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# Universal cutoff for Dyson Ornstein Uhlenbeck process

Jeanne Boursier  (JB) Université Paris-Dauphine, PSL University, UMR 7534,
CNRS, CEREMADE, 75016 Paris, France<EMAIL_ADDRESS>,  Djalil
Chafaï  (DC) École Normale Supérieure, UMR 8553, CNRS, DMA, 75005 Paris,
France & Université Paris-Dauphine, PSL University, UMR 7534, CNRS, CEREMADE,
75016 Paris, France<EMAIL_ADDRESS>https://djalil.chafai.net/ and  Cyril
Labbé  (CL) Université de Paris, Laboratoire de Probabilités, Statistiques et
Modélisation, UMR 8001, F-75205 Paris, France<EMAIL_ADDRESS>
(Date: Summer 2021, revised Winter 2022, revised Summer 2022, compiled )

###### Abstract.

We study the Dyson–Ornstein–Uhlenbeck diffusion process, an evolving gas of
interacting particles. Its invariant law is the beta Hermite ensemble of
random matrix theory, a non-product log-concave distribution. We explore the
convergence to equilibrium of this process for various distances or
divergences, including total variation, relative entropy, and transportation
cost. When the number of particles is sent to infinity, we show that a cutoff
phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical
time. A remarkable feature is that this critical time is independent of the
parameter beta that controls the strength of the interaction, in particular
the result is identical in the non-interacting case, which is nothing but the
Ornstein–Uhlenbeck process. We also provide a complete analysis of the non-
interacting case that reveals some new phenomena. Our work relies among other
ingredients on convexity and functional inequalities, exact solvability, exact
Gaussian formulas, coupling arguments, stochastic calculus, variational
formulas and contraction properties. This work leads, beyond the specific
process that we study, to questions on the high-dimensional analysis of heat
kernels of curved diffusions.

###### Key words and phrases:

Dyson process; Ornstein–Uhlenbeck process; Coulomb gas; Random Matrix Theory;
High dimensional phenomenon; Cutoff phenomenon; High-dimensional probability;
Functional inequalities; Spectral analysis; Stochastic Calculus; Gaussian
analysis; Markov process; Diffusion process; Interacting Particle System

###### 2000 Mathematics Subject Classification:

60J60 (Diffusion processes); 82C22 (Interacting particle systems)

###### Contents

  1. 1 Introduction and main results
  2. 2 Additional comments and open problems
  3. 3 Cutoff phenomenon for the OU
  4. 4 General exactly solvable aspects
  5. 5 The random matrix cases
  6. 6 Cutoff phenomenon for the DOU in TV and Hellinger
  7. 7 Cutoff phenomenon for the DOU in Wasserstein
  8. A Distances and divergences
  9. B Convexity and its dynamical consequences

##  1\. Introduction and main results

Let us consider a Markov process $X={(X_{t})}_{t\geq 0}$ with state space $S$
and invariant law $\mu$ for which

$\lim_{t\to\infty}\mathrm{dist}(\mathrm{Law}(X_{t})\mid\mu)=0$

where $\mathrm{dist}(\cdot\mid\cdot)$ is a distance or divergence on the
probability measures on $S$. Suppose now that $X=X^{n}$ depends on a
dimension, size, or complexity parameter $n$, and let us set $S=S^{n}$,
$\mu=\mu^{n}$, and $X_{0}=x^{n}_{0}\in S^{n}$. For example $X^{n}$ can be a
random walk on the symmetric group of permutations of $\\{1,\ldots,n\\}$,
Brownian motion on the group of $n\times n$ unitary matrices, Brownian motion
on the $n$-dimensional sphere, etc. In many of such examples, it has been
proved that when $n$ is large enough, the supremum over some set of initial
conditions $x^{n}_{0}$ of the quantity
$\mathrm{dist}(\mathrm{Law}(X_{t}^{n})\mid\mu^{n})$ collapses abruptly to $0$
when $t$ passes a critical value $c=c_{n}$ which may depend on $n$. This is
often referred to as a _cutoff phenomenon_. More precisely, if $\mathrm{dist}$
ranges from $0$ to $\max$, then, for some subset $S^{n}_{0}\subset S^{n}$ of
initial conditions, some critical value $c=c_{n}$ and for all
$\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\sup_{x^{n}_{0}\in
S^{n}_{0}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid\mu^{n})=\begin{cases}\max&\text{if
$t_{n}=(1-\varepsilon)c_{n}$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)c_{n}$}\end{cases}.$

It is standard to introduce, for an arbitrary small threshold $\eta>0$, the
quantity $\inf\\{t\geq 0:\sup_{x_{0}\in
S_{0}^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t})\mid\mu^{n})\leq\eta\\}$ known
as the _mixing time_ in the literature. Of course such a definition fully
makes sense as soon as $t\mapsto{\sup_{x_{0}\in
S_{0}^{n}}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t})\mid\mu^{n})$ is non-
increasing.

When $S^{n}$ is finite, it is customary to take $S^{n}_{0}=S^{n}$. When
$S^{n}$ is infinite, it may happen that the supremum over the whole set
$S^{n}$ of the distance to equilibrium remains equal to $\max$ at all times,
in which case one has to consider strict subspaces of initial conditions. For
some processes, it is possible to restrict $S^{n}_{0}$ to a single state in
which case one obtains a very precise description of the convergence to
equilibrium starting from this initial condition. Note that the constraint
over the initial condition can be made compatible with a limiting dynamics,
for instance a mean-field limit when the process describes an exchangeable
interacting particle system.

The _cutoff phenomenon_ was put forward by Aldous and Diaconis at the origin
for random walks on finite sets, see for instance [1, 28, 26, 52] and
references therein. The analysis of the cutoff phenomenon is the subject of an
important activity, still seeking for a complete theory: let us mention that,
for the total variation distance, Peres proposed the so-called product
condition (the mixing time must be much larger than the inverse of the
spectral gap) as a necessary and sufficient condition for a cutoff phenomenon
to hold, but counter-examples were exhibited [52, Sec. 18.3] and the product
condition is only necessary.

The study of the cutoff phenomenon for Markov diffusion processes goes back at
least to the works of Saloff-Coste [62, 64] in relation notably with
Nash–Sobolev type functional inequalities, heat kernel analysis, and
Diaconis–Wilson probabilistic techniques. We also refer to the more recent
work [55] for the case of diffusion processes on compact groups and symmetric
spaces, in relation with group invariance and representation theory, a point
of view inspired by the early works of Diaconis on Markov chains and of
Saloff-Coste on diffusion processes. Even if most of the available results in
the literature on the cutoff phenomenon are related to compact state spaces,
there are some notable works devoted to non-compact spaces such as [47, 19,
10, 8, 9, 11].

Our contribution is an exploration of the cutoff phenomenon for the
Dyson–Ornstein–Uhlenbeck diffusion process, for which the state space is
$\mathbb{R}^{n}$. This process is an interacting particle system. When the
interaction is turned off, we recover the Ornstein–Uhlenbeck process, a
special case that has been considered previously in the literature but for
which we also provide new results.

###  1.1. Distances

As for $\mathrm{dist}$ we use several standard distances or divergences
between probability measures: Wasserstein, total variation (TV), Hellinger,
Entropy, $\chi^{2}$ and Fisher, surveyed in Appendix A. We take the following
convention for probability measures $\mu$ and $\nu$ on the same space:

$\mathrm{dist}(\mu\mid\nu)=\begin{cases}\mathrm{Wasserstein}(\mu,\nu)&\text{when
$\mathrm{dist}=\mathrm{Wasserstein}$}\\\
\left\|\mu-\nu\right\|_{\mathrm{TV}}&\text{when
$\mathrm{dist}=\mathrm{TV}$}\\\ \mathrm{Hellinger}(\mu,\nu)&\text{when
$\mathrm{dist}=\mathrm{Hellinger}$}\\\
\mathrm{Kullback}(\mu\mid\nu)&\text{when $\mathrm{dist}=\mathrm{Kullback}$}\\\
\chi^{2}(\mu\mid\nu)&\text{when $\mathrm{dist}=\chi^{2}$}\\\
\mathrm{Fisher}(\mu\mid\nu)&\text{when
$\mathrm{dist}=\mathrm{Fisher}$}\end{cases},$ (1.1)

see Appendix A for precise definitions. The maximal value $\max$ taken by
$\mathrm{dist}$ is given by

$\max=\begin{cases}1&\text{ if
}\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\},\\\ +\infty&\text{ if
}\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}.\end{cases}$
(1.2)

###  1.2. The Dyson–Ornstein–Uhlenbeck (DOU) process and preview of main
results

The DOU process is the solution $X^{n}={(X^{n}_{t})}_{t\geq 0}$ on
$\mathbb{R}^{n}$ of the stochastic differential equation

$X^{n}_{0}=x^{n}_{0}\in\mathbb{R}^{n},\quad\mathrm{d}X_{t}^{n,i}=\sqrt{\frac{2}{n}}\mathrm{d}B^{i}_{t}-V^{\prime}(X_{t}^{n,i})\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}},\quad 1\leq i\leq n,$ (1.3)

where ${(B_{t})}_{t\geq 0}$ is a standard $n$-dimensional Brownian motion
(BM), and where

  * •

$V(x)=\frac{x^{2}}{2}$ is a “confinement potential” acting through the drift
$-V^{\prime}(x)=-x$

  * •

$\beta\geq 0$ is a parameter tuning the interaction strength.

The notation $X^{n,i}_{t}$ stands for the $i$-th coordinate of the vector
$X^{n}_{t}$. The process $X^{n}$ can be thought of as an interacting particle
system of $n$ one-dimensional Brownian particles $X^{n,1},\ldots,X^{n,n}$,
subject to confinement and singular pairwise repulsion when $\beta>0$
(respectively first and second term in the drift). We take an inverse
temperature of order $n$ in (1.3) in order to obtain a mean-field limit
without time-changing the process, see Section 2.5. The spectral gap is $1$
for all $n\geq 1$, see Section 2.6. We refer to Section 2.9 for other
parametrizations or choices of inverse temperature.

In the special cases $\beta\in\\{0,1,2\\}$, the cutoff phenomenon for the DOU
process can be established by using Gaussian analysis and stochastic calculus,
see Sections 1.4 and 1.5. For $\beta=0$, the process reduces to the
Ornstein–Uhlenbeck process (OU) and its behavior serves as a benchmark for the
interaction case $\beta\neq 0$, while when $\beta\in\\{1,2\\}$, the approach
involves a lift to unitary invariant ensembles of random matrix theory. For a
general $\beta\geq 1$, our main results regarding the cutoff phenomenon for
the DOU process are given in Sections 1.6 and 1.7. We are able, in particular,
to prove the following: for all
$\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{TV},\mathrm{Hellinger}\\}$,
$a>0$, $\varepsilon\in(0,1)$, we have

$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a,a]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases},$

where $P_{n}^{\beta}$ is the invariant law of the process, and where

$c_{n}:=\begin{cases}\log(\sqrt{n}a)&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\\\ \log(na)&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\end{cases}.$

This result is stated in a slightly more general form in Corollary 1.7. Our
proof relies crucially on an exceptional exact solvability of the dynamics,
notably the fact that we know explicitly the optimal long time behavior in
entropy and coupling distance, as well as the eigenfunction associated to the
spectral gap which turns out to be linear and optimal. This comes from the
special choice of $V$ as well as the special properties of the Coulomb
interaction. We stress that such an exact solvability is no longer available
for a general strongly convex $V$, even for instance in the simple example
$V(x)=\frac{x^{2}}{2}+x^{4}$ or for general linear forces. Nevertheless, and
as usual, two other special classical choices of $V$ could be explored,
related to Laguerre and Jacobi weights, see Section 2.8.

###  1.3. Analysis of the Dyson–Ornstein–Uhlenbeck process

The process $X^{n}$ was essentially discovered by Dyson in [33], in the case
$\beta\in\\{1,2,4\\}$, because it describes the dynamics of the eigenvalues of
$n\times n$ symmetric/Hermitian/symplectic random matrices with independent
Ornstein–Uhlenbeck entries, see Lemma 5.1 and Lemma 5.2 below for the cases
$\beta=1$ and $\beta=2$ respectively.

  * •

Case $\beta=0$ (interaction turned off). The particles become $n$ independent
one-dimensional Ornstein–Uhlenbeck processes, and the DOU process $X^{n}$
becomes exactly the $n$-dimensional Ornstein–Uhlenbeck process $Z^{n}$ solving
(1.8). The process lives in $\mathbb{R}^{n}$. The particles collide but since
they do not interact, this does not raise any issue.

  * •

Case $0<\beta<1$. Then with positive probability the particles collide
producing a blow up of the drift, see for instance [22, 25] for a discussion.
Nevertheless, it is possible to define the process for all times, for instance
by adding a local time term to the stochastic differential equation, see [25]
and references therein. It is natural to expect that the cutoff universality
works as for $\beta\not\in(0,1)$, but for simplicity we do not consider this
case here.

  * •

Case $\beta\geq 1$. If we order the coordinates by defining the convex domain

$D_{n}=\\{x\in\mathbb{R}^{n}:x_{1}<\cdots<x_{n}\\},$

and if $x^{n}_{0}\in D_{n}$ then the equation (1.3) admits a unique strong
solution that never exits $D_{n}$, in other words the particles never collide
and the order of the initial particles is preserved at all times, see [60].
Moreover if

$\overline{D}_{n}=\\{x\in\mathbb{R}^{n}:x_{1}\leq\cdots\leq x_{n}\\}$

then it is possible to start the process from the boundary
$\overline{D}_{n}\setminus D_{n}$, in particular from $x^{n}_{0}$ such that
$x^{n,1}_{0}=\cdots=x^{n,n}_{0}$, and despite the singularity of the drift, it
can be shown that with probability one, $X^{n}_{t}\in D_{n}$ for all $t>0$. We
refer to [2, Th. 4.3.2] for a proof in the Dyson Brownian Motion case that can
be adapted _mutatis mutandis_.

In the sequel, we will only consider the cases $\beta=0$ with
$x_{0}^{n}\in\mathbb{R}^{n}$ and $\beta\geq 1$ with
$x^{n}_{0}\in\overline{D}_{n}$.

The drift in (1.3) is the gradient of a function, and (1.3) rewrites

$X_{0}^{n}=x_{0}^{n}\in
D_{n},\quad\mathrm{d}X_{t}^{n}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-\frac{1}{n}\nabla
E(X_{t}^{n})\mathrm{d}t,$ (1.4)

where

$E(x_{1},\ldots,x_{n})=n\sum_{i=1}^{n}V(x_{i})+{\beta}\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}$
(1.5)

can be interpreted as the energy of the configuration of particles
$x_{1},\ldots,x_{n}$.

  * •

If $\beta=0$, then the Markov process $X^{n}$ is an Ornstein–Uhlenbeck
process, irreducible with unique invariant law
$P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$ which is reversible.

