eqref{eq:coldm-coordinate-duhamel}, and comparing with \eqref{eq:coldm-duhamel}, we obtain $\omega_i\le(C_0+1)Ch_i$ for $i\ge2$; similarly $\omega_1=1+(2C_0/N)\int\omega_\Sigma\le C_0Ch_1$. Finally, let $i>n_m$, so that $i-1\ge n_m$. Let $t\le\theta^{-2}$. Then $f_\theta(t)\ge(1+t)^{-1/2}/2$, and using the second and third parts, we find \[ \frac{\omega_i(t)}{\omega_\Sigma(t)}\le\frac{\omega_i(t)}{N\bar h(t)}\le C\Big[\frac{1+t}{N(1+a_it)}+\frac{\sqrt{1+t}}{N^2\theta^2a_i}\Big]\le\frac C{Na_i}\Big(1+\frac1{N\theta^3}\Big)\le\frac C{Na_i}, \] where we used $a_i\le C_*$ and $\sqrt{1+t}\le\sqrt2\theta^{-1}$. Let $t\ge\theta^{-2}$. Then $N\bar h(t)\ge c_1[N\theta^{-2}t^{-3/2}+\theta^{-2}]$, and dividing the three terms of the second part by this bound, we obtain \[ \frac{\omega_i(t)}{N\bar h(t)}\le C\Big[\frac{\theta t^{3/2}e^{-2a_it}}N+\frac1{Na_i}+\frac1{Na_i}\Big]\le C\Big[\frac\theta{Na_i^{3/2}}+\frac1{Na_i}\Big], \] where we used $\sup_tt^{3/2}e^{-2a_it}\le Ca_i^{-3/2}$. Note that $Na_i^{3/2}\ge c_*^{3/2}n_m$ and $Na_i\ge c_*n_m^{2/3}N^{1/3}$, by \eqref{mc-eq:indexed-geometry}. Using $N^{1/3}\theta\ge m/C_*$ and $n_m\le m$, we find $(Na_i)^{-1}\le C\theta m^{-1}n_m^{-2/3}\le C\theta/n_m$. This completes the proof. \end{proof} We record the consequences for the second moments $h_i$ and their mean $\bar h$. \begin{corollary}[Bounds for the mean]\label{lem:coldm-means} For $m\ge m_0$ and $0\le t\le T_{\max}$, the following hold. \begin{enumerate} \item $h_i(t)\le h_1(t)\le\theta^{-1}e^{2t/(N\theta)}\le C/\theta$ for all $i$, and $\theta h_1(t)\ge1-C(\theta^2t)^{-1/2}$ for $t\ge1$. \item $\bar h(t)\le U(t)/2+C/(N\theta^2)$, and $\int_t^\infty U\le C\theta^{-2}t^{-1/2}$ for $t\ge1$. \item $C^{-1}[f_\theta(t)+(N\theta^2)^{-1}]\le\bar h(t)\le C[f_\theta(t)+(N\theta^2)^{-1}]$. \item For $t\ge c_0/a$, we have $\bar h(t)\le C(N\theta^2)^{-1}$ and $h_i(t)\le C[\theta^{-1}e^{-2a_it}+(N\theta^2a_i)^{-1}]$ for $i\ge2$. \item For $0\le t\le\theta^{-2}$, we have $C^{-1}(1+t)^{-1/2}\le U(t)\le C(1+t)^{-1/2}$ and $C^{-1}(1+t)^{-1/2}\le\bar h(t)\le C[(1+t)^{-1/2}+(N\theta^2)^{-1}]$. \end{enumerate} \end{corollary} \begin{proof} Using \eqref{eq:coldm-V-renewal} and the monotonicity of $h_1$, we have $\bar h(t)\le U(t)/2+h_1(t)/(N\theta)$. Integrating and using $1+\int U=\theta^{-1}$, we find $h_1(t)\le\theta^{-1}+(2/(N\theta))\int_0^th_1$, and Gronwall's inequality yields the upper bound in the first part; the bound by $C/\theta$ follows as in Step~2 of the previous proof. For the lower bound, $\theta h_1(t)\ge\theta(1+\int_0^tU)=1-\theta\int_t^\infty U$, and the second part of \cref{lem:coldm-renewal} applies. The second part follows from $\bar h\le U/2+h_1/(N\theta)$ and the first part. The third part is \cref{cpm-lem:weighted-heat} with $C_0=1$ and $C_1=0$. For the fourth part, $c_0/a\ge\theta^{-2}$ when $m$ is large, and the second case of \cref{cpm-lem:weighted-heat} applies, with $\theta^{-2}t^{-3/2}\min\{t,a_i^{-1}\}\le\theta^{-2}(a/c_0)^{3/2}a_i^{-1}\le C(N\theta^2a_i)^{-1}$; the bound on $\bar h$ is part of Step~2 there. For the fifth part, $\theta^{-2}\le c_0/a$, and \cref{lem:coldm-renewal} implies that $C^{-1}f_\theta\le U\le Cf_\theta$ on $[0,\theta^{-2}]$, and $(1+t)^{-1/2}/2\le f_\theta(t)\le(1+t)^{-1/2}$ there; the bounds on $\bar h$ follow from the third part and from $\bar h\ge U/2$. \end{proof} \section{Moments of all orders}\label{sec:coldm-moments} The time-$t$ law depends on the unnormalized process through the ratio $Y/|Y|$, and the path estimates of \cref{sec:coldm-path} require moments of $|Y(t)|^2$ and $Y_i(t)^2$ of every fixed order. The noise amplitude $|Y|$ is random, and the coordinates are not Gaussian. We bound the moments by comparing the exact linear equations they satisfy with an explicit supersolution. The comparison keeps the small deficit $\theta$: the coupling coefficient of the remaining coordinates exceeds one only by $C_1\theta/n_m$, which the bound of \cref{cpm-lem:weighted-heat} on $\omega_i/\omega_\Sigma$ allows. \begin{proposition}[All fixed-order moment bounds for the uniform start]\label{cpm-prop:all-moments} Let $Y$ have the uniform start $Y(0)\sim\nu_N$. For every real $r\ge1$ there exist $C_r$ and $m_0$, depending only on $r$, $c_*$, $C_*$, and $K_0$, such that for $m\ge m_0$, $0\le t\le T_{\max}$, and $1\le i\le N$, \[ \big\||Y(t)|^2\big\|_r\le C_rN\bar h(t),\qquad \big\|Y_i(t)^2\big\|_r\le C_rh_i(t),\qquad \big\||Y(t)|^2Y_i(t)^2\big\|_r\le C_rN\bar h(t)h_i(t), \] where $\|\cdot\|_r$ is the norm of $L^r(\PP)$. \end{proposition} \begin{proof} We first prove the bounds directly for $Y$ and integer orders $p\ge2$. \paragraph{Step 1: the moment equations.} For a multi-index $n\in\N^N$ with $|n|=n_1+\cdots+n_N=p$, let $S_n(t)=\E\prod_iY_i(t)^{2n_i}$. For $f(y)=\prod_iy_i^{2n_i}$, we have $y_i\partial_if=2n_if$ and $y_i^2\partial_i^2f=2n_i(2n_i-1)f$. Applying Itô's formula to $f(Y)$, using \eqref{mc-eq:lift}, and taking expectations, we obtain \begin{equation}\label{eq:coldm-moment-system} S_n'=-4(n\cdot a)\,S_n+\frac2N\sum_{i:n_i\ge1}n_i(2n_i-1)\sum_{j=1}^NS_{n-e_i+e_j}, \end{equation} where $e_i$ is the $i$th unit vector and we used $|Y|^2=\sum_jY_j^2$. The stochastic integrals have mean zero because $Y$ has finite moments of all orders on bounded time intervals. The system \eqref{eq:coldm-moment-system} is linear in $(S_n)_{|n|=p}$, and its off-diagonal coefficients are nonnegative. By independence of Gaussian radius and direction, as in \cref{lem:coldm-gaussian-start}, we have $S_n(0)\le\E\prod_ig_i^{2n_i}=\prod_i(2n_i-1)!!$, with the convention $(-1)!!=1$. \paragraph{Step 2: a supersolution.} Let $C_{\mathrm b}$ be the constant of the last bound in the fourth part of \cref{cpm-lem:weighted-heat} for $C_0=2p-1$; it does not depend on $C_1$. \Cref{cpm-lem:weighted-heat} applies with $C_0=2p-1$ and $C_1=2(p-1)C_{\mathrm b}$. Let $\eta_m=\max_{i>n_m}\omega_i/\omega_\Sigma$, so that $\eta_m\le C_{\mathrm b}\theta/n_m$ on $[0,T_{\max}]$. Increasing $m_0$, we may assume that $(2p-1)\eta_m\le1$. Define \[ \bar S_n(t)=(2p-1)!!\prod_{i=1}^N\omega_i(t)^{n_i}\prod_{i>n_m}(2n_i-1)!! . \] We call the indices $i\le n_m$ low indices and the indices $i>n_m$ high indices. From $\omega_i'=-4a_i\omega_i+(2/N)\kappa_i\omega_\Sigma$, we find \[ \bar S_n'=-4(n\cdot a)\,\bar S_n+\frac2N\sum_{i:n_i\ge1}n_i\kappa_i\frac{\omega_\Sigma}{\omega_i}\,\bar S_n . \] The function $\bar S$ is a supersolution of \eqref{eq:coldm-moment-system} if, for every $n$ and every $i$ with $n_i\ge1$, \begin{equation}\label{eq:coldm-super-condition} \kappa_i\frac{\omega_\Sigma}{\omega_i}\ge(2n_i-1)\sum_{j=1}^N\frac{\bar S_{n-e_i+e_j}}{\bar S_n} . \end{equation} The ratio $\bar S_{n-e_i+e_j}/\bar S_n$ equals one if $j=i$. If $j\ne i$, it equals $\omega_j/\omega_i$ times a factor from the double factorials: a factor $(2n_i-1)^{-1}$ if $i$ is a high index, and a factor $2n_j+1$ if $j$ is a high index. Let $i$ be a high index. Multiplying by $\omega_i/\omega_\Sigma$, the right side of \eqref{eq:coldm-super-condition} becomes \begin{align*} \frac1{\omega_\Sigma}\Big[(2n_i-1)\omega_i+\sum_{j\le n_m}\omega_j+\sum_{j>n_m,\,j\ne i}(2n_j+1)\omega_j\Big] &=1+\frac2{\omega_\Sigma}\Big(\sum_{j>n_m}n_j\omega_j-\omega_i\Big)\\ &\le1+2(p-1)\eta_m , \end{align*} because $\sum_{j>n_m}n_j\omega_j-\omega_i=\sum_{j>n_m}(n_j-\delta_{ij})\omega_j$ and $\sum_j(n_j-\delta_{ij})\le p-1$. The cancellation of the factor $2n_i-1$ is what keeps the coefficient of the high indices close to one. The condition \eqref{eq:coldm-super-condition} holds, because $\kappa_i=1+C_1\theta/n_m\ge1+2(p-1)\eta_m$. Let $i$ be a low index. The same computation gives the right side \[ (2n_i-1)\Big(1+\frac2{\omega_\Sigma}\sum_{j>n_m}n_j\omega_j\Big)\le(2n_i-1)\big(1+2(p-n_i)\eta_m\big)\le2n_i-1+2(p-n_i)=2p-1, \] where we used that the high indices carry at most $p-n_i$ units of $n$, and $(2n_i-1)\eta_m\le1$. Here $\kappa_i=2p-1$, and the condition \eqref{eq:coldm-super-condition} holds. At time zero, $\bar S_n(0)\ge S_n(0)$, because $\prod_{i\le n_m}(2n_i-1)!!