  * •

If $\beta\geq 1$, then the Markov process $X^{n}$ is not irreducible, but
$D_{n}$ is a recurrent class carrying a unique invariant law $P_{n}^{\beta}$,
which is reversible and given by

$P_{n}^{\beta}=\frac{\mathrm{e}^{-E(x_{1},\ldots,x_{n})}}{C_{n}^{\beta}}\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n},$
(1.6)

where $C_{n}^{\beta}$ is the normalizing factor given by

$C_{n}^{\beta}=\int_{\overline{D}_{n}}\mathrm{e}^{-E(x_{1},\ldots,x_{n})}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$
(1.7)

In terms of geometry, it is crucial to observe that since $-\log$ is convex on
$(0,+\infty)$, the map

$(x_{1},\ldots,x_{n})\in
D_{n}\mapsto\mathrm{Interaction}(x_{1},\ldots,x_{n})={\beta}\sum_{i>j}\log\frac{1}{x_{i}-x_{j}},$

is convex. Thus, since $V$ is convex on $\mathbb{R}$, it follows that $E$ is
convex on $D_{n}$. For all $\beta\geq 0$, the law $P_{n}^{\beta}$ is log-
concave with respect to the Lebesgue measure as well as with respect to
$\mathcal{N}(0,\frac{1}{n}I_{n})$.

###  1.4. Non-interacting case and Ornstein–Uhlenbeck benchmark

When we turn off the interaction by taking $\beta=0$ in (1.3), the DOU process
becomes an Ornstein–Uhlenbeck process (OU) $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ on
$\mathbb{R}^{n}$ solving the stochastic differential equation

$Z^{n}_{0}=z^{n}_{0}\in\mathbb{R}^{n},\quad\mathrm{d}Z^{n}_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B^{n}_{t}-Z^{n}_{t}\mathrm{d}t,$
(1.8)

where $B^{n}$ is a standard $n$-dimensional BM. The invariant law of $Z^{n}$
is the product Gaussian law
$P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})=\mathcal{N}(0,\frac{1}{n})^{\otimes
n}$. The explicit Gaussian nature of
$Z_{t}^{n}\sim\mathcal{N}(z_{0}^{n}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n})$,
valid for all $t\geq 0$, allows for a fine analysis of convergence to
equilibrium, as in the following theorem.

######  Theorem 1.1 (Cutoff for OU: mean-field regime).

Let $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ be the OU process (1.8) and let
$P_{n}^{0}$ be its invariant law. Suppose that

$\varliminf_{n\to\infty}\frac{|z^{n}_{0}|^{2}}{n}>0\quad\text{and}\quad\varlimsup_{n\to\infty}\frac{|z^{n}_{0}|^{2}}{n}<\infty$

where $|z|=\sqrt{z_{1}^{2}+\cdots+z_{n}^{2}}$ is the Euclidean norm. Then for
all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z^{n}_{t_{n}})\mid
P_{n}^{0})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$

where

$c_{n}=\begin{cases}\frac{1}{2}\log(n)&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$},\\\ \log(n)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$},\\\
\frac{3}{2}\log(n)&\text{if $\mathrm{dist}=\mathrm{Fisher}$}.\end{cases}$

Theorem 1.1 is proved in Section 3. See Figure 1 and Figure 2 for a numerical
experiment.

Theorem 1.1 constitutes a very natural benchmark for the cutoff phenomenon for
the DOU process. Theorem 1.1 is not a surprise, and actually the TV and
Hellinger cases are already considered in [47], see also [7]. Let us mention
that in [9], a cutoff phenomenon for TV, entropy and Wasserstein is proven for
the OU process of _fixed_ dimension $d$ and vanishing noise. This is to be
compared with our setting where the dimension is sent to infinity: the results
(and their proofs) are essentially the same in these two situations, however
we will see below that if one considers more general initial conditions, there
are some substantial differences according to whether the dimension is fixed
or sent to infinity.

The restriction over the initial condition in Theorem 1.1 is spelled out in
terms of the second moment of the empirical distribution, a natural choice
suggested by the mean-field limit discussed in Section 2.5. It yields a mixing
time of order $\log(n)$, just like for Brownian motion on compact Lie groups,
see [64, 55]. For the OU process and more generally for overdamped Langevin
processes, the non-compactness of the space is replaced by the confinement or
tightness due to the drift.

Actually, Theorem 1.1 is a particular instance of the following, much more
general result that reveals that, except for the Wasserstein distance, a
cutoff phenomenon _always_ occurs.

######  Theorem 1.2 (General cutoff for OU).

Let $Z^{n}={(Z^{n}_{t})}_{t\geq 0}$ be the OU process (1.8) and let
$P_{n}^{0}$ be its invariant law. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}$.
Then, for all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z^{n}_{t_{n}})\mid
P_{n}^{0})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$,}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$

where

$c_{n}=\begin{cases}\log(\sqrt{n}|z_{0}^{n}|)\vee\frac{1}{4}\log(n)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$},\\\
\log(n|z_{0}^{n}|)\vee\frac{1}{2}\log(n)&\text{if
$\mathrm{dist}=\mathrm{Fisher}$}.\end{cases}$

Regarding the Wasserstein distance, the following dichotomy occurs:

  * •

if $\lim_{n\to\infty}|z_{0}^{n}|=+\infty$, then for all $\varepsilon\in(0,1)$,
with $c_{n}=\log|z_{0}^{n}|$,

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(Z_{t_{n}}),P_{n}^{0})=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)c_{n}$,}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)c_{n}$},\end{cases}$

  * •

if $\lim_{n\to\infty}|z_{0}^{n}|=\alpha\in[0,\infty)$ then there is _no cutoff
phenomenon_ namely for any $t>0$

$\lim_{n\to\infty}\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{0})=\alpha^{2}\mathrm{e}^{-2t}+2\Bigr{(}1-\sqrt{1-\mathrm{e}^{-2t}}-\tfrac{1}{2}\mathrm{e}^{-2t}\Bigr{)}.$

Theorem 1.2 is proved in Section 3.

The observation that for every distance or divergence, except for the
Wasserstein distance, a cutoff phenomenon occurs _generically_ seems to be
new.

Let us make a few comments. First, in terms of convergence to equilibrium the
relevant observable in Theorem 1.2 appears to be the Euclidean norm
$|z_{0}^{n}|$ of the initial condition. This quantity differs from the
eigenfunction associated to the spectral gap of the generator, which is given
by $z_{1}+\cdots+z_{n}$ as we will recall later on. Note by the way that (3.4)
and (2.3) are equal! Second, cutoff occurs at a time that is _independent_ of
the initial condition provided that its Euclidean norm is small enough: this
cutoff time appears as the time required to regularize the initial condition
(a Dirac mass) into a sufficiently spread out absolutely continuous
probability measure; in particular this cutoff phenomenon would not hold
generically if we allowed for spread out (non-Dirac) initial conditions. Note
that, for the OU process of _fixed_ dimension and vanishing noise, we would
not observe a cutoff phenomenon when starting from initial conditions with
small enough Euclidean norm: this is a high dimensional phenomenon. In this
respect, the Wasserstein distance is peculiar since it is much less stringent
on the local behavior of the measures at stake: for instance
$\lim_{n\to\infty}\mathrm{Wasserstein}(\delta_{0},\delta_{1/n})=0$ while for
all other distances or divergences considered here, the corresponding quantity
would remain equal to $\max$. This explains the absence of _generic_ cutoff
phenomenon for Wasserstein. Third, the explicit expressions provided in our
proof allow to extract the cutoff profile in each case, but we prefer not to
provide them in our statement and refer the interested reader to the end of
Section 3.

###  1.5. Exactly solvable intermezzo

When $\beta\neq 0$, the law of the DOU process is no longer Gaussian nor
explicit. However several exactly solvable aspects are available. Let us
recall that a Cox–Ingersoll–Ross process (CIR) of parameters $a,b,\sigma$ is
the solution $R=(R_{t})_{t\geq 0}$ on $\mathbb{R}_{+}$ of

$R_{0}=r_{0}\in\mathbb{R}_{+},\quad\mathrm{d}R_{t}=\sigma\sqrt{R_{t}}\mathrm{d}W_{t}+(a-bR_{t})\mathrm{d}t,$
(1.9)

where $W$ is a standard BM. Its invariant law is
$\mathrm{Gamma}(2a/\sigma^{2},2b/\sigma^{2})$ with density proportional to
$r\geq 0\mapsto r^{2a/\sigma^{2}-1}\mathrm{e}^{-2br/\sigma^{2}}$, with mean
$a/b$, and variance $a\sigma^{2}/(2b^{2})$. It was proved by William Feller in
[38] that the density of $R_{t}$ at an arbitrary $t$ can be expressed in terms
of special functions.

If ${(Z_{t})}_{t\geq 0}$ is a $d$-dimensional OU process of parameters
$\theta\geq 0$ and $\rho\in\mathbb{R}$, weak solution of

$\mathrm{d}Z_{t}=\theta\mathrm{d}W_{t}-\rho Z_{t}\mathrm{d}t$ (1.10)

where $W$ is a $d$-dimensional BM, then $R={(R_{t})}_{t\geq 0}$,
$R_{t}:=|Z_{t}|^{2}$, is a CIR process with parameters $a=\theta^{2}d$,
$b=2\rho$, $\sigma=2\theta$. When $\rho=0$ then $Z$ is a BM while $R=|Z|^{2}$
is a squared Bessel process.

The following theorem gathers some exactly solvable aspects of the DOU process
for general $\beta\geq 1$, which are largely already in the statistical
physics folklore, see [58]. It is based on our knowledge of eigenfunctions
associated to the first spectral values of the dynamics, see (2.6), and their
remarkable properties. As in (2.6), we set $\pi(x):=x_{1}+\cdots+x_{n}$ when
$x\in\mathbb{R}^{n}$.

######  Theorem 1.3 (From DOU to OU and CIR).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with $\beta=0$ or
$\beta\geq 1$, and let $P_{n}^{\beta}$ be its invariant law. Then:

  * •

${(\pi(X^{n}_{t}))}_{t\geq 0}$ is a one-dimensional OU process weak solution
of (1.8) with $\theta=\sqrt{2}$, $\rho=1$. Its invariant law is
$\mathcal{N}(0,1)$. It does not depend on $\beta$, and
$\pi(X^{n}_{t})\sim\mathcal{N}(\pi(x^{n}_{0})\mathrm{e}^{-t},1-\mathrm{e}^{-2t})$,
$t\geq 0$. Furthermore $\pi(X^{n}_{t})^{2}$ is a CIR process of parameters
$a=2$, $b=2$, $\sigma=2\sqrt{2}$.

  * •

${(|X^{n}_{t}|^{2})}_{t\geq 0}$ is a CIR process, weak solution of (1.9) with
$a=2+\beta(n-1)$, $b=2$, $\sigma=\sqrt{8/n}$. Its invariant law is
$\mathrm{Gamma}(\frac{1}{2}(n+\beta\frac{n(n-1)}{2}),\frac{n}{2})$ of mean
$1+\frac{\beta}{2}(n-1)$ and variance $\beta+\frac{2-\beta}{n}$. Furthermore,
if $d=n+\beta\frac{n(n-1)}{2}$ is a positive integer, then
${(|X^{n}_{t}|^{2})}_{t\geq 0}$ has the law of ${(|Z_{t}|^{2})}_{t\geq 0}$
where ${(Z_{t})}_{t\geq 0}$ is a $d$-dimensional OU process, weak solution of
(1.8) with $\theta=\sqrt{2/n}$, $\rho=1$, and $Z_{0}=z^{n}_{0}$ for an
arbitrary $z^{n}_{0}\in\mathbb{R}^{d}$ such that $|z^{n}_{0}|=|x^{n}_{0}|$.

At this step it is worth noting that Theorem 1.3 gives in particular, denoting
$\beta_{n}:=1+\frac{\beta}{2}(n-1)$,

$\mathbb{E}[\pi(X^{n}_{t})]=\pi(x^{n}_{0})\mathrm{e}^{-t}\underset{t\to\infty}{\longrightarrow}0\quad\text{and}\quad\mathbb{E}[|X^{n}_{t}|^{2}]=\beta_{n}+(|x^{n}_{0}|^{2}-\beta_{n})\mathrm{e}^{-2t}\underset{t\to\infty}{\longrightarrow}\beta_{n}.$
(1.11)

Following [25, Sec. 2.2], the limits can also be deduced from the
Dumitriu–Edelman tridiagonal random matrix model [32] isospectral to
$\beta$-Hermite. These formulas for the “transient” first two moments
$\mathbb{E}[\pi(X_{t}^{n})]$ and $\mathbb{E}[|X_{t}^{n}|^{2}]$ reveal an
abrupt convergence to their equilibrium values :

  * •

If $\lim_{n\to\infty}\frac{\pi(x^{n}_{0})}{n}=\alpha\neq 0$ then for all
$\varepsilon\in(0,1)$,

$\lim_{n\to\infty}|\mathbb{E}[\pi(X^{n}_{t_{n}})]|=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)\log(n)$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)\log(n)$}\end{cases}.$ (1.12)

  * •

If $\lim_{n\to\infty}\frac{|x^{n}_{0}|^{2}}{n}=\alpha\neq\frac{\beta}{2}$ then
for all $\varepsilon\in(0,1)$, denoting $\beta_{n}:=1+\frac{\beta}{2}(n-1)$,

$\lim_{n\to\infty}\left|\mathbb{E}[|X^{n}_{t_{n}}|^{2}]-\beta_{n}\right|=\begin{cases}+\infty&\text{if
$t_{n}=(1-\varepsilon)\frac{1}{2}\log(n)$}\\\ 0&\text{if
$t_{n}=(1+\varepsilon)\frac{1}{2}\log(n)$}\end{cases}.$ (1.13)

These critical times are universal with respect to $\beta$. The first two
transient moments are related to the eigenfunctions (2.6) associated to the
first two non-zero eigenvalues of the dynamics. Higher order transient moments
are related to eigenfunctions associated to higher order eigenvalues. Note
that $\mathbb{E}[\pi(X^{n}_{t})]$ and $\mathbb{E}[|X^{n}_{t}|^{2}]$ are the
first two moments of the non-normalized mean empirical measure
$\mathbb{E}[\sum_{i=1}^{n}\delta_{X^{n,i}_{t}}]$, and this lack of
normalization is responsible of the critical times of order $\log(n)$. In
contrast, the first two moments of the normalized mean empirical measure
$\mathbb{E}[\frac{1}{n}\sum_{i=1}^{n}\delta_{X^{n,i}_{t}}]$, given by
$\frac{1}{n}\mathbb{E}[\pi(X^{n}_{t})]$ and
$\frac{1}{n}\mathbb{E}[|X^{n}_{t}|^{2}]$ respectively, do not exhibit a
critical phenomenon. This is related to the exponential decay of the first two
moments in the mean-field limit (2.12), as well as the lack of cutoff for
Wasserstein already revealed for OU by Theorem 1.2. This also reminds the high
dimension behavior of norms in the field of the asymptotic geometric analysis
of convex bodies. In another direction, this elementary observation on the
moments also illustrates that the cutoff phenomenon for a given quantity is
not stable under rather simple transformations of this quantity.