\le(2p-1)!!$: the left side counts the pairings of $2\sum_{i\le n_m}n_i$ points that respect a partition into groups of sizes $2n_i$, and these are among the pairings of $2p$ points. \paragraph{Step 3: comparison.} Let $\mathsf A$ be the matrix of the system \eqref{eq:coldm-moment-system}. Then $\bar S'\ge\mathsf A\bar S$ componentwise, $\bar S(0)\ge S(0)$, and $S'=\mathsf AS$, and the difference satisfies $(\bar S-S)'=\mathsf A(\bar S-S)+\phi$ with $\phi\ge0$. Since $\mathsf A$ has nonnegative off-diagonal entries, choosing $\lambda$ with $\mathsf A+\lambda I\ge0$ we have $e^{t\mathsf A}=e^{-\lambda t}\sum_kt^k(\mathsf A+\lambda I)^k/k!\ge0$ entrywise, and therefore \[ \bar S(t)-S(t)=e^{t\mathsf A}\big(\bar S(0)-S(0)\big)+\int_0^te^{(t-s)\mathsf A}\phi(s)\,ds\ge0 . \] This proves the comparison principle used here. \paragraph{Step 4: conclusion.} The double factorials in $\bar S_n$ are at most $((2p-1)!!)^2$. Using the multinomial formula and $S\le\bar S$, we obtain \[ \E|Y(t)|^{2p}=\sum_{|n|=p}\frac{p!}{\prod_in_i!}S_n\le((2p-1)!!)^2\,\omega_\Sigma^p, \] and $\E Y_i^{2p}=S_{pe_i}\le((2p-1)!!)^2\omega_i^p$. Recall that $\omega_\Sigma\le CN\bar h$ and $\omega_i\le Ch_i$, by the fourth part of \cref{cpm-lem:weighted-heat}. This proves the radial and coordinate bounds for integer orders $p\ge2$; the case $p=1$ follows from the identities $\E|Y|^2=N\bar h$ and $\E Y_i^2=h_i$. For real $r\ge1$, we increase $r$ to the integer $\lceil r\rceil$ and use $\||Y|^2Y_i^2\|_r\le\||Y|^2\|_{2r}\|Y_i^2\|_{2r}$. \end{proof} For $p=2$, these are the fourth-moment bounds \begin{equation}\label{mc-eq:fourth-moments} \E|Y(t)|^4\le C\big(N\bar h(t)\big)^2,\qquad \E\big[|Y(t)|^2Y_i(t)^2\big]\le CN\bar h(t)h_i(t)\qquad(0\le t\le T_{\max}). \end{equation} \section{Concentration of the radial part along the path}\label{sec:coldm-path} At late times, the radial part $|Y(t)|^2$ has size $\theta^{-2}$ and the square of the first coordinate has size $\theta^{-1}$. The first coordinate determines the noise amplitude at late times through $\theta|Y|^2\approx Y_1^2$. In this section, we show that after a time of order $a^{-1}$ the quantity $\theta^2|Y(t)|^2$ is close to $H_1^2$, where $H_1=\sqrt\theta\,Y_1(\tau)$ is the scaled first coordinate at an earlier time $\tau$, uniformly in $t$ up to time $T_{\max}$. The difference $\theta^2|Y|^2-H_1^2$ involves the projection of the vector of squares $(Y_i^2)_i$ onto the stable directions of the mean matrix $\mathsf H$; \cref{lem:coldm-projected} collects the properties of the projected kernel, and \cref{cpm-thm:radial-path} proves the concentration. \subsection{The projected kernel} The matrix $\mathsf H$ of \eqref{eq:coldm-mean-matrix} has a positive eigenvalue, because $a_1=0$, and this eigenvalue drives the growth of the mean. We remove the corresponding component. \begin{lemma}[Projected kernel estimates]\label{lem:coldm-projected} There exist $C$ and $m_0$, depending only on $c_*$, $C_*$, and $K_0$, such that the following hold for $m\ge m_0$. \begin{enumerate} \item The matrix $\mathsf H$ has exactly one positive eigenvalue $\lambda_+$. It is the unique positive solution of $(2/N)\sum_i(4a_i+\lambda)^{-1}=1$, with eigenvector $u=((4a_i+\lambda_+)^{-1})_i$, and all other eigenvalues of $\mathsf H$ are smaller than $-2a$. Let $\bar\theta=2/(N\lambda_+)$. Then \[ \theta\le\bar\theta\le\theta\big(1+Cm^{-2}\big),\qquad\lambda_+\le\frac{2a}m . \] \item Let $d(t)=e^{t\mathsf H}(\bar\theta\ind-e_1)$, and let $\widetilde U=U_{\tilde\kappa}$ be the resolvent \eqref{mc-eq:bath-heat} with $\tilde\kappa_i=4a_i/(4a_i+\lambda_+)$. Then $d(t)$ is orthogonal to $u$, and \begin{equation}\label{cpm-eq:projected-exact} \begin{gathered} d_i=\bar\theta\,e^{-4a_i\cdot}*(\delta_0+\widetilde U)\quad(i\ge2),\\ d_1(t)=-\bar\theta\int_t^\infty\widetilde U(s)\,ds,\qquad \sum_{i=1}^Nd_i(t)=\frac{N\bar\theta}2\,\widetilde U(t). \end{gathered} \end{equation} In particular $d_i\ge0$ for $i\ge2$, and $-d_1$ is nonincreasing, with values in $[0,1]$. \item For $t\ge1/a$, we have $\theta\sum_id_i(t)\le Ce^{-2at}$ and \begin{equation}\label{cpm-eq:projected-norm} \|d(t)\|_2\le Ce^{-2at},\qquad|d_1(t)|\le\frac Cm\Big(\sum_{i\ge2}d_i(t)^2\Big)^{1/2}. \end{equation} \item We have \begin{equation}\label{cpm-eq:kernel-energy} \int_0^{T_{\max}}d_1^2\le C\theta^{-2}\log(em),\qquad \int_0^\infty d_i^2\le\frac1{8a_i}\quad(i\ge2),\qquad \int_0^\infty\sum_{i\ge2}\frac{d_i^2}{a_i}\le\frac C{a^2}. \end{equation} \item Extend each $d_i$ by zero on $(-\infty,0)$, and write $\|\cdot\|_2$ for the norm of $L^2(\R)$. For $h>0$, \begin{align} \|d_i(\cdot+h)-d_i\|_2^2&\le2\min\{h,(4a_i)^{-1}\}\quad(i\ge2),\label{cpm-eq:translation}\\ \sum_{i\ge2}\frac{\|d_i(\cdot+h)-d_i\|_2^2}{a_i}&\le CN\sqrt h\le\frac C{a^2}\sqrt{ah}\quad(00$, \begin{equation}\label{cpm-eq:analytic-kernel} \|d'(s)\|_2\le\frac2{es}\,\|d(s/2)\|_2 . \end{equation} \end{enumerate} \end{lemma} \begin{proof} For the first part, let $\varphi$ be an eigenvector of $\mathsf H$ with eigenvalue $\mu$. Then $(\mu+4a_i)\varphi_i=(2/N)\ind\cdot\varphi$ for all $i$. If $\mu\notin\{-4a_i\}$, then $\ind\cdot\varphi\ne0$ and $\Phi(\mu)=1$, where $\Phi(\mu)=(2/N)\sum_i(\mu+4a_i)^{-1}$. On $(0,\infty)$, $\Phi$ decreases strictly from $+\infty$ to $0$, which implies that there exists exactly one positive eigenvalue $\lambda_+$, with eigenvector $u$; it exists because $e_1^\T\mathsf He_1=2/N>0$. On $(-4a,0)$, $\Phi$ decreases strictly from $+\infty$ to $-\infty$, and $\mathsf H$ has at most one eigenvalue $\mu_2$ in this interval. The other eigenvalues are at most $-4a$. With the notation of the proof of \cref{lem:coldm-renewal}, we have $\Phi(-2a)=-(Na)^{-1}+G_1(-2a)\le-\theta/m+1-\theta+C\theta/m<1$ for $m$ large, and since $\Phi$ is decreasing on $(-4a,0)$, any eigenvalue $\mu_2$ in this interval satisfies $\mu_2<-2a$. Next, $\Phi(\lambda_+)=1$ implies that $\bar\theta=2/(N\lambda_+)=1-(2/N)\sum_{i\ge2}(4a_i+\lambda_+)^{-1}\ge\theta$, and hence $\lambda_+\le2/(N\theta)=2a/m$. Also \[ \bar\theta-\theta=\frac2N\sum_{i\ge2}\frac{\lambda_+}{4a_i(4a_i+\lambda_+)}\le\frac{\lambda_+}{8N}\sum_{i\ge2}a_i^{-2}\le\frac{C}{N^2\theta a^2}=\frac{C\theta}{m^2}, \] by \cref{lem:coldm-indexed}. For the second part, the secular equation yields $\sum_iu_i=N/2$ and $u_1=\lambda_+^{-1}=N\bar\theta/2$, and $u\cdot(\bar\theta\ind-e_1)=0$. The matrix $\mathsf H$ is symmetric. Then the orthogonal complement of $u$ is invariant, and $d(t)\perp u$ for all $t$. In coordinates, $d_1/\lambda_++\sum_{i\ge2}d_i/(4a_i+\lambda_+)=0$, that is, $d_1=-\sum_{i\ge2}(1-\tilde\kappa_i)d_i$, and $\sum_id_i=\sum_{i\ge2}\tilde\kappa_id_i$. The vector $d_b=(d_i)_{i\ge2}$ therefore solves $d_b'=[-4\diag(a_2,\dots,a_N)+(2/N)\ind\tilde\kappa^\T]d_b$ with $d_b(0)=\bar\theta\ind$. As in \eqref{eq:coldm-V-renewal}, the sum $\Omega=\sum_{i\ge2}\tilde\kappa_id_i$ solves $\Omega=(N\bar\theta/2)k_{\tilde\kappa}+k_{\tilde\kappa}*\Omega$, which implies that $\Omega=(N\bar\theta/2)\widetilde U$, and variation of constants gives the first identity in \eqref{cpm-eq:projected-exact}. The third identity is $\sum_id_i=\Omega$. Since $d_1'=(2/N)\sum_jd_j=\bar\theta\widetilde U$, $d_1(0)=\bar\theta-1$, and $\int_0^\infty\widetilde U=(1-\theta_{\tilde\kappa})/\theta_{\tilde\kappa}$ with $\theta_{\tilde\kappa}=1-(2/N)\sum_{i\ge2}(4a_i+\lambda_+)^{-1}=\bar\theta$, we obtain the second identity. The coefficients satisfy $1\ge\tilde\kappa_i\ge1-\lambda_+/(4a)\ge1/2$, and $\theta_{\tilde\kappa}=\bar\theta\in[\theta,2\theta]$ with $\bar\theta\le2/3$ for $m$ large, so \cref{lem:coldm-renewal} applies to $\widetilde U$ with $K_1=1$. For the third part, the third identity in \eqref{cpm-eq:projected-exact} and \cref{lem:coldm-renewal} give $\theta\sum_id_i(t)\le CN\theta^2(N\theta^2)^{-1}e^{-2a(t-c_0/a)}\le Ce^{-2at}$ for $t\ge c_0/a$. Let $t_0=c_0/a$. Splitting the convolution in \eqref{cpm-eq:projected-exact} at $t_0/2$ and using the third part of \cref{lem:coldm-renewal}, we find $d_i(t_0)\le\bar\theta e^{-4a_it_0}+e^{-2a_it_0}+C\theta^{-1}t_0^{-3/2}\min\{t_0,a_i^{-1}\}$. Using \cref{lem:coldm-indexed} and $a^{3/2}\le C_*^{3/2}/N$, we have $\sum_{i\ge2}e^{-4a_it_0}\le CNt_0^{-3/2}\le C$ and $\sum_{i\ge2}\min\{t_0,a_i^{-1}\}^2\le CN\sqrt{t_0}$. We obtain $\sum_{i\ge2}d_i(t_0)^2\le C+C\theta^{-2}Nt_0^{-5/2}\le C+C\theta^{-2}a\le C$, since $\theta^2/a\ge m^2/C_*^3$. Let $P=\diag(\sqrt{\tilde\kappa_i})_{i\ge2}$. Then $P[-4\diag(a)+(2/N)\ind\tilde\kappa^\T]P^{-1}=\mathsf H_{\tilde\kappa}$, the symmetric matrix of \cref{lem:coldm-renewal}, all of whose eigenvalues are smaller than $-2a$. Because $\|P\|\,\|P^{-1}\|\le\sqrt2$, we have $\|d_b(t)\|_2\le\sqrt2e^{-2a(t-t_0)}\|d_b(t_0)\|_2\le Ce^{-2at}$ for $t\ge t_0$. Finally, by the Cauchy--Schwarz inequality, $|d_1|\le\lambda_+(\sum_{i\ge2}(4a_i)^{-2})^{1/2}\|d_b\|_2\le(2a/m)\,Ca^{-1}\|d_b\|_2$. For the fourth part, by \eqref{cpm-eq:projected-exact}, $\bar\theta(\delta_0+\widetilde U)$ is a probability measure on $[0,\infty)$, since its mass is $\bar\theta(1+(1-\bar\theta)/\bar\theta)=1$. Using Young's inequality, we have $\|d_i\|_{L^2}\le\|e^{-4a_i\cdot}\|_{L^2}=(8a_i)^{-1/2}$, and summing $a_i^{-1}\|d_i\|^2_{L^2}$ with \cref{lem:coldm-indexed} yields the third bound. For the first bound, we have $|d_1(t)|\le1$ for every $t\ge0$, and the second part of \cref{lem:coldm-renewal} also gives $|d_1(t)|\le C\theta^{-1}t^{-1/2}$ for $t\ge1$. Consequently $|d_1(t)|\le\min\{1,C\theta^{-1}(1+t)^{-1/2}\}$ for all $t\ge0$, after increasing $C$. Integrating its square gives $\int_0^{T_{\max}}d_1(t)^2\,dt\le C\theta^{-2}\log(e+\theta^2T_{\max})\le C\theta^{-2}\log(em)$, since $\theta^2T_{\max}\le Cm^2\log(em)$. For the fifth part, because translation commutes with convolution, we have $d_i(\cdot+h)-d_i=\bar\theta(\delta_0+\widetilde U)*(g_i(\cdot+h)-g_i)$ with $g_i(t)=e^{-4a_it}\ind_{\{t\ge0\}}$, and Young's inequality yields \[ \|d_i(\cdot+h)-d_i\|_2^2\le\|g_i(\cdot+h)-g_i\|_2^2=\int_0^he^{-8a_is}\,ds+\frac{(1-e^{-4a_ih})^2}{8a_i}\le2\min\Big\{h,\frac1{4a_i}\Big\}, \] where in the last step we used $(1-e^{-x})^2\le\min\{x^2,1\}$. Summing with the weights $a_i^{-1}$ and using \cref{lem:coldm-indexed}, we obtain \eqref{cpm-eq:weighted-translation}. The second inequality there uses $N\sqrt h=a^{-2}\sqrt{ah}\,Na^{3/2}$. For \eqref{cpm-eq:top-translation}, on $[-h,0)$ the difference is $d_1(\cdot+h)$, which is bounded by one. On $[0,\infty)$, the difference $d_1(t+h)-d_1(t)$ lies in $[0,1]$, and its square is at most itself, and $\int_0^\infty(d_1(t+h)-d_1(t))\,dt=\int_0^h|d_1|\le h$. For the sixth part, $d(s)=e^{(s/2)\mathsf H}d(s/2)$ and $d(s/2)\perp u$. On the orthogonal complement of $u$, the matrix $\mathsf H$ is symmetric with eigenvalues $-z$, $z>2a$, and $\|\mathsf He^{(s/2)\mathsf H}\varphi\|\le\sup_{z\ge0}ze^{-zs/2}\|\varphi\|=(2/(es))\|\varphi\|$ for $\varphi\perp u$. \end{proof} \subsection{A maximal inequality} The path estimates use a supremum over a time interval of length of order $a^{-1}\log m$. The following deterministic chaining inequality reduces them to bounds at fixed times and for increments. \begin{lemma}[Dyadic maximal inequality]\label{cpm-lem:dyadic-maximal} For every $p>2$ there exists a finite constant $C_p>0$ with the following property. Let $v_0\le v_1$ and $A,B\ge0$, and let $Z$ be a real process with measurable one-time evaluations and almost surely continuous paths on $[v_0,v_1]$. Assume separately that \[ \|Z(t)\|_{2p}\le A\qquad\text{for every }t\in[v_0,v_1], \] and that \[ \|Z(t+h)-Z(t)\|_{2p}\le Bh^{1/4}\qquad\text{whenever }t,t+h\in[v_0,v_1],\quad02$; choose $C_p$ to bound its sum. Finally, the supremum over $[v_0,v_1]$ is the maximum of the $n$ suprema, and bounding the $2p$-th power of the maximum by the sum gives the factor $n^{1/(2p)}$. \end{proof} \subsection{Concentration of the radial part} Fix $\mathfrak a\in(0,1)$. We define the earlier time and the scaled coordinates at that time, \begin{equation}\label{eq:coldm-tau} \tau=\frac{m^{\mathfrak a}}{\theta^2},\qquad H_i=\sqrt\theta\,Y_i(\tau)\quad(1\le i\le N), \end{equation} and, for $8\le\chi\le(K_0/2)\log(em)$, the time $T=\chi/a$. Since $a\tau\le C_*^3m^{\mathfrak a-2}$ by \cref{lem:coldm-indexed}, we have $\tau\le T/2$ for $m\ge m_0$. Let \[ \varepsilon_m=m^{-\mathfrak a/2}+\frac{\log(em)}m,\qquad F_p(m,\chi)=(\log(em))^{1/(2p)}\big(e^{-\chi/4}+m^{-1/2}\big)+\varepsilon_m^{1/2}+m^{-2} . \] \begin{theorem}[Concentration of the radial part along the path]\label{cpm-thm:radial-path} For every integer $p>2$ there exist $C_p$ and $m_0$, depending only on $p$, $\mathfrak a$, $c_*$, $C_*$, and $K_0$, such that for $m\ge m_0$ and every $\chi\in[8,(K_0/2)\log(em)]$, \begin{equation}\label{cpm-eq:radial-path} \begin{gathered} \Big\|\sup_{T\le t\le T_{\max}}\big|\theta^2|Y(t)|^2-H_1^2\big|\Big\|_{2p}\le C_pF_p(m,\chi),\\ \Big\|\sup_{\tau\le t\le T_{\max}}\big|\sqrt\theta\,Y_1(t)-H_1\big|\Big\|_{2p}\le C_p\,\varepsilon_m^{1/2}. \end{gathered} \end{equation} In particular, if $\chi=\mathfrak b\log(em)$ for a fixed $\mathfrak b\in(0,K_0/2]$, then both right sides are at most $C_pm^{-\vartheta}$ for $m\ge m_0$, where $\vartheta=\min\{\mathfrak a,\mathfrak b,1\}/8$ does not depend on $p$, and $C_p,m_0$ depend also on $\mathfrak b$. \end{theorem} The proof writes $\bar\theta|Y|^2-Y_1^2$ as a stochastic convolution against the projected kernel $d$ of \cref{lem:coldm-projected}. We first bound its terms at fixed times and their increments, for the Gaussian start, and apply \cref{cpm-lem:dyadic-maximal}. Then we transfer the bound to the uniform start, treat the first coordinate, which is a martingale, and replace $\bar\theta$ by $\theta$. \begin{proof} Throughout, $C_p$ denotes a constant depending only on $p$, $\mathfrak a$, $c_*$, $C_*$, and $K_0$; $t,t+h\in[T,T_{\max}]$ and $0\varepsilon_0)\le C_Jg(m)^2/\varepsilon_0^2$ by Markov's inequality. \end{proof} \subsection{Residence of the path} Fix $\mathfrak b\in(0,1/4)$, let $\chi=\mathfrak b\log(em)$ and $T=\chi/a$, and fix $J\ge2$ and $0<\ell_-<\ell_+$. We write $|x_{\le J}|^2=\sum_{k\le J}x_k^2$ and let \begin{equation}\label{eq:coldm-Cres} C_{\mathrm{res}}=\frac4{3\ell_-^2}\Big[\Big(\frac{5\ell_+}4\Big)^2+(J-1)(\ell_++1)^2\Big]. \end{equation} \begin{corollary}[Residence of the path]\label{cpm-cor:actual-residence} For every $B_0>0$ there exist $C$ and $m_0$, depending only on $B_0$, $\ell_\pm$, $J$, $\mathfrak a$, $\mathfrak b$, $c_*$, $C_*$, and $K_0$, such that for $m\ge m_0$, \[ \PP\Big(E\cap\Big\{\exists t\in[T,T_{\max}]:\ X_1(t)^2\notin\Big[\frac{\theta N}4,4\theta N\Big]\ \text{or}\ |X_{\le J}(t)|^2>C_{\mathrm{res}}\theta N\Big\}\Big)\le Cm^{-B_0} . \] The probability of $E$ intersected with the event that $X_1$ changes sign on $[\tau,T_{\max}]$ is also at most $Cm^{-B_0}$. If $\PP(E)\ge c_1>0$, the same bounds hold for the law conditioned on $E$, with the right sides divided by $c_1$. \end{corollary} This is a statement about the path started from $\nu_N$ on the event $E$, which is measurable at time $\tau$. It does not concern other initial laws. \begin{proof} Throughout, $C$ denotes a constant depending only on the parameters listed in the statement and on the moment orders chosen below. By \cref{cpm-thm:radial-path} with $\chi=\mathfrak b\log(em)$, the right sides of \eqref{cpm-eq:radial-path} are at most $C_pm^{-\vartheta}$, where $\vartheta>0$ does not depend on $p$. Let $\mathcal G_1$ be the event that \[ \sup_{T\le t\le T_{\max}}\big|\theta^2|Y(t)|^2-H_1^2\big|\le\frac{\ell_-^2}4\qquad\text{and}\qquad\sup_{\tau\le t\le T_{\max}}|\sqrt\theta\,Y_1(t)-H_1|\le\frac{\ell_-}4 . \] We have $\PP(\mathcal G_1^c)\le C_pm^{-2p\vartheta}$ by Markov's inequality. We fix the integer $p>2$ with $2p\vartheta\ge B_0$. On $E\cap\mathcal G_1$, we have $|H_1|\ge\ell_-$, so that $\sqrt\theta Y_1(t)$ has the sign of $H_1$ and $|\sqrt\theta Y_1(t)|\in[3|H_1|/4,5|H_1|/4]$ for $t\in[\tau,T_{\max}]$, and $\theta^2|Y(t)|^2\in[H_1^2-\ell_-^2/4,H_1^2+\ell_-^2/4]$ for $t\in[T,T_{\max}]$. For these $t$, we find \[ \frac{X_1(t)^2}{\theta N}=\frac{\theta Y_1(t)^2}{\theta^2|Y(t)|^2}\in\Big[\frac9{16}\cdot\frac{H_1^2}{H_1^2+\ell_-^2/4},\ \frac{25}{16}\cdot\frac{H_1^2}{H_1^2-\ell_-^2/4}\Big]\subset\Big[\frac14,4\Big] . \] It remains to bound the coordinates $2\le i\le J$. \paragraph{The coordinates at time $T$.} For $2\le i\le J$, the mild form of \eqref{mc-eq:lift} gives $\sqrt\theta Y_i(T)-e^{-2a_i(T-\tau)}H_i=\sqrt{2\theta/N}\int_\tau^Te^{-2a_i(T-s)}|Y(s)|\,dB_i(s)$. By the Burkholder--Davis--Gundy inequality and \cref{cpm-prop:all-moments}, its $L^{2p}$ norm squared is at most $C_p\theta\int_\tau^Te^{-4a_i(T-s)}\bar h(s)\,ds$. We split at $s=c_0/a$. For $s\le c_0/a$, the exponential is at most $Ce^{-4a_iT}$ since $a_i\le C_Ja$, and $2\theta\int_\tau^{c_0/a}\bar h=\theta(h_1(c_0/a)-h_1(\tau))\le C[m^{-\mathfrak a/2}+m^{-1}]$ by \cref{lem:coldm-means}. For $s\ge c_0/a$, we have $\bar h(s)\le C(N\theta^2)^{-1}$, and the integral is at most $C(N\theta a_i)^{-1}\le C/m$. We conclude that \[ \big\|\sqrt\theta Y_i(T)-e^{-2a_i(T-\tau)}H_i\big\|_{2p}^2\le C_p\big[m^{-\mathfrak a/2}e^{-4a_iT}+m^{-1}\big], \] and the event $\mathcal G_2$ that $|\sqrt\theta Y_i(T)-e^{-2a_i(T-\tau)}H_i|\le1/2$ for all $2\le i\le J$ satisfies $\PP(\mathcal G_2^c)\le Cm^{-B_0}$, after increasing $p$. On $E\cap\mathcal G_2$, we have $|\sqrt\theta Y_i(T)|\le\ell_++1/2$. \paragraph{The coordinates after time $T$.} Let $T_{\mathrm{amp}}=\inf\{s\ge T:\theta^2|Y(s)|^2>2\ell_+^2\}$, with $T_{\mathrm{amp}}=T$ if $\theta^2|Y(T)|^2>2\ell_+^2$. For $i\ge2$ and $t\ge T$, define the process with stopped amplitude \begin{equation}\label{eq:coldm-stopped} \widetilde Y_i(t)=e^{-2a_i(t-T)}Y_i(T)+M_i(t),\qquad M_i(t)=\sqrt{\frac2N}\int_T^te^{-2a_i(t-s)}|Y(s)|\ind_{\{sT_{\max}$ and $\widetilde Y_i=Y_i$ on $[T,T_{\max}]$. The integrand of $\sqrt\theta M_i$ is bounded by $2\ell_+(N\theta)^{-1/2}$. In the time variable $v=at$, the process $\sqrt\theta M_i(v/a)$ has the form $\int e^{-2(a_i/a)(v-r)}\phi(r)\,dB'(r)$ for a Brownian motion $B'$ and an adapted $\phi$ with $|\phi|\le2\ell_+m^{-1/2}$. Using the Burkholder--Davis--Gundy inequality, we bound its $L^{2p}$ norm at fixed times by $C_pm^{-1/2}$. For increments of length $h'\le1$, we write the increment as $(e^{-2(a_i/a)h'}-1)$ times the old value plus the integral over the new interval, which yields the bound $C_pm^{-1/2}(h'+h'^{1/2})\le C_pm^{-1/2}h'^{1/4}$. Applying \cref{cpm-lem:dyadic-maximal} on an interval of length at most $K_0\log(em)$, we obtain $\|\sup_{[T,T_{\max}]}\sqrt\theta|M_i|\|_{2p}\le C_p(\log(em))^{1/(2p)}m^{-1/2}$. Let $\mathcal G_3$ be the event that $\sup_{[T,T_{\max}]}\sqrt\theta|M_i|\le1/2$ for $2\le i\le J$; then $\PP(\mathcal G_3^c)\le Cm^{-B_0}$ after increasing $p$. \paragraph{Conclusion.} Let $\mathcal G=\mathcal G_1\cap\mathcal G_2\cap\mathcal G_3$, so that $\PP(\mathcal G^c)\le Cm^{-B_0}$. On $E\cap\mathcal G$, for $t\in[T,T_{\max}]$ and $2\le i\le J$, we have $|\sqrt\theta Y_i(t)|\le|\sqrt\theta Y_i(T)|+\sqrt\theta|M_i(t)|\le\ell_++1$, and therefore \[ \frac{|X_{\le J}(t)|^2}{\theta N}=\sum_{i\le J}\frac{\theta Y_i(t)^2}{\theta^2|Y(t)|^2}\le\frac{(5\ell_+/4)^2+(J-1)(\ell_++1)^2}{3\ell_-^2/4}=C_{\mathrm{res}} . \] This proves the first claim. We have already proved that $X_1$, which has the sign of $Y_1$, does not change sign on $[\tau,T_{\max}]$ on $E\cap\mathcal G_1$. \end{proof} \subsection{The endpoint measure weighted by \texorpdfstring{$\ind_E$}{1\_E} and killed on leaving \texorpdfstring{$D$}{D}} We now define the endpoint subprobability measure to which \cref{cce-thm:consumer} is applied. Fix $L_0,L_1,L$ with $C_{\mathrm{res}}t,\ X(t)\in A\big). \end{equation} This is the endpoint subprobability measure \eqref{eq:score-sigma} of \cref{sec:score-setting}, weighted by $\Psi=\ind_E$ and killed on leaving $D$. It is dominated by the time-$t$ law: $\sigma_t\le\rho_t$ as measures. The weight and the killing change the mass of the measure, not the sample paths. We also use the weighted ratio of the remaining coordinates to the first ones, \[ \xi(x)=\frac{a_{J+1}\sum_{i>J}x_i^2/a_i}{|x_{\le J}|^2}, \] with $\xi(x)=0$ when $|x_{\le J}|=0$, as in \eqref{mc-eq:te2}. We write $\alpha_J=a_{J+1}/a$. \begin{corollary}[Moments, survival, and the mass where the cutoff may differ from one]\label{cpm-cor:killed-collars} For every $p\ge1$ and $B_0>0$ there exist $C$ and $m_0$, depending only on $p$, $B_0$, $\ell_\pm$, $J$, $L$, $\mathfrak a$, $\mathfrak b$, $c_*$, $C_*$, and $K_0$, such that the following hold for $m\ge m_0$ and $t\in[T,T_{\max}]$. \begin{enumerate} \item The killing loss satisfies \begin{equation}\label{cpm-eq:killing-loss} \PP(E,\ T_{\mathrm{exit}}\le T_{\max})\le Cm^{-B_0}. \end{equation} \item For $2\le j\le J$, \begin{equation}\label{cpm-eq:killed-head-moments} \Big(\int|x_j|^{2p}\,d\sigma_t\Big)^{1/p}\le C\big[\theta Ne^{-4a_j(t-T)}+a_j^{-1}\big]+C\theta Nm^{-B_0/p} . \end{equation} If $B_0\ge p$, the last term is at most $Ca_j^{-1}$. \item For $0<\delta\le1$, \begin{equation}\label{cpm-eq:endpoint-cone} \sigma_t\{\xi>\delta/2\}\le Cm^{-B_0}+C\delta^{-p}\big[e^{-2\alpha_J\chi}+m^{-1}\big]^p . \end{equation} \item Let $0L_0\theta N\}\cup\{\xi>\delta/2\}. \] Then $\sigma_t(\mathcal C)\le Cm^{-B_0}+\sigma_t\{\xi>\delta/2\}$, and for every measurable $\varphi:S_N\to[0,1]$ that equals one outside $\mathcal C$, \[ \int\varphi\,d\sigma_t\ge\PP(E)-Cm^{-B_0}-\sigma_t\{\xi>\delta/2\} . \] \end{enumerate} \end{corollary} In the third part, the bound concerns each fixed time $t$. It does not state that the path stays in $\{\xi\le\delta/2\}$. If $\alpha_J\ge\alpha_*>0$ and $\delta=m^{-\zeta}$ with $0<\zeta<\min\{1,2\alpha_*\mathfrak b\}$, the second term in \eqref{cpm-eq:endpoint-cone} is at most $Cm^{-p\min\{2\alpha_*\mathfrak b-\zeta,1-\zeta\}}$, which is smaller than any prescribed power of $m^{-1}$ once $p$ is large. For example, if $\alpha_J\ge6$ and $\mathfrak b=\zeta=1/100$, then the exponent is $p\min\{11/100,99/100\}$, and with $p=200$ we obtain $m^{-22}$. \begin{proof} Let $\mathcal G$ be the event of the proof of \cref{cpm-cor:actual-residence}, for the exponent $B_0$. Recall that $\PP(\mathcal G^c)\le Cm^{-B_0}$. On $E\cap\mathcal G$, the path satisfies $|X_{\le J}|^2\le C_{\mathrm{res}}\theta NT_{\max}$, and this proves the first part. We keep the definition of $\sigma_t$, and use the event $\mathcal G$ only to split integrals. For the second part, let $2\le j\le J$. On $E\cap\mathcal