From the first part of Theorem 1.3 and contraction properties available for
_some_ distances or divergences, see Lemma A.2, we obtain the following lower
bound on the mixing time for the DOU, which is independent of $\beta$:

######  Corollary 1.4 (Lower bound on the mixing time).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Set

$c_{n}:=\begin{cases}\log(|\pi(x_{0}^{n})|)&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log\Big{(}\frac{|\pi(x_{0}^{n})|}{\sqrt{n}}\Big{)}&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases},$

and assume that $\lim_{n\to\infty}c_{n}=\infty$. Then, for all
$\varepsilon\in(0,1)$, we have

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=\max.$

Theorem 1.3 and Corollary 1.4 are proved in Section 4.

The derivation of an upper bound on the mixing time is much more delicate:
once again recall that the case $\beta=0$ covered by Theorem 1.2 is specific
as it relies on exact Gaussian computations which are no longer available for
$\beta\geq 1$. In the next subsection, we will obtain results for general
values of $\beta\geq 1$ via more elaborate arguments.

In the specific cases $\beta\in\\{1,2\\}$, there are some exactly solvable
aspects that one can exploit to derive, in particular, precise upper bounds on
the mixing times. Indeed, for these values of $\beta$, the DOU process is the
process of eigenvalues of the matrix-valued OU process:

$M_{0}=m_{0},\quad\mathrm{d}M_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-M_{t}\mathrm{d}t,$

where $B$ is a BM on the symmetric $n\times n$ matrices if $\beta=1$ and on
Hermitian $n\times n$ matrices if $\beta=2$, see (5.4) and (5.16) for more
details. Based on this observation, we can deduce an upper bound on the mixing
times by contraction (for _most_ distances or divergences).

######  Theorem 1.5 (Upper bound on mixing time in matrix case).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with
$\beta\in\\{0,1,2\\}$, and invariant law $P_{n}^{\beta}$, and
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Set

$c_{n}:=\begin{cases}\log(\sqrt{n}|x_{0}^{n}|)\vee\log(\sqrt{n})&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log(|x_{0}^{n}|)&\text{if $\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases},$

and assume that $\lim_{n\to\infty}c_{n}=\infty$ if
$\mathrm{dist}=\mathrm{Wasserstein}$. Then, for all $\varepsilon\in(0,1)$, we
have

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$

Combining this upper bound with the lower bound already obtained above, we
derive a cutoff phenomenon in this particular matrix case.

######  Corollary 1.6 (Cutoff for DOU in the matrix case).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with
$\beta\in\\{0,1,2\\}$, and invariant law $P_{n}^{\beta}$. Let
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Wasserstein}\\}$.
Let ${(a_{n})}_{n}$ be a real sequence satisfying $\inf_{n}\sqrt{n}a_{n}>0$,
and assume further that $\lim_{n\to\infty}\sqrt{n}a_{n}=\infty$ if
$\mathrm{dist}=\mathrm{Wasserstein}$. Then, for all $\varepsilon\in(0,1)$, we
have

$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$

where

$c_{n}:=\begin{cases}\log(na_{n})&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$}\\\
\log(\sqrt{n}a_{n})&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$

Theorem 1.5 and Corollary 1.6 are proved in Section 5.

It is worth noting that $d=n+\beta\frac{n(n-1)}{2}$ in Theorem 1.3 is indeed
an integer in the “random matrix” cases $\beta\in\\{1,2\\}$, and corresponds
then exactly to the degree of freedom of the Gaussian random matrix models GOE
and GUE respectively. More precisely, if we let $X^{n}_{\infty}\sim
P_{n}^{\beta}$ then:

  * •

If $\beta=1$ then $P_{n}^{\beta}$ is the law of the eigenvalues of
$S\sim\mathrm{GOE}_{n}$, and $|X^{n}_{\infty}|^{2}=\sum_{j,k=1}^{n}S_{jk}^{2}$
which is the sum of $n$ squared Gaussians of variance $v=1/n$ (diagonal) plus
twice the sum of $\frac{n^{2}-n}{2}$ squared Gaussians of variance
$\frac{v}{2}$ (off-diagonal) all being independent. The duplication has the
effect of renormalizing the variance from $\frac{v}{2}$ to $v$. All in all we
have the sum of $d=\frac{n^{2}+n}{2}$ independent squared Gaussians of same
variance $v$. See Section 5.

  * •

If $\beta=2$ then $P_{n}^{\beta}$ is the law of the eigenvalues of
$H\sim\mathrm{GUE}_{n}$, and
$|X^{n}_{\infty}|^{2}=\sum_{j,k=1}^{n}|H_{jk}|^{2}$ is the sum of $n$ squared
Gaussians of variance $v=1/n$ (diagonal) plus twice the sum of $n^{2}-n$
squared Gaussians of variance $\frac{v}{2}$ (off-diagonal) all being
independent. All in all we have the sum of $d=n^{2}$ independent squared
Gaussians of same variance $v$. See Section 5.

Another manifestation of exact solvability lies at the level of functional
inequalities. Indeed, and following [25], the optimal Poincaré constant of
$P_{n}^{\beta}$ is given by $1/n$ and does not depend on $\beta$, and the
extremal functions are tranlations/dilations of
$x\mapsto\pi(x)=x_{1}+\cdots+x_{n}$. This corresponds to a spectral gap of the
dynamics equal to $1$ and its associated eigenfunction. Moreover, the optimal
logarithmic Sobolev inequality of $P_{n}^{\beta}$ (Lemma B.1) is given by
$2/n$ and does not depend on $\beta$, and the extremal functions are of the
form $x\mapsto\mathrm{e}^{c(x_{1}+\cdots+x_{n})}$, $c\in\mathbb{R}$. This
knowledge of the optimal constants and extremal functions and their
independence with respect to $\beta$ is truly remarkable. It plays a crucial
role in the results presented in this article. More precisely, the optimal
Poincaré inequality is used for the lower bound via the first eigenfunctions
while the optimal logarithmic Sobolev inequality is used for the upper bound
via exponential decay of the entropy.

###  1.6. Cutoff in the general interacting case

Our main contribution consists in deriving an upper bound on the mixing times
in the general case $\beta\geq 1$: the proof relies on the logarithmic Sobolev
inequality, some coupling arguments and a regularization procedure.

######  Theorem 1.7 (Upper bound on the mixing time: the general case).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3), with $\beta=0$ or
$\beta\geq 1$ and invariant law $P_{n}^{\beta}$. Take
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Wasserstein}\\}$.
Set

$c_{n}:=\begin{cases}\log(\sqrt{n}|x_{0}^{n}|)\vee\log({n})&\text{if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\\\
\log(|x_{0}^{n}|)\vee\log(\sqrt{n})&\text{if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$

Then, for all $\varepsilon\in(0,1)$, we have

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$

Combining this upper bound with the general lower bound that we obtained in
Corollary 1.4, we deduce the following cutoff phenomenon. Observe that it
holds both for $\beta=0$ and $\beta\geq 1$, and that the expression of the
mixing time does not depend on $\beta$.

######  Corollary 1.8 (Cutoff for DOU in the general case).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$ and invariant law $P_{n}^{\beta}$. Take
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Wasserstein}\\}$.
Let ${(a_{n})}_{n}$ be a real sequence satisfying $\inf_{n}a_{n}>0$. Then, for
all $\varepsilon\in(0,1)$, we have

$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{t_{n}})\mid
P_{n}^{\beta})=\begin{cases}\max&\text{if $t_{n}=(1-\varepsilon)c_{n}$}\\\
0&\text{if $t_{n}=(1+\varepsilon)c_{n}$}\end{cases}$

where

$c_{n}:=\begin{cases}\log(na_{n})&\text{ if
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$}\\\
\log(\sqrt{n}a_{n})&\text{ if
$\mathrm{dist}=\mathrm{Wasserstein}$}\end{cases}.$

The proofs of Theorem 1.7 and Corollary 1.8 for the TV and Hellinger distances
are presented in Section 6. The Wasserstein distance is treated in Section 7.
Let us make a comment on the assumptions made on $a_{n}$ in Corollaries 1.6
and 1.8. They are dictated by the upper bounds established in Theorems 1.5 and
1.7, which take the form of maxima of two terms: one that depends on the
initial condition, and another one which is a power of a logarithm of $n$. The
logarithmic term is an upper bound on the time required to regularize a
pointwise initial condition, its precise expression varies according to the
method of proof we rely on: in the matrix case, it is the time required to
regularize a larger object, the matrix-valued OU process; in the general case,
it is related to the time it takes to make the entropy of a pointwise initial
condition small. These bounds are not optimal for $\beta=0$ (compare with
Theorem 1.2), and probably neither for $\beta\geq 1$.

A natural, but probably quite difficult, goal would be to establish a cutoff
phenomenon in the situation where the set of initial conditions is reduced to
_any_ given singleton, as in Theorem 1.2 for the case $\beta=0$. Recall that
in that case, the asymptotic of the mixing time is dictated by the Euclidean
norm of the initial condition. In the case $\beta\geq 1$, this _cannot_ be the
right observable since the Euclidean norm does not measure the distance to
equilibrium. Instead one should probably consider the Euclidean norm
$|x_{0}^{n}-\rho_{n}|$, where $\rho_{n}$ is the vector of the quantiles of
order $1/n$ of the semi-circle law that arises in the mean-field limit
equilibrium (see Subsection 2.5). More precisely

$\rho_{n,i}=\inf\left\\{t\in\mathbb{R}:\int_{-\infty}^{t}\frac{\sqrt{2\beta-x^{2}}}{\beta\pi}\mathbf{1}_{x\in[-\sqrt{2\beta},\sqrt{2\beta}]}\mathrm{d}x\geq\frac{i}{n}\right\\},\quad
i\in\\{1,\ldots,n\\}.$ (1.14)

Note that $\rho_{n}=0$ when $\beta=0$.

A first step in this direction is given by the following result:

######  Theorem 1.9 (DOU in the general case and pointwise initial condition).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. There hold

  * •

If $\lim_{n\to\infty}|x^{n}_{0}-\rho_{n}|=+\infty$, then, denoting
$t_{n}=\log(|x_{0}^{n}-\rho_{n}|)$, for all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X_{(1+\varepsilon)t_{n}}),P_{n}^{\beta})=0.$

  * •

If $\lim_{n\to\infty}|x^{n}_{0}-\rho_{n}|=\alpha\in[0,\infty)$, then, for all
$t>0$,

$\varlimsup_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X_{t}),P_{n}^{\beta})^{2}\leq\alpha^{2}\mathrm{e}^{-2t}.$

Theorem 1.9 is proved in Section 7.

###  1.7. Non-pointwise initial conditions

It is natural to ask about the cutoff phenomenon when the initial conditions
$X^{n}_{0}$ is not pointwise. Even if we turn off the interaction by taking
$\beta=0$, the law of the process at time $t$ is then no longer Gaussian in
general, which breaks the method of proof used for Theorem 1.1 and Theorem
1.2. Nevertheless, Theorem 1.10 below provides a universal answer, that is
both for $\beta=0$ and $\beta\geq 1$, at the price however of introducing
several objects and notations. More precisely, for any probability measure
$\mu$ on $\mathbb{R}^{n}$, we introduce

$S(\mu)=\begin{cases}\displaystyle\int\frac{\mathrm{d}\mu}{\mathrm{d}x}\log\frac{\mathrm{d}\mu}{\mathrm{d}x}\mathrm{d}x=\text{``}\mathrm{Kullback}(\mu\mid\mathrm{d}x)\text{''}&\text{if
$\displaystyle\frac{\mathrm{d}\mu}{\mathrm{d}x}\log\frac{\mathrm{d}\mu}{\mathrm{d}x}\in
L^{1}(\mathrm{d}x)$}\\\ +\infty&\text{otherwise}\end{cases}.$ (1.15)

Note that $S$ takes its values in the whole $(-\infty,+\infty]$, and when
$S(\mu)<+\infty$ then $-S(\mu)$ is the _Boltzmann–Shannon entropy_ of the law
$\mu$. For all $x\in\mathbb{R}^{n}$ with $x_{i}\neq x_{j}$ for all $i\neq j$,
we have

$E(x_{1},\ldots,x_{n})=n^{2}\iint\Phi(x,y)\mathbf{1}_{\\{x\neq
y\\}}L_{n}(\mathrm{d}x)L_{n}(\mathrm{d}y)$ (1.16)

where $\displaystyle L_{n}:=\frac{1}{n}\sum_{i=1}^{n}\delta_{x_{i}}$ and where
$\displaystyle\Phi(x,y):=\frac{n}{n-1}\frac{V(x)+V(y)}{2}+\frac{\beta}{2}\log\frac{1}{|x-y|}$.

Let us define the map $\Psi:\mathbb{R}^{n}\mapsto\overline{D}_{n}$ by

$\Psi(x_{1},\ldots,x_{n}):=(x_{\sigma(1)},\ldots,x_{\sigma(n)}).$ (1.17)

where $\sigma$ is any permutation of $\\{1,\ldots,n\\}$ that reorders the
particles non-decreasingly.

######  Theorem 1.10 (Cutoff for DOU with product smooth initial conditions).

Let ${(X^{n}_{t})}_{t\geq 0}$ be the DOU process (1.3) with $\beta=0$ or
$\beta\geq 1$, and invariant law $P_{n}^{\beta}$. Let $S$, $\Phi$, and $\Psi$
be as in (1.15), (1.16), and (1.17). Let us assume that
$\mathrm{Law}(X_{0}^{n})$ is the image law or push forward of a product law
$\mu_{1}\otimes\cdots\otimes\mu_{n}$ by $\Psi$ where $\mu_{1},\ldots,\mu_{n}$
are laws on $\mathbb{R}$. Then:

  1. (1)

If $\displaystyle\varliminf_{n\to\infty}\Bigr{|}\frac{1}{n}\sum_{i=1}^{n}\int
x\mu_{i}(\mathrm{d}x)\Bigr{|}\neq 0$ then, for all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1-\varepsilon)\log(n)})\mid
P_{n}^{\beta})=+\infty.$

  2. (2)

If
$\displaystyle\varlimsup_{n\to\infty}\frac{1}{n^{2}}\sum_{i=1}^{n}S(\mu_{i})<\infty$
and $\displaystyle\varlimsup_{n\to\infty}\frac{1}{n^{2}}\sum_{i\neq
j}\iint\Phi\,\mathrm{d}\mu_{i}\otimes\mathrm{d}\mu_{j}<\infty$, then, for all
$\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1+\varepsilon)\log(n)})\mid
P_{n}^{\beta})=0.$

Theorem 1.10 is proved in Section 6.3.