G$, we have $\theta^2|Y(t)|^2\ge3\ell_-^2/4$ and $Y_j=\widetilde Y_j$. Then $X_j(t)^2=NY_j(t)^2/|Y(t)|^2\le(4N\theta^2/(3\ell_-^2))\widetilde Y_j(t)^2$. On $E\setminus\mathcal G$, the measure $\sigma_t$ only sees the event $\{T_{\mathrm{exit}}>t\}$, on which $X_j(t)^2\le|X_{\le J}(t)|^2J}Y_i(t)^2/a_i}{\sum_{i\le J}Y_i(t)^2}. \] On $E\cap\mathcal G$, the denominator is at least $Y_1(t)^2\ge9\ell_-^2/(16\theta)$, and $Y_i=\widetilde Y_i$ for all $i\ge2$. By Minkowski's inequality and \eqref{cpm-eq:stopped-ou-moments}, \[ \Big\|a_{J+1}\theta\sum_{i>J}\frac{\widetilde Y_i(t)^2}{a_i}\Big\|_p\le C_pa_{J+1}\theta\sum_{i>J}\frac1{a_i}\Big[e^{-4a_i(t-T)}h_i(T)+\frac1{N\theta^2a_i}\Big]. \] Using \cref{lem:coldm-means} and $T\ge c_0/a$, we have $h_i(T)\le C[\theta^{-1}e^{-2a_iT}+(N\theta^2a_i)^{-1}]$. The terms with $(N\theta^2a_i)^{-1}$ contribute at most $C(a_{J+1}/(N\theta))\sum_{i\ge2}a_i^{-2}\le C\alpha_J/m$, by \cref{lem:coldm-indexed}. For the terms with $e^{-2a_iT}$, the indices with $a_{J+1}\le a_i\le2a_{J+1}$ number at most $(2C_*/c_*)^{3/2}J$, by \eqref{mc-eq:indexed-geometry}, and each contributes at most $e^{-2a_{J+1}T}$. For the indices with $a_i>2a_{J+1}$, we have $e^{-2a_iT}\le e^{-2a_{J+1}T}e^{-a_iT}$, and $a_{J+1}\sum_{i\ge2}e^{-a_iT}/a_i\le CNa_{J+1}T^{-1/2}\le C_J$, by the fourth sum in \cref{lem:coldm-indexed} and $T\ge1/a$. Using $a_{J+1}\le C_Ja$, we have \[ \Big\|a_{J+1}\theta\sum_{i>J}\frac{\widetilde Y_i(t)^2}{a_i}\Big\|_p\le C\big[e^{-2\alpha_J\chi}+m^{-1}\big], \] where we used $a_{J+1}T=\alpha_J\chi$. Markov's inequality on $E\cap\mathcal G$ and the bound on $\PP(\mathcal G^c)$ give \eqref{cpm-eq:endpoint-cone}. For the fourth part, on $E\cap\mathcal G$ we have $x_1^2\ge\theta N/4>s\theta N$ and $|x_{\le J}|^2\le C_{\mathrm{res}}\theta N\delta/2\}$. Finally, $\int\varphi\,d\sigma_t\ge\sigma_t(S_N)-\sigma_t(\mathcal C)$ and $\sigma_t(S_N)\ge\PP(E)-\PP(E,T_{\mathrm{exit}}\le T_{\max})$. \end{proof} \section{The limit of finitely many coordinates}\label{sec:coldm-weak}\label{mc-subsec:seed-means}\label{mc-subsec:weak-seeds} In this section, we identify the limit in distribution of finitely many standardized coordinates at the times $t_m(u)$, and we deduce the lower bound on the distance. For the rest of this section, we fix the following choices of the parameters of \cref{sec:coldm-path,sec:coldm-residence}: \begin{equation}\label{mc-eq:weak-profile-prefix} \begin{gathered} K_0=1,\qquad\mathfrak a=\frac1{10},\qquad\mathfrak b=\frac1{100},\\ \tau=\frac{m^{1/10}}{\theta^2},\qquad\chi=\frac{\log(em)}{100},\qquad T=\frac\chi a,\qquad H_i=\sqrt\theta\,Y_i(\tau) . \end{gathered} \end{equation} For a fixed $u\in\R$, the time $t_m(u)$ of \eqref{mc-eq:cp2} lies in $[T+1/a,T_{\max}]$ for $m\ge m_0(u)$, because $at_m(u)=\log(2m)/4+u$. For $J\ge2$, we write $\mathbf z(x)=(\sqrt{2a_2}\,x_2,\dots,\sqrt{2a_J}\,x_J)\in\R^{J-1}$ for the standardized coordinates, as in \eqref{mc-eq:hv2}, and we let \begin{equation}\label{mc-eq:cp10} \mu_u=\Law\big(G_2+e^{-2u}C,\,G_3,\dots,G_J\big), \end{equation} where $G_2,\dots,G_J$ are independent standard Gaussian variables and $C$ is an independent standard Cauchy variable. \begin{theorem}[The limit of finitely many coordinates and the lower bound]\label{mesocold-thm:weak-profile} Let $a^{(n)}$ be as in \cref{ccf-thm:full-profile}. For every fixed $u\in\R$ and $J\ge2$, the law of $\mathbf z(X(t_{m_n}(u)))$, where $X$ is the diffusion for $a^{(n)}$ started from $\nu_{N_n}$, converges weakly to $\mu_u$. Moreover, for every compact interval $I\subset\R$, \[ \sup_{u\in I}\big(\cV(u)-D_n(t_{m_n}(u))\big)_+\longrightarrow0 . \] \end{theorem} We describe the proof. For $2\le i\le J$, the mild form of \eqref{mc-eq:lift} on $[\tau,t]$ splits $Y_i(t)$ into the inherited part $e^{-2a_i(t-\tau)}Y_i(\tau)$, the noise on $[\tau,T]$, and the noise on $[T,t]$. After normalization by $|Y(t)|$, the inherited part of the second coordinate is $e^{-2u}H_2/|H_1|$, up to a vanishing error, and the inherited parts of the other coordinates vanish because $a_i\ge(1+\varepsilon_*)a$. The noise on $[\tau,T]$ is negligible. On $[T,t]$, we replace the amplitude $\theta|Y(s)|$ of the noise by the variable $|H_1|$, which is measurable at time $\tau$; after division by $\theta|Y(t)|\approx|H_1|$, this noise becomes a centered Gaussian variable independent of $(H_1,\dots,H_J)$. No Gaussian property of a stochastic integral with random amplitude is used. We do not claim that $H_2/|H_1|$ has a Cauchy law for finite $N$. It converges in distribution to the ratio of two independent standard Gaussian variables, which is standard Cauchy \cite[Section~1, p.~193]{Marsaglia1965}. \begin{proof} We fix $u$ and $J$, drop the index $n$, and write $t=t_m(u)$. Throughout, $C$ denotes a constant depending only on $u$, $J$, $c_*$, $C_*$, and $\varepsilon_*$, and $\alpha_i=a_i/a$, so that $1\le\alpha_i\le C$ for $i\le J$ and $\alpha_i\ge1+\varepsilon_*$ for $3\le i\le J$. \paragraph{Step 1: decomposition.} For $2\le i\le J$, we have $\sqrt{2a_i}X_i(t)=\sqrt{2a_iN}\,\theta Y_i(t)/(\theta|Y(t)|)$, and the mild form of \eqref{mc-eq:lift} gives $\sqrt{2a_iN}\,\theta Y_i(t)=I_i+Q_i+R_i$, where \begin{gather*} I_i=\sqrt{2a_iN\theta}\,e^{-2a_i(t-\tau)}H_i,\\ Q_i=\sqrt{4a_i}\int_\tau^Te^{-2a_i(t-s)}\theta|Y(s)|\,dB_i(s),\qquad R_i=\sqrt{4a_i}\int_T^te^{-2a_i(t-s)}\theta|Y(s)|\,dB_i(s). \end{gather*} \paragraph{Step 2: the inherited part.} Since $2a_iN\theta=2\alpha_im$ and $e^{-2a_it}=(2m)^{-\alpha_i/2}e^{-2\alpha_iu}$, we have \[ I_i=\sqrt{\alpha_i}\,(2m)^{(1-\alpha_i)/2}e^{-2\alpha_iu}e^{2a_i\tau}H_i . \] Here $a_i\tau\le C_*^3\alpha_im^{\mathfrak a-2}\to0$. For $i=2$, we have $\alpha_2=1$ and $I_2=e^{-2u}e^{2a\tau}H_2$. For $3\le i\le J$, the coefficient is at most $C(2m)^{-\varepsilon_*/2}\to0$. By \cref{cpm-lem:finite-seed}, $(H_1,\dots,H_J)$ converges in distribution to a standard Gaussian vector. This sequence is tight, and $I_2-e^{-2u}H_2\to0$ and $I_i\to0$ ($i\ge3$) in probability. \paragraph{Step 3: the noise on $[\tau,T]$.} Using the Itô isometry and $\E|Y(s)|^2=N\bar h(s)$, we have \[ \E Q_i^2=4a_i\theta^2e^{-4a_it}\int_\tau^Te^{4a_is}N\bar h(s)\,ds\le Cm^{1-\alpha_i}\,\theta\int_\tau^Te^{4a_is}\bar h(s)\,ds, \] where we used $4a_i\theta^2Ne^{-4a_it}=4\alpha_i\theta m(2m)^{-\alpha_i}e^{-4\alpha_iu}$. We split the integral at $s=c_0/a$. For $s\le c_0/a$, we have $e^{4a_is}\le C$ and, by \cref{lem:coldm-means}, $\bar h\le U/2+C(N\theta^2)^{-1}$ and $\theta\int_\tau^\infty U\le C\theta^{-1}\tau^{-1/2}=Cm^{-\mathfrak a/2}$; also $\theta\cdot(N\theta^2)^{-1}c_0/a\le C/m$. For $c_0/a\le s\le T$, we have $\bar h(s)\le C(N\theta^2)^{-1}$ and $\theta\int^Te^{4a_is}(N\theta^2)^{-1}ds\le Ce^{4a_iT}/(N\theta a_i)\le Cm^{-1}e^{4\alpha_i\chi}$. Since $\alpha_i\ge1$ and $e^{4\alpha_i\chi}=(em)^{\alpha_i/25}$, we obtain \[ \E Q_i^2\le C\big[m^{-\mathfrak a/2}+m^{-1}+m^{-\alpha_i}(em)^{\alpha_i/25}\big]\longrightarrow0 . \] \paragraph{Step 4: the noise on $[T,t]$.} The variable $|H_1|$ is measurable at time $\tau\ell/2)\to0$ by \cref{mesocold-prop:late-radial}. We conclude that, for every $\varepsilon>0$, \[ \limsup\PP\big(|\mathbf z(X(t))-\Phi_u(H,\mathsf N)|>\varepsilon\big)\le\limsup\PP(|H_1|<\ell)=\PP(|G_1|<\ell), \] where $G_1$ is standard Gaussian, and letting $\ell\to0$ shows that $\mathbf z(X(t))-\Phi_u(H,\mathsf N)\to0$ in probability. The vectors $H$ and $\mathsf N$ are independent, $H$ converges in distribution to a standard Gaussian vector $(G_1',\dots,G_J')$ by \cref{cpm-lem:finite-seed}, and $\mathsf N$ converges in distribution to a standard Gaussian vector $(G_2,\dots,G_J)$. The map $\Phi_u$ is continuous on $\{\mathsf h_1\ne0\}$, which has full measure for the limit, and the continuous mapping theorem implies that $\Phi_u(H,\mathsf N)\Rightarrow\Phi_u(G',G)$. The ratio $G_2'/|G_1'|$ has the law of $G_2'/G_1'$, the standard Cauchy law, because $G_2'$ is symmetric and independent of $|G_1'|$; it is also independent of $(G_2,\dots,G_J)$. We conclude that $\Phi_u(G',G)$ has law $\mu_u$, and Slutsky's lemma proves the first claim. \paragraph{Step 6: the lower bound.