It is likely that Theorem 1.10 can be extended to the case
$\mathrm{dist}\in\\{\mathrm{Wasserstein},\mathrm{Hellinger},\mathrm{Fisher}\\}$.

###  1.8. Structure of the paper

  * •

Section 2 provides additional comments and open problems.

  * •

Section 3 focuses on the OU process ($\beta=0$) and gives the proofs of
Theorems 1.1 and 1.2.

  * •

Section 4 concerns the exact solvability of the DOU process for all $\beta$,
and provides the proofs of Theorem 1.3 and Corollary 1.4.

  * •

Section 5 is about random matrices and gives the proofs of Theorem 1.5 and
Corollary 1.6.

  * •

Section 6 deals with the DOU process for all $\beta$ with the TV and Hellinger
distances, and provides the proofs of Theorem 1.7 and Corollary 1.8.

  * •

Section 7 gives the Wasserstein counterpart of Section 6 and the proof of
Theorem 1.9.

  * •

Appendix A provides a survey on distances and divergences, with new results.

  * •

Appendix B gathers useful dynamical consequences of convexity.

##  2\. Additional comments and open problems

###  2.1. About the results and proofs

The proofs of our results rely among other ingredients on convexity and
optimal functional inequalities, exact solvability, exact Gaussian formulas,
coupling arguments, stochastic calculus, variational formulas, contraction
properties and regularization.

The proofs of Theorems 1.1 and 1.2 are based on the explicit Gaussian nature
of the OU process, which allows to use Gaussian formulas for all the distances
and divergences that we consider (the Gaussian formula for $\mathrm{Fisher}$
seems to be new). Our analysis of the convergence to equilibrium of the OU
process seems to go beyond what is already known, see for instance [47] and
[10, 8, 9, 11].

Theorem 1.3 is a one-dimensional analogue of [15, Th. 1.2]. The proof exploits
the explicit knowledge of eigenfunctions of the dynamics (2.6), associated
with the first two non-zero spectral values, and their remarkable properties.
The first one is associated to the spectral gap and the optimal Poincaré
inequality. It implies Corollary 1.4, which is the provider of all our lower
bounds on the mixing time for the cutoff.

The proof of Theorem 1.5 is based on a contraction property and the upper
bound for matrix OU processes. It is not available beyond the matrix cases.
All the other upper bounds that we establish are related to an optimal
exponential decay which comes from convexity and involves sometimes coupling,
the simplest instance being Theorem 1.7 about the Wasserstein distance. The
usage of the Wasserstein metrics for Dyson dynamics is quite natural, see for
instance [13].

The proof of Theorem 1.7 for the $\mathrm{TV}$ and $\mathrm{Hellinger}$ relies
on the knowledge of the optimal exponential decay of the entropy (with respect
to equilibrium) related to the optimal logarithmic Sobolev inequality. Since
pointwise initial conditions have infinite entropy, the proof proceeds in
three steps: first we regularize the initial condition to make its entropy
finite, second we use the optimal exponential decay of the entropy of the
process starting from this regularized initial condition, third we control the
distance between the processes starting from the initial condition and its
regularized version. This last part is inspired by a work of Lacoin [48] for
the simple exclusion process on the segment, subsequently adapted to
continuous state-spaces [18, 19], where one controls an _area_ between two
versions of the process.

The (optimal) exponential decay of the entropy (Lemma B.2) is equivalent to
the (optimal) logarithmic Sobolev inequality (Lemma B.1). For the DOU process,
the optimal logarithmic Sobolev inequality provided by Lemma B.1 achieves also
the universal bound with respect to the spectral gap, just like for Gaussians.
This sharpness between the best logarithmic Sobolev constant and the spectral
gap also holds for instance for the random walk on the hypercube, a discrete
process for which a cutoff phenomenon can be established with the optimal
logarithmic Sobolev inequality, and which can be related to the OU process,
see for instance [30, 29] and references therein. If we generalize the DOU
process by adding an arbitrary convex function to $V$, then we will still have
a logarithmic Sobolev inequality – see [25] for several proofs including the
proof via the Bakry–Émery criterion – however the optimal logarithmic Sobolev
constant will no longer be explicit nor sharp with respect to the spectral
gap, and the spectral gap will no longer be explicit.

The proof of Theorem 1.10 relies crucially on the tensorization property of
$\mathrm{Kullback}$ and on the asymptotics on the normalizing constant
$C_{n}^{\beta}$ at equilibrium.

###  2.2. Analysis and geometry of the equilibrium

The full space $\mathbb{R}^{n}$ is, up to a bunch of hyperplanes, covered with
$n!$ disjoint isometric copies of the convex domain $D_{n}$ obtained by
permuting the coordinates (simplices or Weyl chambers). Following [25], for
all $\beta\geq 0$ let us define the law $P_{*n}^{\beta}$ on $\mathbb{R}^{n}$
with density proportional to $\mathrm{e}^{-E}$, just like for $P_{n}^{\beta}$
in (1.6) but without the
$\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}$.

If $\beta=0$ then $P_{*n}^{0}=P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$
according to our definition of $P_{n}^{0}$.

If $\beta>0$ then $P_{*n}^{\beta}$ has density
$(C_{*n}^{\beta})^{-1}\mathrm{e}^{-E}$ with $C_{*n}^{\beta}=n!C_{n}^{\beta}$
where $C_{n}^{\beta}$ is the normalization of $P_{n}^{\beta}$. Moreover
$P_{*n}^{\beta}$ is a mixture of $n!$ isometric copies of $P_{n}^{\beta}$,
while $P_{n}^{\beta}$ is the image law or push forward of $P_{*n}^{\beta}$ by
the map $\Psi_{n}:\mathbb{R}^{n}\to\overline{D}_{n}$ defined in (1.17).
Furthermore for all bounded measurable $f:\mathbb{R}^{n}\to\mathbb{R}$,
denoting $\Sigma_{n}$ the symmetric group of permutations of
$\\{1,\ldots,n\\}$,

$\int f\mathrm{d}P_{*n}^{\beta}=\int
f_{\mathrm{sym}}\mathrm{d}P_{n}^{\beta}\quad\text{with}\quad
f_{\mathrm{sym}}(x_{1},\ldots,x_{n}):=\frac{1}{n!}\sum_{\sigma\in\Sigma_{n}}f(x_{\sigma(1)},\ldots,x_{\sigma(n)}).$

Regarding log-concavity, it is important to realize that if $\beta=0$ then $E$
is convex on $\mathbb{R}^{n}$, while if $\beta>0$ then $E$ is convex on
$D_{n}$ but is not convex on $\mathbb{R}^{n}$ and has $n!$ isometric local
minima.

  * •

The law $P_{*n}^{\beta}$ is centered but is not log-concave when $\beta>0$
since $E$ is not convex on $\mathbb{R}^{n}$.
As $\beta\to 0^{+}$ the law $P_{*n}^{\beta}$ tends to
$P_{*n}^{0}=P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})$ which is log-concave.

  * •

The law $P_{n}^{\beta}$ is not centered but is log-concave for all $\beta\geq
0$.
Its density vanishes at the boundary of $D_{n}$ if $\beta>0$.
As $\beta\to 0^{+}$ the law $P_{n}^{\beta}$ tends to the law of the order
statistics of $n$ i.i.d. $\mathcal{N}(0,\frac{1}{n})$.

###  2.3. Spectral analysis of the generator: the non-interacting case

This subsection and the next deal with analytical aspects of our dynamics. We
start with the OU process ($\beta=0$) for which everything is explicit; the
next subsection deals with the DOU process ($\beta\geq 1$).

The infinitesimal generator of the OU process is given by

$\operatorname{G}f=\frac{1}{n}\Bigr{(}\Delta-\nabla
E\cdot\nabla\Bigr{)}=\frac{1}{n}\sum_{i=1}^{n}\partial_{i}^{2}-\sum_{i=1}^{n}V^{\prime}(x_{i})\partial_{i}.$
(2.1)

It is a self-adjoint operator on $L^{2}(\mathbb{R}^{n},P_{n}^{0})$ that leaves
globally invariant the set of polynomials. Its spectrum is the set of all non-
positive integers, that is,
$\lambda_{0}=0>\lambda_{1}=-1>\lambda_{2}=-2>\cdots$. The corresponding
eigenspaces $F_{0},F_{1},F_{2},\cdots$ are finite dimensional: $F_{m}$ is
spanned by the multivariate Hermite polynomials of degree $m$, in other words
tensor products of univariate Hermite polynomials. In particular, $F_{0}$ is
the vector space of constant functions while $F_{1}$ is the $n$-dimensional
vector space of all linear functions.

Let us point out that $\operatorname{G}$ can be restricted to the set of
$P_{n}^{0}$ square integrable _symmetric_ functions: it leaves globally
invariant the set of _symmetric_ polynomials, its spectrum is unchanged but
the associated eigenspaces $E_{m}$ are the restrictions of the vector spaces
$F_{m}$ to the set of symmetric functions, in other words, $E_{m}$ is spanned
by the multivariate _symmetrized_ Hermite polynomials of degree $m$. Note that
$E_{1}$ is the one-dimensional space generated by $\pi(x)=x_{1}+\cdots+x_{n}$.

The Markov semigroup ${(\mathrm{e}^{t\operatorname{G}})}_{t\geq 0}$ generated
by $\operatorname{G}$ admits $P_{n}^{0}$ as a reversible invariant law since
$\operatorname{G}$ is self-adjoint in $L^{2}(P_{n}^{0})$. Following [62], let
us introduce the _heat kernel_ $p_{t}(x,y)$ which is the density of
$\mathrm{Law}(X^{n}_{t}\mid X^{n}_{0}=x)$ with respect to the invariant law
$P_{n}^{0}$. The long-time behavior reads $\lim_{t\to\infty}p_{t}(x,\cdot)=1$
for all $x\in\mathbb{R}^{n}$. Let $\left\|\cdot\right\|_{p}$ be the norm of
$L^{p}=L^{p}(P_{n}^{0})$. For all $1\leq p\leq q$, $t\geq 0$,
$x\in\mathbb{R}^{n}$, we have

$2\|\mathrm{Law}(X^{n}_{t}\mid
X^{n}_{0}=x)-P_{n}^{0}\|_{\mathrm{TV}}=\|p_{t}(x,\cdot)-1\|_{1}\leq\|p_{t}(x,\cdot)-1\|_{p}\leq\|p_{t}(x,\cdot)-1\|_{q}.$
(2.2)

In the particular case $p=2$ we can write

$\|p_{t}(x,\cdot)-1\|_{2}^{2}=\sum_{m=1}^{\infty}\mathrm{e}^{-2mt}\sum_{\psi\in
B_{m}}|\psi(x)|^{2}.$ (2.3)

where $B_{m}$ is an orthonormal basis of $F_{m}\subset L^{2}(P_{n}^{0})$,
hence

$\|p_{t}(x,\cdot)-1\|_{2}^{2}\geq\mathrm{e}^{-2t}\sum_{\psi\in
B_{1}}|\psi(x)|^{2},$ (2.4)

which leads to a lower bound on the $\chi^{2}$ (in other words $L^{2}$)
cutoff, provided one can estimate $\sum_{\psi\in B_{1}}|\psi(x)|^{2}$ which is
the square of the norm of the projection of $\delta_{x}$ on $B_{1}$.

Following [62, Th. 6.2], an upper bound would follow from a Bakry–Émery
curvature–dimension criterion $\mathrm{CD}(\rho,d)$ with a finite dimension
$d$, in relation with Nash–Sobolev inequalities and dimensional pointwise
estimates on the heat kernel $p_{t}(x,\cdot)$ or ultracontractivity of the
Markov semigroup, see for instance [63, Sec. 4.1]. The OU process satisfies to
$\mathrm{CD}(\rho,\infty)$ but never to $\mathrm{CD}(\rho,d)$ with $d$ finite
and is not ultracontractive. Actually the OU process is a critical case, see
[3, Ex. 2.7.3].

###  2.4. Spectral analysis of the generator: the interacting case

We now assume that $\beta\geq 1$. The infinitesimal generator of the DOU
process is the operator

$\operatorname{G}f=\frac{1}{n}\Bigr{(}\Delta-\nabla
E\cdot\nabla\Bigr{)}=\frac{1}{n}\sum_{i=1}^{n}\partial_{i}^{2}-\sum_{i=1}^{n}V^{\prime}(x_{i})\partial_{i}+\frac{\beta}{2n}\sum_{j\neq
i}\frac{\partial_{i}-\partial_{j}}{x_{i}-x_{j}}.$ (2.5)

Despite the interaction term, the operator leaves globally invariant the set
of _symmetric_ polynomials. Following Lassalle in [49, 4], see also [25], the
operator $\operatorname{G}$ is a self-adjoint operator on the space of
$P_{*n}^{\beta}$ square integrable _symmetric_ functions of $n$ variables, its
spectrum does not depend on $\beta$ and matches the spectrum of the OU process
case $\beta=0$. In particular the spectral gap is $1$. The eigenspaces $E_{m}$
are spanned by the generalized symmetrized Hermite polynomials of degree $m$.
For instance, $E_{1}$ is the one-dimensional space generated by
$\pi(x)=x_{1}+\cdots+x_{n}$ while $E_{2}$ is the two-dimensional space spanned
by

$(x_{1}+\cdots+x_{n})^{2}-1\quad\text{and}\quad
x_{1}^{2}+\cdots+x_{n}^{2}-1-\frac{\beta}{2}(n-1).$ (2.6)

From the isometry between $L^{2}(\overline{D}_{n},P_{n}^{\beta})$ and
$L^{2}_{\mathrm{sym}}(\mathbb{R}^{n},P_{*n}^{\beta})$, the above explicit
spectral decomposition applies to the semigroup of the DOU on
$\overline{D}_{n}$. Formally, the discussion presented at the end of the
previous subsection still applies. However, in the present interacting case
the integrability properties of the heat kernel are not known: in particular,
we do not know whether $p_{t}(x,\cdot)$ lies in $L^{p}(P_{n}^{\beta})$ for
$t>0$, $x\in\overline{D}_{n}$ and $p>1$. This leads to the question, of
independent interest, of pointwise upper and lower Gaussian bounds for heat
kernels similar to the OU process, with explicit dependence of the constants
over the dimension. We refer for example to [65, 36, 41] for some results in
this direction.

###  2.5. Mean-field limit

The measure $P_{n}^{\beta}$ is log-concave since $E$ is convex, and its
density writes

$x\in\mathbb{R}^{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}|x|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(x_{i}-x_{j})^{\beta}\mathbf{1}_{x_{1}\leq\cdots\leq
x_{n}}.$ (2.7)

See [25, Sec. 2.2] for a high-dimensional analysis. The Boltzmann–Gibbs
measure $P_{n}^{\beta}$ is known as the $\beta$-Hermite ensemble or H$\beta$E.
When $\beta=2$, it is better known as the Gaussian Unitary Ensemble (GUE). If
$X^{n}\sim P_{n}^{\beta}$ then the Wigner theorem states that the empirical
measure with atoms distributed according to $P_{n}^{\beta}$ converges in
distribution to a semi-circle law, namely

$\frac{1}{n}\sum_{i=1}^{n}\delta_{X^{n,i}}\underset{n\to\infty}{\overset{\text{weak}}{\longrightarrow}}\frac{\sqrt{2\beta-x^{2}}}{\beta\pi}\mathbf{1}_{x\in[-\sqrt{2\beta},\sqrt{2\beta}]}\mathrm{d}x,$
(2.8)

and this can be deduced in this Coulomb gas context from a large deviation
principle as in [12].

Let ${(X^{n})}_{t\geq 0}$ be the process solving (1.3) with $\beta\geq 0$ or
$\beta\geq 1$, and let

$\mu^{n}_{t}=\frac{1}{n}\sum_{k=1}^{n}\delta_{X^{n,i}_{t}}$ (2.9)

be the empirical measure of the particles at time $t$. Following notably [60,
14, 21, 20, 53, 31], if the sequence of initial conditions
${(\mu_{0}^{n})}_{n\geq 1}$ converges weakly as $n\to\infty$ to a probability
measure $\mu_{0}$, then the sequence of measure valued processes
${({(\mu_{t}^{n})}_{t\geq 0})}_{n\geq 1}$ converges weakly to the unique
probability measure valued deterministic process ${(\mu_{t})}_{t\geq 0}$
satisfying the evolution equation

$\langle\mu_{t},f\rangle=\langle\mu_{0},f\rangle-\int_{0}^{t}\int
V^{\prime}(x)f^{\prime}(x)\mu_{s}(\mathrm{d}x)\mathrm{d}s+\frac{\beta}{2}\int_{0}^{t}\int_{\mathbb{R}^{2}}\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{s}(\mathrm{d}x)\mu_{s}(\mathrm{d}y)\mathrm{d}s\quad$
(2.10)

for all $t\geq 0$ and $f\in\mathcal{C}^{3}_{b}(\mathbb{R},\mathbb{R})$. The
equation (2.10) is a weak formulation of a McKean–Vlasov equation or free
Fokker–Planck equation associated to a free OU process. Moreover, if $\mu_{0}$
has all its moments finite, then for all $t\geq 0$, we have the free Mehler
formula

$\mu_{t}=\mathrm{dil}_{\mathrm{e}^{-2t}}\mu_{0}\boxplus\mathrm{dil}_{\sqrt{1-\mathrm{e}^{-2t}}}\mu_{\infty},$
(2.11)

where $\mathrm{dil}_{\sigma}\mu$ is the law of $\sigma X$ when $X\sim\mu$,
where “$\boxplus$” stands for the free convolution of probability measures of
Voiculescu free probability theory, and where $\mu_{\infty}$ is the semi-
circle law of variance $\frac{\beta}{2}$. In particular, if $\mu_{0}$ is a
semi-circle law then $\mu_{t}$ is a semi-circle law for all $t\geq 0$.

Let us introduce the $k$-th moment $m_{k}(t):=\displaystyle\int
x^{k}\mu_{t}(\mathrm{d}x)$ of $\mu_{t}$. The first and second moments satisfy
the differential equations $m_{1}^{\prime}=-m_{1}$ and
$m_{2}^{\prime}=-2m_{2}+\beta$ respectively, which give

$m_{1}(t)=\mathrm{e}^{-t}m_{1}(0)\underset{t\to\infty}{\longrightarrow}0\quad\text{and}\quad
m_{2}(t)=m_{2}(0)\mathrm{e}^{-2t}+\frac{\beta}{2}(1-\mathrm{e}^{-2t})\underset{t\to\infty}{\longrightarrow}\frac{\beta}{2}.$
(2.12)

More generally, beyond the first two moments, the Cauchy–Stieltjes transform

$z\in\mathbb{C}_{+}=\\{z\in\mathbb{C}:\Im z>0\\}\mapsto
s_{t}(z)=\int_{\mathbb{R}}\frac{\mu_{t}(\mathrm{d}x)}{x-z}$ (2.13)

of $\mu_{t}$ is the solution of the following complex Burgers equation

$\partial_{t}s_{t}(z)=s_{t}(z)+z\partial_{z}s_{t}(z)+\beta
s_{t}(z)\partial_{z}s_{t}(z),\quad t\geq 0,z\in\mathbb{C}_{+}.$ (2.14)

The semi-circle law on $[-c,c]$ has density $\frac{2\sqrt{c^{2}-x^{2}}}{\pi
c^{2}}\mathbf{1}_{x\in[-c,c]}$, mean $0$, second moment or variance
$\frac{c^{2}}{4}$, and Cauchy–Stieltjes transform
$s_{t}(z)=\frac{\sqrt{4z^{2}-4c^{2}}-2z}{c^{2}}$, $t\geq
0,z\in\mathbb{C}_{+}$.

The cutoff phenomenon is in a sense a diagonal $(t,n)$ estimate, melting long
time behavior and high dimension. When $|z_{0}^{n}|$ is of order $n$, cutoff
occurs at a time of order $\approx\log(n)$: this informally corresponds to
taking $t\to\infty$ in $(\mu_{t})_{t\geq 0}$.

When $\mu_{0}$ is centered with same second moment $\frac{\beta}{2}$ as
$\mu_{\infty}$, then there is a Boltzmann H-theorem interpretation of the
limiting dynamics as $n\to\infty$: the steady-state is the Wigner semi-circle
law $\mu_{\infty}$, the second moment is conserved by the dynamics, the
Voiculescu entropy is monotonic along the dynamics, grows exponentially, and
is maximized by the steady-state.

###  2.6. $L^{p}$ cutoff

Following [26], we can deduce an $L^{p}$ cutoff started from $x$ from an
$L^{1}$ cutoff by showing that the heat kernel $p_{t}(x,\cdot)$ is in
$L^{p}(P_{n}^{\beta})$ for some $t>0$. Thanks to the Mehler formula, it can be
checked that this holds for the OU case, despite the lack of
ultracontractivity. The heat kernel of the DOU process is less accessible.

In another exactly solvable direction, the $L^{p}$ cutoff phenomenon has been
studied for instance in [62, 64] for Brownian motion on compact simple Lie
groups, and in [64, 55] for Brownian motion on symmetric spaces, in relation
with representation theory, an idea which goes back at the origin to the works
of Diaconis on random walks on groups.

###  2.7. Cutoff window and profile

Once a cutoff phenomenon is established, one can ask for a finer description
of the pattern of convergence to equilibrium. The _cutoff window_ is the order
of magnitude of the transition time from the value $\max$ to the value $0$:
more precisely, if cutoff occurs at time $c_{n}$ then we say that the cutoff
window is $w_{n}$ if

$\displaystyle\varlimsup_{b\to+\infty}\varlimsup_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})$ $\displaystyle=0,$
$\displaystyle\varliminf_{b\to-\infty}\varliminf_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})$ $\displaystyle=\max,$

and for any $b\in\mathbb{R}$

$0<\varliminf_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})\leq\varlimsup_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})<\max.$

Note that necessarily $w_{n}=o(c_{n})$ by definition of the cutoff phenomenon.
Note also that $w_{n}$ is unique in the following sense: $w^{\prime}_{n}$ is a
cutoff window if and only if $w_{n}/w^{\prime}_{n}$ remains bounded from above
and below as $n\to\infty$.
We say that the _cutoff profile_ is given by $\varphi:\mathbb{R}\to[0,1]$ if

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X_{c_{n}+bw_{n}})\mid
P_{n}^{\beta})=\varphi(b).$

The analysis of the OU process carried out in Theorems 1.1 and 1.2 can be
pushed further to establish the so-called cutoff profiles, we refer to the end
of Section 3 for details.

Regarding the DOU process, such a detailed description of the convergence to
equilibrium does not seem easily accessible. However it is straightforward to
deduce from our proofs that the cutoff window is of order $1$, in other words
the inverse of the spectral gap, in the setting of Corollary 1.6. This is also
the case in the setting of Corollary 1.8 for the Wasserstein distance.
We believe that this remains true in the setting of Corollary 1.8 for the TV
and Hellinger distances: actually, a lower bound of the required order can be
derived from the calculations in the proof of Corollary 1.4; on the other
hand, our proof of the upper bound on the mixing time does not allow to give a
precise enough upper bound on the window.

###  2.8. Other potentials

It is natural to ask about the cutoff phenomenon for the process solving (1.3)
when $V$ is a more general $\mathcal{C}^{2}$ function. The invariant law
$P_{n}^{\beta}$ of this Markov diffusion writes

$\frac{\mathrm{e}^{-n\sum_{i=1}^{n}V(x_{i})}}{C_{n}^{\beta}}\prod_{i>j}(x_{i}-x_{j})^{\beta}\mathbf{1}_{(x_{1},\ldots,x_{n})\in\overline{D}_{n}}\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}.$
(2.15)

The case where $V-\frac{\rho}{2}\left|\cdot\right|^{2}$ is convex for some
constant $\rho\geq 0$ generalizes the DOU case and has exponential convergence
to equilibrium, see [25]. Three exactly solvable cases are known:

  * •

$\mathrm{e}^{-V(x)}=\mathrm{e}^{-\frac{x^{2}}{2}}$: the DOU process associated
to the Gaussian law weight and the $\beta$-Hermite ensemble including
HOE/HUE/HSE when $\beta\in\\{1,2,4\\}$,

  * •

$\mathrm{e}^{-V(x)}=x^{a-1}\mathrm{e}^{-x}\mathbf{1}_{x\in[0,\infty)}$: the
Dyson–Laguerre process associated to the Gamma law weight and the
$\beta$-Laguerre ensembles including LOE/LUE/LSE when $\beta\in\\{1,2,4\\}$,

  * •

$\mathrm{e}^{-V(x)}=x^{a-1}(1-x)^{b-1}\mathbf{1}_{x\in[0,1]}$: the
Dyson–Jacobi process associated to the Beta law weight and the $\beta$-Jacobi
ensembles including JOE/JUE/JSE when $\beta\in\\{1,2,4\\}$,

up to a scaling. Following Lassalle [49, 51, 50, 4] and Bakry [5], in these
three cases, the multivariate orthogonal polynomials of the invariant law
$P_{n}^{\beta}$ are the eigenfunctions of the dynamics of the process. We
refer to [35, 32, 54] for more information on (H/L/J)$\beta$E random matrix
models.

The contraction property or spectral projection used to pass from a matrix
process to the Dyson process can be used to pass from BM on the unitary group
to the Dyson circular process for which the invariant law is the Circular
Unitary Ensemble (CUE). This provides an upper bound for the cutoff
phenomenon. The cutoff for BM on the unitary group is known and holds at
critical time or order $\log(n)$, see for instance [64, 62, 55].

More generally, we could ask about the cutoff phenomenon for a McKean–Vlasov
type interacting particle system ${(X^{n}_{t})}_{t\geq 0}$ in
$(\mathbb{R}^{d})^{n}$ solution of the stochastic differential equation of the
form

$\mathrm{d}X^{n,i}_{t}=\sigma_{n,t}(X^{n})\mathrm{d}B^{n}_{t}-\sum_{i=1}^{n}\nabla
V_{n,t}(X^{n,i}_{t})\mathrm{d}t-\sum_{j\neq i}\nabla
W_{n,t}(X^{n,i}_{t}-X^{n,j}_{t})\mathrm{d}t,\quad 1\leq i\leq n,$ (2.16)

for various types of confinement $V$ and interaction $W$ (convex, repulsive,
attractive, repulsive-attractive, etc), and discuss the relation with the
propagation of chaos. The case where $V$ and $W$ are both convex and constant
in time is already very well studied from the point of view of long-time
behavior and mean-field limit in relation with convexity, see for instance
[21, 20, 53].

Regarding universality, it is worth noting that if $V=\left|\cdot\right|^{2}$
and if $W$ is convex then the proof by factorization of the optimal Poincaré
and logarithmic Sobolev inequalities and their extremal functions given in
[25] remains valid, paving the way to the generalization of many of our
results in this spirit. On the other hand, the convexity of the limiting
energy functional in the mean-field limit is of Bochner type and suggests to
take for $W$ a power, in other words a Riesz type interaction.

###  2.9. Alternative parametrization

If ${(X^{n}_{t})}_{t\geq 0}$ is the process solution of the stochastic
differential equation (1.3), then for all real parameters $\alpha>0$ and
$\sigma>0$, the space scaled and time changed stochastic process
${(Y^{n}_{t})}_{t\geq 0}={(\sigma X^{n}_{\alpha t})}_{t\geq 0}$ solves the
stochastic differential equation

$Y^{n}_{0}=\sigma
x^{n}_{0},\quad\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2\alpha\sigma^{2}}{n}}\mathrm{d}B^{i}_{t}-\alpha
Y_{t}^{n,i}\mathrm{d}t+\frac{\alpha\beta\sigma^{2}}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n,$ (2.17)

where ${(B_{t})}_{t\geq 0}$ is a standard $n$-dimensional BM. The invariant
law of ${(Y^{n}_{t})}_{t\geq 0}$ is

$\frac{\mathrm{e}^{-\frac{n}{2\sigma^{2}}|y|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(y_{i}-y_{j})^{\beta}\mathbf{1}_{(y_{1},\ldots,y_{n})\in\overline{D}_{n}}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}$
(2.18)

where $C_{n}^{\beta}$ is the normalizing constant. This law and its
normalization $C_{n}^{\beta}$ depend on the “shape parameter” $\beta$, the
“scale parameter” $\sigma$, and does not depend on the “speed parameter”
$\alpha$. When $\beta>0$, taking $\sigma^{2}=\beta^{-1}$, the stochastic
differential equation (2.17) boils down to

$Y^{n}_{0}=\frac{x^{n}_{0}}{\sqrt{\beta}},\quad\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2\alpha}{n\beta}}\mathrm{d}B^{i}_{t}-\alpha
Y_{t}^{n,i}\mathrm{d}t+\frac{\alpha}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n$ (2.19)

while the invariant law becomes

$\frac{\mathrm{e}^{-\frac{n\beta}{2}|y|^{2}}}{C_{n}^{\beta}}\prod_{i>j}(y_{i}-y_{j})^{\beta}\mathbf{1}_{(y_{1},\cdots,y_{n})\in\overline{D}_{n}}\mathrm{d}y_{1}\cdots\mathrm{d}y_{n}.$
(2.20)

The equation (2.19) is the one considered in [37, Eq. (12.4)] and in [46, Eq.
(1.1)]. The advantage of (2.19) is that $\beta$ can be now truly interpreted
as an inverse temperature and the right-hand side in the analogue of (2.8)
does not depend on $\beta$, while the drawback is that we cannot turn off the
interaction by setting $\beta=0$ and recover the OU process as in (1.3). It is
worthwhile mentioning that for instance Theorem 1.7 remains the same for the
process solving (2.19) in particular the cutoff threshold is at critical time
$\frac{c_{n}}{\alpha}$ and does not depend on $\beta$.