} Let $\varphi:\R\to[0,1]$ be continuous. By \cref{mesocold-prop:stationary-graph}, the law of $\sqrt{2a}\,x_2$ under $\pi$ converges in total variation to the standard Gaussian law. Its hypotheses hold with constants depending only on $c_*$ and $C_*$, by \cref{lem:coldm-indexed}: $a_N\le C_*$, $a_3\le2^{2/3}(C_*/c_*)a$, and $a^2\sum_{i=2}^Na_i^{-2}\le C$. By the first claim with $J=2$, and the second part of \cref{lem:model-tv}, \[ \liminf D(t)\ge\lim\Big|\E\varphi\big(\sqrt{2a}\,X_2(t)\big)-\E_\pi\varphi\big(\sqrt{2a}\,x_2\big)\Big|=\big|\E\varphi(G+e^{-2u}C)-\E\varphi(G)\big| . \] The two limiting laws have continuous densities, and the supremum of the right side over continuous $\varphi$ with values in $[0,1]$ is their total variation distance $\cV(u)$, by the first part of \cref{lem:model-tv} and the density of continuous functions in $L^1(\R)$. This proves $\liminf D(t_m(u))\ge\cV(u)$ for each $u$. For the uniformity, fix $\delta>0$ and a mesh $u_0<\dots= theta to the power -2}}\label{sec:coldm-renyi}\label{mc-renyi-section} In this section, we bound all moments of the density of the time-$t$ law with respect to the Gibbs measure, for $t\ge\theta^{-2}$. The bound $e^{O(N\theta^3)}=e^{O(m^3)}$ is far from one. It is used in \cref{sec:coldm-entropy} only as the initial value of a conditional relative entropy that then decays. The proof is an application of \cref{thm:dens-main}, after the quadratic potential is shifted so that it has the form considered in \cref{sec:density}. \begin{theorem}[Density moment bounds for $t\ge\theta^{-2}$]\label{mc-renyi-thm} Suppose that $a$ satisfies the indexed bounds \eqref{mc-eq:indexed-geometry} and that $0<\theta\le1/2$. For every $p>1$ there exist $C_p$ and $m_0$, depending only on $p$, $c_*$, and $C_*$, such that if $m\ge m_0$, then for every $t\ge\theta^{-2}$, \begin{equation}\label{mc-eq:mr2} \log\int_{S_N}\Big(\frac{d\rho_t}{d\pi}\Big)^p\,d\pi\le C_pN\theta^3,\qquad \KL(\rho_t\Vert\pi)\le C_2N\theta^3 . \end{equation} \end{theorem} \begin{proof} Let $\lambda_+$ be the eigenvalue of \cref{lem:coldm-projected}, and let $r_i=a_i+\lambda_+/4$. The secular equation $(2/N)\sum_i(4a_i+\lambda_+)^{-1}=1$ says that $\sum_i(2r_i)^{-1}=N$, that is, $r=(r_1,\dots,r_N)$ satisfies \eqref{eq:dens-trace}. On $S_N$, we have $\sum_ir_ix_i^2=F(x)+\lambda_+N/4$, so that the generator, the semigroup, and the Gibbs measure of \eqref{eq:dens-dynamics} are $L$, $P_t$, and $\pi$. Let $S=\theta^{-2}$. We verify the hypotheses \textup{(H1)}--\textup{(H5)} of \cref{def:dens-hyp} on $[0,S]$, with a constant $C_{\mathrm H}$ depending only on $c_*$ and $C_*$. Then \cref{thm:dens-main} yields $\log\int(d\rho_S/d\pi)^pd\pi\le CN(1+S)^{-3/2}\le CN\theta^3$, the monotonicity of the $p$-th moment in $t$, and the bound on the relative entropy with $p=2$. Throughout, $C$ denotes a constant depending only on $c_*$ and $C_*$. \paragraph{\textup{(H1)} and \textup{(H4)}.} We have $\lambda_+\le2a/m$ by \cref{lem:coldm-projected}. Then $\max_ir_i\le C_*+1$ and $\lambda_+S\le2/(N\theta^3)\le2C_*^3/m^3$. Also $N(1+S)^{-3/2}\ge2^{-3/2}N\theta^3\ge2^{-3/2}m^3/C_*^3\ge1$ for large $m$. For \textup{(H4)}, the indices $i=1,2,3$ satisfy $r_iS\le(a_3+\lambda_+/4)\theta^{-2}\le C(a/\theta^2)+C/m^3\le C$, by \cref{lem:coldm-indexed}. \paragraph{\textup{(H3)}.} For $i\ge2$, we have $r_i\ge a_i\ge c_*((i-1)/N)^{2/3}$, and \cref{lem:dens-count} applies with $k_0=1$ and gives $\sum_i(1+r_it)^{-2}\le1+CN(1+t)^{-3/2}$; the term $1$ is at most $N(1+t)^{-3/2}$ for $t\le S$, by \textup{(H1)}. \paragraph{\textup{(H2)}.} The heat vector of \cref{sec:density} for the half-precisions $r$ is $e^{t(\mathsf H-\lambda_+I)}\ind=e^{-\lambda_+t}h(t)$, since $-4\diag(r)+(2/N)\ind\ind^\T=\mathsf H-\lambda_+I$. For $t\le S$, we have $e^{-\lambda_+t}\in[e^{-1},1]$, and it suffices to prove the bounds of \textup{(H2)} for $\bar h$ and $h_i$. From the fifth part of \cref{lem:coldm-means}, we have $C^{-1}(1+t)^{-1/2}\le\bar h(t)\le C[(1+t)^{-1/2}+(N\theta^2)^{-1}]$, and $(N\theta^2)^{-1}\le(1+t)^{-1/2}$ for $t\le S$ because $N\theta^3\ge2$. This proves the bounds on $\bar h$, and the following bounds hold for $t\le S$: \begin{equation}\label{mc-eq:mr1} C^{-1}(1+t)^{-1/2}\le U(t)\le C(1+t)^{-1/2},\qquad C^{-1}(1+t)^{-1/2}\le\bar h(t)\le C(1+t)^{-1/2}. \end{equation} Since $\lambda_+t\le1$, we have $1+a_it\le1+r_it\le(5/4)(1+a_it)$, and we may replace $r_i$ by $a_i$ in the bounds on $h_i$. For the upper bound with $i\ge2$, the first case of \cref{cpm-lem:weighted-heat} (with $C_0=1$, $C_1=0$) gives $h_i(t)\le C[\sqrt{1+t}/(1+a_it)+\min\{t,a_i^{-1}\}/(N\theta^2)]$, and $\min\{t,a_i^{-1}\}\le2(1+t)/(1+a_it)$ and $\sqrt{1+t}\le2/\theta$ bound the second term by the first. For $i=1$, we have $h_1(t)=1+2\int_0^t\bar h\le C\sqrt{1+t}$, by \eqref{mc-eq:mr1}. For the lower bound, let $\ell_i=\min\{t,(4a_i)^{-1}\}$, with $\ell_1=t$; by \eqref{eq:coldm-duhamel} and \eqref{mc-eq:mr1}, \[ h_i(t)\ge e^{-4a_it}+2\int_{t-\ell_i}^te^{-4a_i(t-s)}\bar h(s)\,ds\ge e^{-4a_it}+\frac{2\ell_i}{Ce}(1+t)^{-1/2}. \] If $a_it\le1/4$, the right side is at least $c_1\sqrt{1+t}$; otherwise it is at least $c_1/(a_i\sqrt{1+t})\ge c_1'\sqrt{1+t}/(1+a_it)$, since $a_i(1+t)\le(4C_*+1)a_it$ when $a_it\ge1/4$. Together, we obtain \begin{equation}\label{mc-eq:mr3} C^{-1}\frac{\sqrt{1+t}}{1+r_it}\le e^{-\lambda_+t}h_i(t)\le C\frac{\sqrt{1+t}}{1+r_it}\qquad(0\le t\le S,\ 1\le i\le N). \end{equation} \paragraph{\textup{(H5)}.} Let $v_i(t)$ be the normalized variances of \eqref{eq:dens-heat} for the half-precisions $r$, and $\mu_t=\mu_{v(t)}$ the Bingham comparison law, whose half-precisions are $k_i(t)=(2v_i(t))^{-1}$. By \textup{(H2)} and the first part of \cref{lem:dens-coefficients}, there exists $C_3\ge1$, depending only on $c_*$ and $C_*$, with $C_3^{-1}(1+t)/(1+r_it)\le v_i(t)\le C_3(1+t)/(1+r_it)$. In particular, $\max_iv_i(t)\le C_3(1+t)$ and $k_{\min}(t)\ge(2C_3(1+t))^{-1}$, and every index with $r_iS\le1$ satisfies $v_i(t)\ge(1+t)/(2C_3)$, that is, $k_i(t)\le L_{\mathrm H}k_{\min}(t)$ with $L_{\mathrm H}=2C_3^2$. Let $k_*=\lceil64L_{\mathrm H}^2\rceil+2$. Using \cref{lem:coldm-indexed}, we have $r_{k_*}S\le C(k_*)^{2/3}a\theta^{-2}+C/m^3\le1$ and $Nk_{\min}(t)\ge N\theta^2/(4C_3)\ge4$ for large $m$. \Cref{cg-many-soft-pi}, in its form for half-precisions applied to the indices $1,\dots,k_*$, gives $C_{\mathrm P}(\mu_t)\le1/k_{\min}(t)\le2C_3(1+t)$. The indices $1,2,3$ satisfy $v_i(t)\ge(2C_3^2)^{-1}\max_jv_j(t)$, and \cref{cg-three-soft-lsi} with $\delta=(2C_3^2)^{-1}$ gives \begin{equation}\label{mc-eq:mr5} C_{\mathrm{LS}}(\mu_t)\le C_\delta\max_iv_i(t)+C_\delta C_{\mathrm P}(\mu_t)\le C(1+t)\qquad(0\le t\le S). \end{equation} This is \textup{(H5)}, and the logarithmic Sobolev inequality refers to the spherical Dirichlet form of $\mu_t$. The constants $L_{\mathrm H}$ and $k_*$ depend only on $c_*$ and $C_*$, and they are fixed before the threshold for $m$ is chosen. This completes the proof. \end{proof} The proof of \cref{thm:dens-main} compares the time-$t$ law with the comparison law $\mu_{\gamma t}$ at the earlier time $\gamma t$. The coefficients of the comparison