###  2.10. Discrete models

There are several discrete space Markov processes admitting the OU process as
a scaling limit, such as for instance the random walk on the discrete
hypercube, related to the Ehrenfest model, for which the cutoff has been
studied in [30, 29], and the M/M/$\infty$ queuing process, for which a
discrete Mehler formula is available [24]. Certain discrete space Markov
processes incorporate a singular repulsion mechanism, such as for instance the
exclusion process on the segment, for which the study of the cutoff in [48]
shares similarities with our proof of Theorem 1.7. It is worthwhile noting
that there are discrete Coulomb gases, related to orthogonal polynomials for
discrete measures, suggesting to study discrete Dyson processes. More
generally, it could be natural to study the cutoff phenomenon for Markov
processes on infinite discrete state spaces, under curvature condition, even
if the subject is notoriously disappointing in terms of high-dimensional
analysis. We refer to the recent work [61] for the finite state space case.

##  3\. Cutoff phenomenon for the OU

In this section, we prove Theorems 1.1 and 1.2: actually we only prove the
latter since it implies the former. We start by recalling a well-known fact.

######  Lemma 3.1 (Mehler formula).

If ${(Y_{t})}_{t\geq 0}$ is an OU process in $\mathbb{R}^{d}$ solution of the
stochastic differential equation $Y_{0}=y_{0}\in\mathbb{R}^{d}$ and
$\mathrm{d}Y_{t}=\sigma\mathrm{d}B_{t}-\mu Y_{t}\mathrm{d}t$ for parameters
$\sigma>0$ and $\mu>0$ where $B$ is a standard $d$-dimensional Brownian motion
then

${(Y_{t})}_{t\geq 0}={\Bigr{(}y_{0}\mathrm{e}^{-\mu
t}+\sigma\int_{0}^{t}\mathrm{e}^{\mu(s-t)}\mathrm{d}B_{s}\Bigr{)}}_{t\geq
0}\text{ hence }Y_{t}\sim\mathcal{N}\Bigr{(}y_{0}\mathrm{e}^{-\mu
t},\frac{\sigma^{2}}{2}\frac{1-\mathrm{e}^{-2\mu
t}}{\mu}\mathrm{I}_{d}\Bigr{)}\text{ for all $t\geq 0$.}$

Moreover its coordinates are independent one-dimensional OU processes with
initial condition $y_{0}^{i}$ and invariant law
$\mathcal{N}(0,\frac{\sigma^{2}}{2\mu})$, $1\leq i\leq d$.

###### Proof of Theorem 1.1 and Theorem 1.2.

By using Lemma 3.1, for all $n\geq 1$ and $t\geq 0$,

$Z^{n}_{t}\sim\mathcal{N}\Bigr{(}z^{n}_{0}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n}\Bigr{)}=\otimes_{i=1}^{n}\mathcal{N}\Bigr{(}z^{n,i}_{0}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}\Bigr{)},\
P_{n}^{0}=\mathcal{N}\Bigr{(}0,\frac{I_{n}}{n}\Bigr{)}=\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)}^{\otimes
n}.$ (3.1)

_Hellinger, Kullback, $\chi^{2}$, Fisher, and Wasserstein cutoffs._ A direct
computation from (3.1) or Lemma A.5 either from multivariate Gaussian formulas
or univariate via tensorization gives

$\displaystyle\mathrm{Hellinger}^{2}(\mathrm{Law}(Z^{n}_{t}),P_{n}^{0})$
$\displaystyle=1-\exp\Bigr{(}-\frac{n}{4}\frac{|z^{n}_{0}|^{2}\mathrm{e}^{-2t}}{2-\mathrm{e}^{-2t}}+\frac{n}{4}\log\Bigr{(}4\frac{1-\mathrm{e}^{-2t}}{(2-\mathrm{e}^{-2t})^{2}}\Bigr{)}\Bigr{)},$
(3.2) $\displaystyle 2\mathrm{Kullback}(\mathrm{Law}(Z^{n}_{t})\mid
P_{n}^{0})$
$\displaystyle=n|z^{n}_{0}|^{2}\mathrm{e}^{-2t}-n\mathrm{e}^{-2t}-n\log(1-\mathrm{e}^{-2t}),$
(3.3) $\displaystyle\chi^{2}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})$
$\displaystyle=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{n/2}}\exp\Bigr{(}n|z_{0}^{n}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\Bigr{)},$
(3.4) $\displaystyle\mathrm{Fisher}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})$
$\displaystyle=n^{2}|z^{n}_{0}|^{2}\mathrm{e}^{-2t}+n^{2}\frac{\mathrm{e}^{-4t}}{1-\mathrm{e}^{-2t}},$
(3.5)
$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z^{n}_{t}),P_{n}^{0})$
$\displaystyle=|z^{n}_{0}|^{2}\mathrm{e}^{-2t}+2(1-\sqrt{1-\mathrm{e}^{-2t}}-\frac{1}{2}\mathrm{e}^{-2t}),$
(3.6)

which gives the desired lower and upper bounds as before by using the
hypothesis on $z^{n}_{0}$.

_Total variation cutoff._ By using the comparison between total variation and
Hellinger distances (Lemma A.1) we deduce from (3.2) the cutoff in total
variation distance at the same critical time. The upper bound for the total
variation distance can alternatively be obtained by using the
$\mathrm{Kullback}$ estimate (3.3) and the Pinsker–Csiszár–Kullback inequality
(Lemma A.1). Since both distributions are tensor products, we could use
alternatively the tensorization property of the total variation distance
(Lemma A.4) together with the one-dimensional version of the Gaussian formula
for $\mathrm{Kullback}$ (Lemma A.1) to obtain the result for the total
variation. ∎

######  Remark 3.2 (Competition between bias and variance mixing).

From the computations of the proof of Theorem 1.2, we can show that for
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\chi^{2}\\}$

$A_{t}:=\mathrm{dist}(\mathrm{Law}(Z^{n}_{t})\mid\mathrm{Law}(Z^{n}_{t}-z^{n}_{0}\mathrm{e}^{-t}))$

has a cutoff at time $c_{n}^{A}=\log(\sqrt{n}|z^{n}_{0}|)$, while

$B_{t}:=\mathrm{dist}(\mathrm{Law}(Z^{n}_{t}-z^{n}_{0}\mathrm{e}^{-t})\mid
P_{n}^{0})$

admits a cutoff at time $c_{n}^{B}=\frac{1}{4}\log(n)$. The triangle
inequality for $\mathrm{dist}$ yields

$|A_{t}-B_{t}|\leq\mathrm{dist}(\mathrm{Law}(Z^{n}_{t})\mid P_{n}^{0})\leq
A_{t}+B_{t}.$

Therefore the critical time of Theorem 1.2 is dictated by either $A_{t}$ or
$B_{t}$, according to whether $c_{n}^{A}\gg c_{n}^{B}$ or $c_{n}^{A}\ll
c_{n}^{B}$. This can be seen as a competition between bias and variance
mixing.

######  Remark 3.3 (Total variation discriminating event for small initial
conditions).

Let us introduce the random variable $Z^{n}_{\infty}\sim
P_{n}^{0}=\mathcal{N}(0,\frac{1}{n}I_{n})=\mathcal{N}(0,\frac{1}{n})^{\otimes
n}$, in accordance with (3.1). There holds

$S_{t}^{n}:=\sum_{i=1}^{n}(Z_{t}^{n,i}-z_{0}^{n,i}\mathrm{e}^{-t})^{2}\sim\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2(1-\mathrm{e}^{-2t})}\Bigr{)}\quad\text{and}\quad|Z^{n}_{\infty}|^{2}\sim\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2}\Bigr{)}.$

We can check, using an explicit computation of Hellinger and Kullback between
Gamma distributions and the comparison between total variation and Hellinger
distances (Lemma A.1), that

$C_{t}:=\mathrm{dist}(\mathrm{Law}(S^{n}_{t})\mid\mathrm{Law}(|Z^{n}_{\infty}|^{2}))$

admits a cutoff at time $c_{n}^{C}=c_{n}^{B}=\frac{1}{4}\log(n)$. Moreover,
one can exhibit a discriminating event for the TV distance. Namely, we can
observe that

$\Bigr{\|}\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2(1-\mathrm{e}^{-2t})}\Bigr{)}-\mathrm{Gamma}\Bigr{(}\frac{n}{2},\frac{n}{2}\Bigr{)}\Bigr{\|}_{\mathrm{TV}}=\mathbb{P}(|Z^{n}_{\infty}|^{2}\geq\alpha_{t})-\mathbb{P}(S_{t}^{n}\geq\alpha_{t})$

with $\alpha_{t}$ the unique point where the two densities meet, which happens
to be

$\alpha_{t}=-\mathrm{e}^{2t}\log(1-\mathrm{e}^{-2t})(1-\mathrm{e}^{-2t}).$

From the explicit expressions (3.2), (3.3), (3.4), (3.5), (3.6), it is
immediate to extract the cutoff profile associated to the convergence of
$\mathrm{Law}(Z_{t}^{n})$ to $P_{n}^{0}$ in Hellinger, Kullback, $\chi^{2}$,
Fisher and Wasserstein. For Wasserstein we already know by Theorem 1.2 that a
cutoff occurs if and only if $|z_{0}^{n}|\underset{n\to\infty}{\to}\infty$. In
this case, regarding the profile, we have

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(Z_{t}^{n}),P_{n}^{0})=\phi(b),$
(3.7)

where for all $b\in\mathbb{R}$,

$t_{n,b}=\log|z_{0}^{n}|+b\quad\text{and}\quad\phi(b)=\mathrm{e}^{-b}.$ (3.8)

For the other distances and divergences, let us assume that the following
limit exists

$a:=\lim_{n\to\infty}\sqrt{n}|z_{0}^{n}|^{2}\in[0,+\infty].$ (3.9)

This quantity can be related with

$c_{n}^{A}:=\log(|z_{0}^{n}|\sqrt{n})\quad\text{and}\quad
c_{n}^{B}:=\frac{\log n}{4}$ (3.10)

which were already introduced in Remark 3.2. Indeed

$a=0\Longleftrightarrow c_{n}^{A}\ll c_{n}^{B},\quad
a=+\infty\Longleftrightarrow c_{n}^{A}\gg c_{n}^{B},$

while $a\in(0,\infty)$ is equivalent to $c_{n}^{A}\asymp c_{n}^{B}$.
Then, for
$\mathrm{dist}\in\\{\mathrm{Hellinger},\mathrm{Kullback},\chi^{2},\mathrm{Fisher}\\}$,
we have, for all $b\in\mathbb{R}$,

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(Z_{t_{n,b}})\mid
P_{n}^{0})=\phi(b),$ (3.11)

where $t_{n,b}$ and $\phi(b)$ are as in Table 1. The cutoff window is always
of size $1$.

| $a=+\infty$ | $a=0$ | $a\in(0,+\infty)$
---|---|---|---
$t_{n,b}$ |  |  |
Hellinger | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
Kullback | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
$\chi^{2}$ | $\log(|z_{0}^{n}|\sqrt{n})+b$ | $\frac{\log n}{4}+b$ | $\frac{\log n}{4}+b$
Fisher | $\log(|z_{0}^{n}|n)+b$ | $\frac{\log n}{2}+b$ | $\frac{\log n}{2}+b$
$\phi(b)$ |  |  |
Hellinger | $\sqrt{1-\mathrm{e}^{-\frac{1}{8}\mathrm{e}^{-2b}}}$ | $\sqrt{1-\mathrm{e}^{-\frac{1}{16}\mathrm{e}^{-4b}}}$ | $\sqrt{1-\mathrm{e}^{-\frac{1}{8}{a\mathrm{e}^{-2b}}-\frac{1}{16}\mathrm{e}^{-4b}}}$
Kullback | $\frac{1}{2}\mathrm{e}^{-2b}$ | $\frac{1}{4}\mathrm{e}^{-4b}$ | $\frac{1}{2}a\mathrm{e}^{-2b}+\frac{1}{4}\mathrm{e}^{-4b}$
$\chi^{2}$ | $\mathrm{e}^{\mathrm{e}^{-2b}}-1$ | $\mathrm{e}^{\frac{1}{2}\mathrm{e}^{-4b}}-1$ | $\mathrm{e}^{a\mathrm{e}^{-2b}+\frac{1}{2}\mathrm{e}^{-4b}}-1$
Fisher | $\mathrm{e}^{-2b}$ | $\mathrm{e}^{-4b}$ | $a\mathrm{e}^{-2b}+\mathrm{e}^{-4b}$

Table 1. Values of $t_{n,b}$ and $\phi(b)$ for the cutoff profile of the OU
process in (3.11).

Since the total variation distance is not expressed in a simple explicit
manner, further computations are needed to extract the precise cutoff profile,
which is given in the following lemma:

######  Lemma 3.4 (Cutoff profile in $\mathrm{TV}$ for OU).

Let $Z^{n}=(Z_{t}^{n})_{t\geq 0}$ be the OU process (1.8), started from
$z_{0}^{n}\in\mathbb{R}^{n}$, and let $P_{n}^{0}$ be its invariant law. Assume
as in (3.9) that $a:=\lim_{n\to\infty}|z_{0}^{n}|^{2}\sqrt{n}\in[0,+\infty]$,
and let $t_{n,b}$ be as in Table (1) for Hellinger. Then, for all
$b\in\mathbb{R}$, we have

$\lim_{n\to\infty}\|\mathrm{Law}({Z}_{t_{n,b}}^{n})-P_{n}^{0}\|_{\mathrm{TV}}=\phi(b),$

where

$\phi(b):=\begin{cases}\displaystyle\mathrm{erf}\Bigr{(}\frac{\mathrm{e}^{-b}}{2\sqrt{2}}\Bigr{)}&\text{if
$a=+\infty$}\\\\[10.00002pt]
\displaystyle\mathrm{erf}\Bigr{(}\frac{\mathrm{e}^{-2b}}{4}\Bigr{)}&\text{if
$a=0$}\\\\[10.00002pt]
\displaystyle\mathrm{erf}\Bigr{(}\frac{\sqrt{2a\mathrm{e}^{-2b}+\mathrm{e}^{-4b}}}{4}\Bigr{)}&\text{if
$a\in(0,+\infty)$}\end{cases},$

where $\displaystyle\mathrm{erf}(u):=\frac{1}{\sqrt{\pi}}\int_{|t|\leq
u}\mathrm{e}^{-t^{2}}\mathrm{d}t=\mathbb{P}(|X|\leq\sqrt{2}u)$ with
$X\sim\mathcal{N}(0,1)$ is the _error function_.