law are controlled by \cref{lem:dens-coefficients}, and its exponential moments by \cref{lem:dens-tilt}. In the present setting, the counting bound \textup{(H3)} contains the extra index $i=1$, with $r_1=\lambda_+/4$ of order $(N\theta)^{-1}$. Its contribution is absorbed because $N(1+S)^{-3/2}$ is of order $N\theta^3\ge1$. The next corollary bounds the relative entropy of a law obtained from the time-$t$ law by an endpoint cutoff, relative to the Gaussian product law in the chart. It is not used in the proof of the main results, where the corresponding bound is obtained inside \cref{cce-thm:consumer}. Let $G$ be the centered Gaussian measure on $\R^{N-1}$ with independent coordinates of variances $(2a_i)^{-1}$, $2\le i\le N$, as in \eqref{eq:score-gaussian}, and for a measure $\rho$ on $S_N$ let $\rho^+$ be its image under $x\mapsto(x_2,\dots,x_N)$. \begin{corollary}[Relative entropy after an endpoint cutoff]\label{mc-renyi-cut-entropy} In the setting of \cref{mc-renyi-thm}, let $t\ge\theta^{-2}$, $s>0$, $z_0>0$, and let $\varphi:S_N\to[0,1]$ be measurable and equal to zero on $\{x_1^2T$, $f_t$ is the density of $\sigma_t^+$ with respect to $G=G_{\mathrm z}\otimes G_{\mathrm v}$, $Z_t=\int\eta f_t\,dG$, $w_tG$ with $w_t=\eta f_t/Z_t$ is the image of the normalized cutoff-weighted law $\eta\sigma_t/Z_t$, with normalizing mass $Z_t$, $\bar w_t$ is the density of its first marginal, and $\mathscr H(t)$ and $\mathscr D(t)$ are the conditional relative entropy of the remaining coordinates and its dissipation, as in \eqref{eq:score-H-D} and \eqref{mc-eq:te5}. \paragraph{The filtration.} The weights below depend on the unnormalized process. Let $\mathcal F_s$ be the $\sigma$-algebra generated by $g$ and $(B_r)_{r\le s}$. The process $B$ is an $(\mathcal F_s)$-Brownian motion, because $g$ is independent of $B$, and $Y$, $Y^G$, and $X=\sqrt NY/|Y|$ are $(\mathcal F_s)$-adapted. Itô's formula for $\sqrt NY_i/|Y|$ shows that $X$ solves the equation \eqref{mc-eq:sr1} driven by the same Brownian motion $B$, with noise $\sqrt2\,\Pi(X)\,dB$; this is the diffusion of the second part of \cref{cd-lift}. In particular, the setting of \cref{sec:score-setting} holds with this filtration, and the event $E$ of \eqref{eq:coldm-event}, which is determined by $Y(\tau)$, belongs to $\mathcal F_\tau$. \begin{lemma}[Mass, collar, and fourth-moment bounds for the entropy estimate]\label{lem:coldm-entropy-hyp} Let $a$ satisfy the indexed bounds \eqref{mc-eq:indexed-geometry} with $0<\theta\le1/2$, fix $2\le J0,\\0,&v\le0,\end{cases} \qquad \psi(v)=\frac{b_0(v)}{b_0(v)+b_0(1-v)}, \] and use the cutoff \eqref{mc-eq:te3} with \[ \chi_1(v)=\psi(16v-1),\qquad \chi_2(v)=1-\psi(v-C_{\mathrm{res}}-1),\qquad \chi_3(v)=1-\psi(2v-1). \] Let $D$ and $\sigma_t$ be as in \eqref{eq:coldm-sigma}, let $\mathcal C$ be as in \eqref{mc-eq:te4}, and write $Z_t=\int\eta\,d\sigma_t$. There exist $C$ and $m_0$, depending only on $J$, $\ell_\pm$, $c_*$, and $C_*$, such that for $m\ge m_0$ and every $t\in[T+1/a,T_{\max}]$, \[ Z_t\ge\PP(E)-Cm^{-20},\qquad \sigma_t(\mathcal C)\le Cm^{-20}, \] and, for $2\le j\le J$, \[ \Big(\int x_j^4\,d\sigma_t\Big)^{1/2}\le C\big\{\theta Ne^{-4a_j(t-T)}+a_j^{-1}\big\}. \] \end{lemma} \begin{proof} Apply \cref{cpm-cor:killed-collars} with $B_0=20$ and $p=200$. Since $\alpha_J=b/a\ge6$ and $\mathfrak b=\zeta=1/100$, the example after that corollary bounds the cone term by $Cm^{-22}$. Thus $\sigma_t\{\xi>\delta/2\}\le Cm^{-20}$. The cutoff $\eta$ takes values in $[0,1]$ and equals one outside $\mathcal C$, by the displayed definitions of $\chi_1,\chi_2,\chi_3$. The fourth part of \cref{cpm-cor:killed-collars} therefore gives both the mass and collar bounds. The fourth-moment bound is the second part of that corollary with $p=2$ and $B_0=20$; its last term is absorbed into $Ca_j^{-1}$ because $B_0\ge p$. \end{proof} We now integrate the conclusion of \cref{cce-thm:consumer} over the time interval from $t_0=T+1/a$ to the times $t_m(u)$. The integration uses only \eqref{mc-eq:te8}, \eqref{mc-eq:te9}, and the absolute continuity of $\mathscr H$, and it is deterministic. \begin{corollary}[Scalar entropy bound at the times $t_m(u)$]\label{cce-cor:quantitative} Fix $\mathfrak b\in(0,1/8)$, $\zeta>0$, $K>0$, $C_0>0$, $\alpha_0>\max\{2,3/(1-4\mathfrak b)\}$, and a compact interval $I=[u_-,u_+]$. There exist $C$ and $m_0$, depending only on these parameters, with the following property. Let $a>0$ and $m\ge m_0$, let $T=\mathfrak b\log(em)/a$, $t_0=T+1/a$, $t_m(u)=\log(2m)/(4a)+u/a$, and $t_1=t_m(u_+)$. Suppose that $\mathscr H,D:[t_0,t_1]\to\R$, that $\mathscr H$ is absolutely continuous, that $\mathscr H(t_0)\le C_0m^3$, and that, for every $t\in[t_0,t_1]$, \[ 0\le\mathscr H(t),\qquad 0\le D(t),\qquad 4a\alpha_0\mathscr H(t)\le D(t), \] and, for almost every $t\in[t_0,t_1]$, \[ \mathscr H'(t)\le-D(t)+\frac{C_0a}m+C_0am e^{-8a(t-T)}+C_0am^{5+2\zeta-K/2}. \] Then, for every $u\in I$, \begin{equation}\label{mc-eq:te29} \mathscr H(t_m(u))\le E_m,\qquad E_m=C\big[m^{3-\alpha_0(1-4\mathfrak b)}+m^{-1+8\mathfrak b}+m^{5+2\zeta-K/2}\big]. \end{equation} \end{corollary} For example, with $\mathfrak b=\zeta=1/100$, $K=20$, and $\alpha_0=4$, the three exponents in \eqref{mc-eq:te29} are $3-4\cdot24/25=-21/25$, $-1+8/100=-23/25$, and $5+2/100-10=-249/50$, so that $E_m\le Cm^{-4/5}$. To apply the corollary to \cref{cce-thm:consumer}, take $D=(1-2\varepsilon)\mathscr D$ and $\delta=m^{-\zeta}$. If $p_*\le C_5m^{-K}$, its conclusions give the scalar hypotheses, with a fixed $C_0$, provided that \begin{equation}\label{mc-eq:te28} \alpha_m=\frac{(1-2\varepsilon)\,b}{a\,(1+2L_1\delta/s)}\ge\alpha_0 . \end{equation} The values in the example are used in \cref{sec:coldm-proofs}; there $b\ge6a$ and $\varepsilon=1/16$, so that $\alpha_m\ge(7/8)\cdot6/(1+2L_1m^{-1/100}/s)\ge4$ for large $m$. \begin{proof} Throughout, $C$ denotes a constant depending only on the parameters listed in the statement. For $m$ large, $t_m(u_-)>t_0$, since $a(t_m(u_-)-t_0)=\log(2m)/4+u_--\mathfrak b\log(em)-1\to\infty$. Using $D\ge4a\alpha_0\mathscr H$ in the assumed differential inequality, we obtain, for almost every $t\in[t_0,t_1]$, \[ \mathscr H'(t)\le-4a\alpha_0\mathscr H(t)+\frac{Ca}m+Came^{-8a(t-T)}+Cam^{5+2\zeta-K/2} . \] Since $\mathscr H$ is absolutely continuous, we may multiply by $e^{4a\alpha_0t}$ and integrate from $t_0$ to $t=t_m(u)$: \[ \mathscr H(t)\le\mathscr H(t_0)e^{-4a\alpha_0(t-t_0)}+\int_{t_0}^te^{-4a\alpha_0(t-r)}\Big[\frac{Ca}m+Came^{-8a(r-T)}+Cam^{5+2\zeta-K/2}\Big]dr . \] We bound the four terms. By the initial bound, $\mathscr H(t_0)\le Cm^3$, and $a(t-t_0)\ge\log(2m)/4+u_--\mathfrak b\log(em)-1$, which bounds the first term by $Cm^{3-\alpha_0(1-4\mathfrak b)}$. The constant forcing $Ca/m$ contributes at most $C/(4\alpha_0m)$, and the last forcing contributes at most $Cm^{5+2\zeta-K/2}$. For the third forcing, since $\alpha_0>2$, \[ Cam\int_{t_0}^te^{-4a\alpha_0(t-r)}e^{-8a(r-T)}\,dr\le\frac{Cm}{4\alpha_0-8}\,e^{-8a(t-T)}\le Cm^{-1+8\mathfrak b}, \] because $e^{-8a(t-T)}=e^{-2\log(2m)-8u+8\mathfrak b\log(em)}\le Cm^{-2+8\mathfrak b}$. The term $C/m$ is at most $Cm^{-1+8\mathfrak b}$, and \eqref{mc-eq:te29} follows. \end{proof} \section{Proof of the main results}\label{sec:coldm-proofs}\label{ccf-sec:full-profile} \subsection{The profile under the scaled-gap bounds} \begin{proof}[Proof of \cref{ccf-thm:full-profile}] For each $n$, the function $t\mapsto D_n(t)$ is nonincreasing and tends to zero as $t\to\infty$. This follows as in the proof of \cref{cor:model-profile-cutoff}, since $\pi$ is the reversible invariant measure of $P_t$ and $\nu_N$ has a bounded density with respect to $\pi$. Discarding finitely many $n$, we may assume that $2m_n>1$. Then the centers $b_n=\log(2m_n)/(4a^{(n)}_2)$ and windows $w_n=1/a^{(n)}_2$ are positive and satisfy $b_n/w_n=\log(2m_n)/4\to\infty$, and $\cV$ is continuous and strictly decreasing from one to zero (\cref{cot-profile}). It therefore suffices, by the deterministic version of \cref{hp-monotone-profile-transfer}, to prove that $D_n(t_{m_n}(u))\to\cV(u)$ for each fixed $u\in\R$. The lower bound $\liminf_nD_n(t_{m_n}(u))\ge\cV(u)$ is part of \cref{mesocold-thm:weak-profile}, and it remains to prove the upper bound. We fix $u$, drop the index $n$, and write $t=t_m(u)$. We choose the parameters in the order of \cref{tab:coldm-parameters}, and all thresholds $m_0$ below depend only on them and on $u$. All estimates below hold for $n$ large, because $m\to\infty$. \paragraph{Step 1: parameters.