###### Proof of Lemma 3.4.

The idea is to exploit the fact that we consider Gaussian product measures
(the covariance matrices are multiple of the identity), which allows a finer
analysis than for instance in [27, Le. 3.1]. We begin with a rather general
step. Let $\mu$ and $\nu$ be two probability measures on $\mathbb{R}^{n}$ with
densities $f$ and $g$ with respect to the Lebesgue measure $\mathrm{d}x$. We
have then

$\|\mu-\nu\|_{\mathrm{TV}}=\frac{1}{2}\int|f-g|\mathrm{d}x=\frac{1}{2}\Bigr{(}\int(f-g)\mathbf{1}_{g\leq
f}\mathrm{d}x-\int(f-g)\mathbf{1}_{f\leq g}\mathrm{d}x\Bigr{)},$ (3.12)

and since

$-\int(f-g)\mathbf{1}_{f\leq
g}\mathrm{d}x=-\int(f-g)(1-\mathbf{1}_{g<f})\mathrm{d}x=\int(f-g)\mathbf{1}_{g<f}\mathrm{d}x=\int(f-g)\mathbf{1}_{g\leq
f}\mathrm{d}x,$

we obtain

$\|\mu-\nu\|_{\mathrm{TV}}=\int(f-g)\mathbf{1}_{g\leq f}\mathrm{d}x=\mu(g\leq
f)-\nu(g\leq f).$ (3.13)

In particular, when $\mu=\mathcal{N}(m_{1},\sigma_{1}^{2}I_{n})$ and
$\nu=\mathcal{N}(m_{2},\sigma_{2}^{2}I_{n})$ then $g(x)\leq f(x)$ is
equivalent to

$\psi(x):=\frac{|x-m_{1}|^{2}}{\sigma_{1}^{2}}-\frac{|x-m_{2}|^{2}}{\sigma_{2}^{2}}\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}},$ (3.14)

for all $x\in\mathbb{R}^{n}$, and therefore, with $Z_{1}\sim\mu$ and
$Z_{2}\sim\nu$, we get

$\|\mu-\nu\|_{\mathrm{TV}}=\mathbb{P}\Bigr{(}\psi(Z_{1})\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\Bigr{)}-\mathbb{P}\Bigr{(}\psi(Z_{2})\leq
n\log\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\Bigr{)}.$ (3.15)

Let us assume from now on that $m_{2}=0$ and $\sigma_{1}\neq\sigma_{2}$. We
can then gather the quadratic terms as

$\psi(x)=\Bigr{(}1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}\Bigr{)}\frac{|x-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}+\Bigr{(}\frac{1}{\sigma_{1}^{2}}-\frac{1}{(1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}})\sigma_{1}^{2}}\Bigr{)}|m_{1}|^{2}\quad\text{where}\quad\tilde{m}_{1}:=\frac{1}{1-\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}}m_{1}.$
(3.16)

We observe at this step that the random variable
$\frac{|Z_{1}-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}$ follows a noncentral chi-
squared distribution, which depends only on $n$ and on the noncentrality
parameter

$\lambda_{1}:=\frac{|m_{1}-\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}=\frac{\sigma_{1}^{2}}{(\sigma_{2}^{2}-\sigma_{1}^{2})^{2}}|m_{1}|^{2}.$
(3.17)

Similarly, the random variable
$\frac{|Z_{2}-\tilde{m}_{1}|^{2}}{\sigma_{2}^{2}}$ follows a noncentral chi-
squared distribution, which depends only on $n$ and on the noncentrality
parameter

$\lambda_{2}:=\frac{|\tilde{m}_{1}|^{2}}{\sigma_{1}^{2}}=\frac{\sigma_{2}^{4}}{\sigma_{1}^{2}(\sigma_{2}^{2}-\sigma_{1}^{2})^{2}}|m_{1}|^{2}.$
(3.18)

It follows that the law of $\psi(Z_{1})$ and the law of $\psi(Z_{2})$ depend
over $m_{1}$ only via $|m_{1}|$. Hence

$\psi(Z_{1})\overset{\mathrm{d}}{=}X_{1}+\cdots+X_{n}\quad\text{and}\quad\psi(Z_{2})\overset{\mathrm{d}}{=}Y_{1}+\cdots+Y_{n}$
(3.19)

where $X_{1},\ldots,X_{n}$ and $Y_{1},\ldots,Y_{n}$ are two sequences of
i.i.d. random variables for which the mean and variance depends only (and
explicitly) on $|m_{1}|$, $\sigma_{1}$, $\sigma_{2}$. Note in particular that
these means and variances are given by $\frac{1}{n}$ the ones of $\psi(Z_{1})$
and $\psi(Z_{2})$. Now we specialize to the case where
$\mu=\mathrm{Law}(Z^{n}_{t})=\mathcal{N}(z_{0}^{n}\mathrm{e}^{-t},\frac{1-\mathrm{e}^{-2t}}{n}I_{n})$
and
$\nu=\mathrm{Law}(Z^{n}_{\infty})=\mathcal{N}(0,\frac{1}{n}I_{n})=P_{n}^{0}$,
and we find

$\mathbb{E}[\psi(Z_{1})]=n\Bigr{(}1-\frac{\sigma_{t}^{2}}{\sigma_{\infty}^{2}}\Bigr{)}-\frac{|z_{0}^{n}|^{2}\mathrm{e}^{-2t}}{\sigma_{\infty}^{2}},\quad\mathbb{E}[\psi(Z_{2})]=n\Bigr{(}\frac{\sigma_{\infty}^{2}}{\sigma_{t}^{2}}-1\Bigr{)}+\frac{|z_{0}^{n}|^{2}\mathrm{e}^{-2t}}{\sigma_{t}^{2}}$

while

$\mathrm{Var}[\psi(Z_{1})]=2n\Bigr{(}\frac{1}{\sigma_{t}^{2}}-\frac{1}{\sigma_{\infty}^{2}}\Bigr{)}^{2}\sigma_{t}^{4}+4\frac{\sigma_{t}^{2}}{\sigma_{\infty}^{4}}|z_{0}^{n}|^{2}\mathrm{e}^{-2t},\quad\mathrm{Var}[\psi(Z_{2})]=2n\Bigr{(}\frac{1}{\sigma_{t}^{2}}-\frac{1}{\sigma_{\infty}^{2}}\Bigr{)}^{2}\sigma_{\infty}^{4}+4\frac{\sigma_{\infty}^{2}}{\sigma_{t}^{4}}|z_{0}^{n}|^{2}\mathrm{e}^{-2t}.$

Let $t=t_{n,b}$ be as in Table 1 for Hellinger. Using (3.15) and the central
limit theorem for the i.i.d. random variables $X_{1},\ldots,X_{n}$ and
$Y_{1},\ldots,Y_{n}$, we get, with $Z\sim\mathcal{N}(0,1)$,

$\bigr{\|}\mathrm{Law}({Z}_{t}^{n})-P_{n}^{0}\bigr{\|}_{\mathrm{TV}}=\mathbb{P}(Z\leq\gamma_{n,t})-\mathbb{P}(Z\leq\tilde{\gamma}_{n,t})+o_{n}(1),$

where

$\gamma_{n,t}:=\frac{-n\log(1-\mathrm{e}^{-2t})-\mathbb{E}[\psi(Z^{n}_{t})]}{\sqrt{\mathrm{Var}[\psi(Z^{n}_{t})]}},\quad\tilde{\gamma}_{n,t}:=\frac{-n\log(1-\mathrm{e}^{-2t})-\mathbb{E}[\psi(Z^{n}_{\infty})]}{\sqrt{\mathrm{Var}[\psi(Z^{n}_{\infty})]}}.$

Expanding $\gamma_{n,t_{n,b}}$ gives the cutoff profile. Let us detail the
computations in the most involved case
$\lim_{n\to\infty}|z_{0}^{n}|^{2}\sqrt{n}=a\in(0,+\infty)$. For all
$b\in\mathbb{R}$, recall $t_{n,b}=\frac{\log n}{4}+b$. One may check that

$-n\log(1-\mathrm{e}^{-2t_{n,b}})-\mathbb{E}[\psi(Z^{n}_{t_{n,b}})]=\frac{1}{2}\mathrm{e}^{-4b}+a\mathrm{e}^{-2b}+o_{n}(1),$
$-n\log(1-\mathrm{e}^{-2t_{n,b}})-\mathbb{E}[\psi(Z^{n}_{\infty})]=-\frac{1}{2}\mathrm{e}^{-4b}-a\mathrm{e}^{-2b}+o_{n}(1),$
$\mathrm{Var}[\psi(Z^{n}_{t_{n,b}})]=2\mathrm{e}^{-4b}+4a\mathrm{e}^{-2b}+o_{n}(1),\quad\mathrm{Var}[\psi(Z^{n}_{\infty})]=2\mathrm{e}^{-4b}+4a\mathrm{e}^{-2b}+o_{n}(1).$

It follows that

$\lim_{n\to\infty}\bigr{\|}\mathrm{Law}({Z}_{t_{n,b}}^{n})-P_{n}^{0}\bigr{\|}_{\mathrm{TV}}=\mathbb{P}\Bigr{(}|Z|\leq\frac{1}{2\sqrt{2}}\sqrt{\mathrm{e}^{-4b}+2a\mathrm{e}^{-2b}}\Bigr{)}=\mathrm{erf}\Bigr{(}\frac{1}{4}\sqrt{\mathrm{e}^{-4b}+2a\mathrm{e}^{-2b}}\Bigr{)}.$

The other cases are similar. ∎

##  4\. General exactly solvable aspects

In this section, we prove Theorem 1.3 and Corollary 1.4.

The proof of Theorem 1.3 is based on the fact that the polynomial functions
$\pi(x)=x_{1}+\cdots+x_{n}$ and $|x|^{2}=x_{1}^{2}+\cdots+x_{n}^{2}$ are, up
to an additive constant for the second, eigenfunctions of the dynamics
associated to the spectral values $-1$ and $-2$ respectively, and that their
“carré du champ” is affine. In the matrix cases $\beta\in\\{1,2\\}$, these
functions correspond to the dynamics of the trace, the dynamics of the squared
Hilbert–Schmidt trace norm, and the dynamics of the squared trace. It is
remarkable that this phenomenon survives beyond these matrix cases, yet
another manifestation of the Gaussian “ghosts” concept due to Edelman, see for
instance [34].

###### Proof of Theorem 1.3.

The process $Y_{t}:=\pi(X_{t}^{n})$ solves

$\mathrm{d}Y_{t}=\sum_{i=1}^{n}\mathrm{d}X_{t}^{n,i}=\sqrt{\frac{2}{n}}\sum_{i=1}^{n}\mathrm{d}B^{i}_{t}-\sum_{i=1}^{n}X_{t}^{n,i}\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}}.$

By symmetry, the double sum vanishes. Note that the process
$W_{t}:=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}B^{i}_{t}$ is a standard one
dimensional BM, so that
$\mathrm{d}Y_{t}=\sqrt{2}\mathrm{d}W_{t}-Y_{t}\mathrm{d}t.$ This proves the
first part of the statement.
We turn to the second part. Recall that $X_{t}\in D_{n}$ for all $t>0$. By
Itô’s formula

$\mathrm{d}(X_{t}^{n,i})^{2}=\sqrt{\frac{8}{n}}X_{t}^{n,i}\mathrm{d}B^{i}_{t}-2(X_{t}^{n,i})^{2}\mathrm{d}t+2\frac{\beta}{n}X_{t}^{n,i}\sum_{j:j\neq
i}\frac{\mathrm{d}t}{X_{t}^{n,i}-X_{t}^{n,j}}+\frac{2}{n}\mathrm{d}t.$

Set
$W_{t}:=\sum_{i=1}^{n}\int_{0}^{t}\frac{X_{s}^{n,i}}{|X^{n}_{s}|}\mathrm{d}B^{i}_{s}$.
The process ${(W_{t})}_{t\geq 0}$ is a BM by the Lévy characterization since

$\langle
W\rangle_{t}=\int_{0}^{t}\frac{\sum_{i=1}^{n}(X^{n,i}_{s})^{2}}{|X^{n}_{s}|^{2}}\mathrm{d}s=t.$

Furthermore, a simple computation shows that

$\sum_{i=1}^{n}X_{t}^{n,i}\sum_{j:j\neq
i}\frac{1}{X_{t}^{n,i}-X_{t}^{n,j}}=\frac{n(n-1)}{2}.$

Consequently the process $R_{t}:=|X_{t}^{n}|^{2}$ solves

$\mathrm{d}R_{t}=\sqrt{\frac{8}{n}R_{t}}\mathrm{d}W_{t}+\Big{(}2+\beta(n-1)-2R_{t}\Big{)}\mathrm{d}t,$

and is therefore a CIR process of parameters $a=2+\beta(n-1)$, $b=2$, and
$\sigma=\sqrt{8/n}$.
When $d=\frac{\beta}{2}n^{2}+(1-\frac{\beta}{2})n$ is a positive integer, the
last property of the statement follows from the connection between OU and CIR
recalled right before the statement of the theorem. ∎

The last proof actually relies on the following general observation. Let $X$
be an $n$-dimensional continuous semi-martingale solution of

$\mathrm{d}X_{t}=\sigma(X_{t})\mathrm{d}B_{t}+b(X_{t})\mathrm{d}t$

where $B$ is a $n$-dimensional standard BM, and where

$x\in\mathbb{R}^{n}\mapsto\sigma(x)\in\mathcal{M}_{n,n}(\mathbb{R})\quad\text{and}\quad
x\in\mathbb{R}^{n}\mapsto b(x)\in\mathbb{R}^{n}$

are Lipschitz. The infinitesimal generator of the Markov semigroup is given by

$\operatorname{G}(f)(x)=\frac{1}{2}\sum_{i,j=1}^{n}a_{i,j}(x)\partial_{i,j}f(x)+\sum_{i=1}^{n}b_{i}(x)\partial_{i}f(x),\quad\text{where}\quad
a(x)=\sigma(x)(\sigma(x))^{\top},$

for all $f\in\mathcal{C}^{2}(\mathbb{R}^{n},\mathbb{R})$ and
$x\in\mathbb{R}^{n}$. Then, by Itô’s formula, the process
$M^{f}={(M^{f}_{t})}_{t\geq 0}$ given by

$M^{f}_{t}=f(X_{t})-f(X_{0})-\int_{0}^{t}(\operatorname{G}f)(X_{s})\mathrm{d}s=\sum_{i,k=1}^{n}\int_{0}^{t}\partial_{i}f(X_{s})\sigma_{i,k}(X_{s})\mathrm{d}B_{s}^{k}$

is a local martingale, and moreover, for all $t\geq 0$,

$\langle
M^{f}\rangle_{t}=\int_{0}^{t}\Gamma(f)(X_{s})\mathrm{d}s\quad\text{where}\quad\Gamma(f)(x)=|\sigma(x)^{\top}\nabla
f(x)|^{2}=a(x)\nabla f\cdot\nabla f.$

The functional quadratic form $\Gamma$ is known as the “carré du champ”
operator.
If $f$ is an eigenfunction of $\operatorname{G}$ associated to the spectral
value $\lambda$ in the sense that $\operatorname{G}f=\lambda f$ (note by the
way that $\lambda\leq 0$ since $\operatorname{G}$ generates a Markov process),
then we get

$f(X_{t})=f(X_{0})+\lambda\int_{0}^{t}f(X_{s})\mathrm{d}s+M_{t}^{f},\quad\text{in
other words}\quad\mathrm{d}f(X_{t})=\mathrm{d}M^{f}_{t}+\lambda
f(X_{t})\mathrm{d}t.$

Now if $\Gamma(f)=c$ (as in the first part of the theorem), then by the Lévy
characterization of Brownian motion, the continuous local martingale
$W:=\frac{1}{\sqrt{c}}M^{f}$ starting from the origin is a standard BM and we
recover the result of the first part of the theorem. On the other hand, if
$\Gamma(f)=cf$ (as in the second part of the theorem), then by the Lévy
characterization of BM the local martingale

$W:=\int_{0}^{t}\frac{1}{\sqrt{cf(X_{s})}}\mathrm{d}M_{s}^{f}$

is a standard BM and we recover the result of the second part.