} Let $J\ge2$ be the smallest integer with $(c_*/C_*)J^{2/3}\ge6$. Since $m\le(C_*/2)N^{1/3}$, we have $N\to\infty$ and thus $Jt\}}$ and $\eta^0=\eta$. \begin{enumerate} \item The weight $W$ equals $\Phi(\xi',X|_{[t-1/a,t]})$, where $\xi'=\ind_E\ind_{\{X(r)\in D\text{ for }r\in[T,t-1/a]\}}$ is $\mathcal F_{t-1/a}$-measurable (since $t-1/a\ge T$) and $\Phi(e,\omega)=e\,\ind_{\{\omega_r\in D\text{ for all }r\}}$. The set $D$ is open, invariant under all rotations of $(x_1,\dots,x_J)$, and contained in $\{|x_{\le J}|^20$ depending only on $c_*$ and $C_*$. Inserting these bounds and \eqref{mc-eq:cp9} into \eqref{ccf-eq:finite-profile-closure}, we obtain \begin{equation}\label{mc-eq:cp12} \big|D(t_m(u))-\cV(u)\big|\le2d_t+C_\gamma m^{-2/5}+\|\mu_N-\mu_u\|_\TV+\frac Cm+Ce^{-c_1m} . \end{equation} The limit superior of the right side is at most $2\gamma$ by \eqref{mc-eq:cp6} and \eqref{mc-eq:cp11}. Since $\delta'\in(0,1)$ is arbitrary and $\gamma\le\delta'$, this proves $D_n(t_{m_n}(u))\to\cV(u)$. \end{proof} The proof uses only finite bounds in which the tolerance $\gamma$ and the parameters of \cref{tab:coldm-parameters} are fixed before $m\to\infty$, together with two convergences along the sequence: the weak convergence of \cref{mesocold-thm:weak-profile}, and its upgrade \eqref{mc-eq:cp11}. No step exchanges these limits. \subsection{Uniformity and the GOE}\label{ccf-sec:outer-uniformity} The next lemma converts convergence along every sequence of vectors in $\mathcal A_N$ into the uniform statement \eqref{mc-eq:cp14} and into convergence in probability for the GOE. \begin{lemma}[From sequences to uniform bounds]\label{lem:coldm-transfer} Fix $00$, and for each $N\ge3$ let $Q_N:\mathcal A_N(c_*,C_*,\varepsilon_*)\to[0,\infty]$. Suppose that $Q_{N_n}(a^{(n)})\to0$ for every sequence $a^{(n)}\in\mathcal A_{N_n}(c_*,C_*,\varepsilon_*)$ with $\theta_n>0$, $\theta_n\to0$, and $m_n\to\infty$. Then \[ \lim_{M\to\infty}\limsup_{N\to\infty}\sup\big\{Q_N(a):a\in\mathcal A_N(c_*,C_*,\varepsilon_*),\ 0<\theta\le1/M,\ m\ge M\big\}=0, \] where the supremum of the empty set is zero. \end{lemma} \begin{proof} Suppose the conclusion fails. Since the quantity inside the limit in $M$ is nonincreasing in $M$, there exists $\varepsilon>0$ such that, for every $M$, the limit superior in $N$ exceeds $\varepsilon$. We can therefore choose $M_l\to\infty$, dimensions $N_l\to\infty$, and vectors $a^{(l)}\in\mathcal A_{N_l}$ with $0<\theta_l\le1/M_l$, $m_l\ge M_l$, and $Q_{N_l}(a^{(l)})>\varepsilon$. This sequence satisfies the hypothesis, and $Q_{N_l}(a^{(l)})\not\to0$, a contradiction. \end{proof} \begin{proof}[Proof of \cref{ccf-cor:uniform-envelope}] We use \cref{lem:coldm-transfer} with $Q_N(a)=\sup\{|D_a(t_m(u))-\cV(u)|:u\in I,\ t_m(u)\ge0\}$ when $\theta>0$, and $Q_N(a)=0$ otherwise; the supremum over an empty set is also set to $0$. Here $D_a$ is the distance for the vector $a$. The hypothesis of the lemma is \cref{ccf-thm:full-profile}, because along a sequence with $m_n\to\infty$ the times $t_{m_n}(u)$, $u\in I$, are positive for large $n$. If $\log(2M)+4\min I>0$, then all times in the definition of $\mathcal E_N(M)$ are positive, and $\mathcal E_N(M)$ is at most the supremum in the lemma. This proves \eqref{mc-eq:cp14}. For \eqref{mc-eq:cp15}, fix $M\ge1$ with $\log(2M)+4\min I>0$. For every $S$ with $m_S>M$, we have $\theta_{N,S}\le1/M$ and $m_{N,S}\ge M$ for all large $N$, so that the error at $(N,S)$ is at most $\mathcal E_N(M)$, and hence \[ \limsup_{N\to\infty}\sup_{u\in I}\big|D_{N,S}(t_{m_{N,S}}(u))-\cV(u)\big|\le\limsup_{N\to\infty}\mathcal E_N(M). \] This holds for all large $S$, because $m_S\to\infty$. Letting $S\to\infty$ and then $M\to\infty$ in \eqref{mc-eq:cp14}, we obtain \eqref{mc-eq:cp15}. \end{proof} The same argument applies to random families. Let the vectors $a^{(N,S)}$ be random, and let $e_{N,S}=Q_N(a^{(N,S)})$, with the preceding nonnegative-time and empty-supremum conventions. Suppose that for every $\eta>0$ there exist constants $c_*,C_*,\varepsilon_*$, not depending on $N$ and $S$, and events $\mathcal G^\eta_{N,S}$ on which $a^{(N,S)}\in\mathcal A_N(c_*,C_*,\varepsilon_*)$, such that \[ \limsup_{S\to\infty}\limsup_{N\to\infty}\PP\big((\mathcal G^\eta_{N,S})^c\big)\le\eta, \] and suppose that for every fixed $M\ge1$, \[ \lim_{S\to\infty}\limsup_{N\to\infty}\PP\big(\theta_{N,S}\le0\text{ or }\theta_{N,S}>1/M\text{ or }m_{N,S}r)=0$ for every $r>0$. Indeed, fix $r>0$ and $\eta>0$, and use \eqref{mc-eq:cp14}, with the constants for $\eta$, to choose $M$ with $\log(2M)+4\min I>0$ and $\limsup_N\mathcal E_N(M)r)\le\eta$. Then let $\eta\to0$. This is a statement in probability, and the limit in $N$ is taken before the limit in $S$. \begin{proof}[Proof of \cref{thm:coldm-goe}] By \cref{sp-condensed-geometry}, case (M), we have $\PP(\theta_N>0)\to1$, $\theta_N\to0$, and $m_N\to\infty$ in probability. In particular, the event $\mathcal L_N=\{\theta_N>0,\ 2m_N>1\}$ has probability tending to one, and $a_2T_N=\log(2m_N)/4$ on $\mathcal L_N$, which implies that $a_2T_N\to\infty$ in probability. In the eigencoordinates, the model is the diffusion \eqref{mc-eq:model} with $a_i=c_N(\lambda_1-\lambda_i)/2$, by \eqref{eq:model-raw}, and $D_N(t)$ is the distance $D(t)$ of this vector $a$. Fix $\eta\in(0,1)$, and let $\mathcal G_N^\eta$ be the events of the second part of \cref{sp-condensed-geometry}, with constants $c_*,C_*,\varepsilon_*$ depending only on $\eta$; on $\mathcal G_N^\eta$, the vector $a$ belongs to $\mathcal A_N(c_*,C_*,\varepsilon_*)$ for all large $N$. Fix a compact interval $I$. The assertion is immediate when $I=\emptyset$; assume $I\ne\emptyset$. Fix $\varepsilon>0$, and let $e_N=\sup_{u\in I}|D_N(T_N+u/a_2)-\cV(u)|$, with the convention $D_N(t)=D_N(0)$ for $t<0$. By \cref{ccf-cor:uniform-envelope}, there exists $M\ge\exp(4\max_I|u|)$ with $\limsup_N\mathcal E_N(M)<\varepsilon$. On the event $\{0<\theta_N\le1/M,\ m_N\ge M\}$, which is contained in $\mathcal L_N$, we have $T_N+u/a_2=t_{m_N}(u)>0$ for $u\in I$. Then $e_N\le\mathcal E_N(M)<\varepsilon$ on $\mathcal G_N^\eta\cap\{0<\theta_N\le1/M,\ m_N\ge M\}$ for all large $N$, and we obtain \[ \limsup_N\PP\big(\mathcal G_N^\eta\cap\{e_N\ge\varepsilon\}\big)\le\limsup_N\PP\big(\theta_N\le0\text{ or }\theta_N>1/M\text{ or }m_N0$. Since $D_N$ is nonincreasing on $\R$, we have $D_N((1+\eta)T_N)\le D_N(T_N+u_0/a_2)$ on $\{\omega_N\ge u_0\}$, and \begin{align*} \PP\big(D_N((1+\eta)T_N)>\cV(u_0)+\varepsilon\big)&\le\PP\big(\omega_N\varepsilon\big)\longrightarrow0, \end{align*} by \cref{thm:coldm-goe}. Finally, $\cV(u_0)\to0$ as $u_0\to\infty$. The limit of $a_2(t_N(\eps_1)-t_N(\eps_2))$ follows as in the first part. \end{proof}