At this point, we observe that the infinitesimal generator of the CIR process
$R$ is the Laguerre partial differential operator

$L(f)(x)=\frac{4}{n}xf^{\prime\prime}(x)+(2+\beta(n-1)-2x)f^{\prime}(x).$
(4.1)

This operator leaves invariant the set of polynomials of degree less than or
equal to $k$, for all integer $k\geq 0$, a property inherited from (2.5). We
will use this property in the following proof.

###  4.1. Proof of Corollary 1.4

By Theorem 1.3, $Z=\pi(X^{n})$ is an OU process in $\mathbb{R}$ solution of
the stochastic differential equation

$Z_{0}=\pi(X_{0}^{n}),\quad\mathrm{d}Z_{t}=\sqrt{2}\mathrm{d}B_{t}-Z_{t}\mathrm{d}t,$

where $B$ is a standard one-dimensional BM. By Lemma 3.1,
$Z_{t}\sim\mathcal{N}(Z_{0}\mathrm{e}^{-t},1-\mathrm{e}^{-2t})$ for all $t\geq
0$ and the equilibrium distribution is
$P_{n}^{\beta}\circ\pi^{-1}=\mathcal{N}(0,1)$. Using the contraction property
stated in Lemma A.2, the comparison between Hellinger and TV of Lemma A.1 and
the explicit expressions for Gaussian distributions of Lemma A.5, we find

$\displaystyle\|\mathrm{Law}(X^{n}_{t})-P_{n}^{\beta}\|_{\mathrm{TV}}$
$\displaystyle\geq\|\mathrm{Law}(Z_{t})-P_{n}^{\beta}\circ\pi^{-1}\|_{\mathrm{TV}}$
$\displaystyle\geq\mathrm{Hellinger}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=1-\frac{(1-\mathrm{e}^{-2t})^{1/4}}{(1-\frac{1}{2}\mathrm{e}^{-2t})^{1/2}}\exp\Bigr{(}-\frac{\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}}{4(2-\mathrm{e}^{-2t})}\Bigr{)}.$

Setting $c_{n}:=\log(|\pi(X^{n}_{0})|)$ and assuming that
$\lim_{n\to\infty}c_{n}=\infty$, we deduce that for all $\varepsilon\in(0,1)$

$\lim_{n\to\infty}\|\mathrm{Law}(X^{n}_{c_{n}(1-\varepsilon)})-P_{n}^{\beta}\|_{\mathrm{TV}}=1.$

The comparison between $\mathrm{Hellinger}$ and $\mathrm{TV}$ of Lemma A.1
allows to deduce that this remains true for the Hellinger distance.

We turn to Kullback. The contraction property stated in Lemma A.2 and the
explicit expressions for Gaussian distributions of Lemma A.5 yield

$\displaystyle 2\mathrm{Kullback}(\mathrm{Law}(X^{n}_{t})\mid P_{n}^{\beta})$
$\displaystyle\geq 2\mathrm{Kullback}(\mathrm{Law}(Z_{t})\mid
P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}-\mathrm{e}^{-2t}-\log(1-\mathrm{e}^{-2t}).$

This is enough to deduce that

$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=+\infty.$

The situation is similar for $\chi^{2}$: the contraction property stated in
Lemma A.2 and the explicit expressions for Gaussian distributions of Lemma A.5
yield

$\displaystyle\chi^{2}(\mathrm{Law}(X^{n}_{t})\mid P_{n}^{\beta})$
$\displaystyle\geq\chi^{2}(\mathrm{Law}(Z_{t})\mid
P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=-1+\frac{1}{\sqrt{1-\mathrm{e}^{-4t}}}\exp\left(\frac{1}{1+\mathrm{e}^{-2t}}(1-\pi(X^{n}_{0})\mathrm{e}^{-t})^{2}\right),$

so that

$\lim_{n\to\infty}\chi^{2}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=+\infty.$

Regarding the Wasserstein distance, we have
$\left\|\pi\right\|_{\mathrm{Lip}}:=\sup_{x\neq
y}\frac{|\pi(x)-\pi(y)|}{|x-y|}\leq\sqrt{n}$ from the Cauchy–Schwarz
inequality, and by Lemma A.2, for all probability measures $\mu$ and $\nu$ on
$\mathbb{R}^{n}$,

$\mathrm{Wasserstein}(\mu\circ\pi^{-1},\nu\circ\pi^{-1})\leq\sqrt{n}\mathrm{Wasserstein}(\mu,\nu).$
(4.2)

Using the explicit expressions for Gaussian distributions of Lemma A.5, we
thus find

$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})$
$\displaystyle\geq\frac{1}{n}\mathrm{Wasserstein}^{2}(\mathrm{Law}(Z_{t}),P_{n}^{\beta}\circ\pi^{-1})$
$\displaystyle=\frac{1}{n}\Bigr{(}\pi(X^{n}_{0})^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}\Bigr{)}.$

Setting $c_{n}:=\log\Big{(}\frac{|\pi(x_{0}^{n})|}{\sqrt{n}}\Big{)}$ and
assuming $c_{n}\to\infty$ as $n\to\infty$, we thus deduce that for all
$\varepsilon\in(0,1)$

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}}),P_{n}^{\beta})=+\infty.$

##  5\. The random matrix cases

In this section, we prove Theorem 1.5 and Corollary 1.6 that cover the matrix
cases $\beta\in\\{1,2\\}$. For these values of $\beta$, the DOU process is the
image by the spectral map of a matrix OU process, connected to the random
matrix models $\mathrm{GOE}$ and $\mathrm{GUE}$. We could consider the case
$\beta=4$ related to $\mathrm{GSE}$. Beyond these three algebraic cases, it
could be possible for an arbitrary $\beta\geq 1$ to use random tridiagonal
matrices dynamics associated to $\beta$ Dyson processes, see for instance
[44].

The next two subsections are devoted to the proof of Theorem 1.5 in the
$\beta=2$ and $\beta=1$ cases respectively. The third section provides the
proof of Corollary 1.6.

###  5.1. Hermitian case ($\beta=2$)

Let $\mathrm{Herm}_{n}$ be the set of $n\times n$ complex Hermitian matrices,
namely the set of $h\in\mathcal{M}_{n,n}(\mathbb{C})$ with
$h_{i,j}=\overline{h_{j,i}}$ for all $1\leq i,j\leq n$. An element
$h\in\mathrm{Herm}_{n}$ is parametrized by the $n^{2}$ real variables
$(h_{i,i})_{1\leq i\leq n}$, $(\Re h_{i,j})_{1\leq i<j\leq n}$, $(\Im
h_{i,j})_{1\leq i<j\leq n}$. We define, for $h\in\mathrm{Herm}_{n}$ and $1\leq
i,j\leq n$,

$\pi_{i,j}(h)=\begin{cases}h_{i,i}&\text{if $i=j$}\\\ \sqrt{2}\,\Re
h_{i,j}&\text{if $i<j$}\\\ \sqrt{2}\,\Im h_{j,i}&\text{if $i>j$}\end{cases}.$
(5.1)

Note that

$\mathrm{Tr}(h^{2})=\sum_{i,j=1}^{n}|h_{i,j}|^{2}=\sum_{i=1}^{n}h_{i,i}^{2}+2\sum_{i<j}(\Re
h_{i,j})^{2}+2\sum_{i<j}(\Im h_{i,j})^{2}=\sum_{i,j}\pi_{i,j}(h)^{2}.$

We thus identify $\mathrm{Herm}_{n}$ with
$\mathbb{R}^{n}\times\mathbb{R}^{2\frac{n^{2}-n}{2}}=\mathbb{R}^{n^{2}}$, this
identification is isometrical provided $\mathrm{Herm}_{n}$ is endowed with the
norm $\sqrt{\mathrm{Tr}(h^{2})}$ and $\mathbb{R}^{n^{2}}$ with the Euclidean
norm.

The Gaussian Unitary Ensemble $\mathrm{GUE}_{n}$ is the Gaussian law on
$\mathrm{Herm}_{n}$ with density

$h\in\mathrm{Herm}_{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(h^{2})}}{C_{n}}\quad\text{where}\quad
C_{n}:=\int_{\mathbb{R}^{n^{2}}}\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(h^{2})}\prod_{i=1}^{n}\mathrm{d}h_{i,i}\prod_{i<j}\mathrm{d}\Re
h_{i,j}\prod_{i<j}\mathrm{d}\Im h_{i,j}.$ (5.2)

If $H$ is a random $n\times n$ Hermitian matrix then $H\sim\mathrm{GUE}_{n}$
if and only if the $n^{2}$ real random variables $\pi_{i,j}(H)$, $1\leq
i,j\leq n$, are independent Gaussian random variables with

$\pi_{i,j}(H)\sim\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)},\quad 1\leq i,j\leq
n.$ (5.3)

The law $\mathrm{GUE}_{n}$ is the unique invariant law of the Hermitian matrix
OU process ${(H_{t})}_{t\geq 0}$ on $\mathrm{Herm}_{n}$ solution of the
stochastic differential equation

$H_{0}=h_{0}\in\mathrm{Herm}_{n},\quad\mathrm{d}H_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-H_{t}\mathrm{d}t,$
(5.4)

where $B={(B_{t})}_{t\geq 0}$ is a Brownian motion on $\mathrm{Herm}_{n}$, in
the sense that the stochastic processes $(\pi_{i,j}(B_{t}))_{t\geq 0}$, $1\leq
i\neq j\leq n$, are independent standard one-dimensional BM. The coordinates
stochastic processes ${(\pi_{i,j}(H_{t}))}_{t\geq 0}$, $1\leq i,j\leq n$, are
independent real OU processes.

For any $h$ in $\mathrm{Herm}_{n}$, we denote by $\Lambda(h)$ the vector of
the eigenvalues of $h$ ordered in non-decreasing order. Lemma 5.1 below is an
observation which dates back to the seminal work of Dyson [33], hence the name
DOU for $X^{n}$. We refer to [37, Ch. 12] and [2, Sec. 4.3] for a mathematical
approach using modern stochastic calculus.

######  Lemma 5.1 (From matrix OU to DOU).

The image of $\mathrm{GUE}_{n}$ by the map $\Lambda$ is the Coulomb gas
$P_{n}^{\beta}$ given by (1.6) with $\beta=2$. Moreover the stochastic process
$X^{n}={(X^{n}_{t})}_{t\geq 0}={(\Lambda(H_{t}))}_{t\geq 0}$ is well-defined
and solves the stochastic differential equation (1.3) with $\beta=2$ and
$x_{0}^{n}=\Lambda(h_{0})$.

Let $\beta=2$. Let us assume from now on that the initial value
$h_{0}\in\mathrm{Herm}_{n}$ of ${(H_{t})}_{t\geq 0}$ has eigenvalues
$x_{0}^{n}$ where $x^{n}_{0}$ is as in Theorem 1.5. We start by proving the
upper bound on the $\chi^{2}$ distance stated in Theorem 1.5: it will be an
adaptation of the proof of the upper bound of Theorem 1.1 applied to the
Hermitian matrix OU process ${(H_{t})}_{t\geq 0}$ combined with the
contraction property of the $\chi^{2}$ distance. Indeed, by Lemma 5.1 and the
contraction property of Lemma A.2

$\chi^{2}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GUE}_{n}).$ (5.5)

We claim now that the right-hand side tends to $0$ as $n\to\infty$ when
$t=t_{n}$ is well chosen. Indeed, using the identification between
$\mathrm{Herm}_{n}$ and $\mathbb{R}^{n^{2}}$ mentioned earlier, we have
$\mathrm{GUE}_{n}=\mathcal{N}(m_{2},\Sigma_{2})$ where $m_{2}=0$ and where
$\Sigma_{2}$ is an $n^{2}\times n^{2}$ diagonal matrix with

$(\Sigma_{2})_{(i,j),(i,j)}=\frac{1}{n}.$ (5.6)

On the other hand, the Mehler formula (Lemma 3.1) gives
$\mathrm{Law}(H_{t})=\mathcal{N}(m_{1},\Sigma_{1})$ where
$m_{1}=\mathrm{e}^{-t}h_{0}$ and where $\Sigma_{1}$ is an $n^{2}\times n^{2}$
diagonal matrix with

$(\Sigma_{1})_{(i,j),(i,j)}=\frac{1-\mathrm{e}^{-2t}}{n}.$ (5.7)

Therefore, using Lemma A.5, the analogue of (3.3) reads

$\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GUE}_{n})=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{n^{2}/2}}\exp\left(n|h_{0}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right).$
(5.8)

where

$|h_{0}|^{2}=\sum_{1\leq i,j\leq n}\pi_{i,j}(h_{0})^{2}=\sum_{1\leq i,j\leq
n}|(h_{0})_{i,j}|^{2}=\mathrm{Tr}(h_{0}^{2})=|x_{0}^{n}|^{2}.$ (5.9)

Taking now $c_{n}:=\log(\sqrt{n}|x_{0}^{n}|)\vee\log(\sqrt{n})$, for any
$\varepsilon\in(0,1)$, we get

$\chi^{2}(\mathrm{Law}(X_{(1+\varepsilon)c_{n}}^{n})\mid
P_{n}^{\beta})\leq\chi^{2}(\mathrm{Law}(H_{(1+\varepsilon)c_{n}})\mid\mathrm{GUE}_{n})\underset{n\to\infty}{\longrightarrow}0.$
(5.10)

In the right-hand side of (5.8), the factor $n^{2}$ is the dimension of the
$\mathbb{R}^{n^{2}}$ to which $\mathrm{Herm}_{n}$ is identified, while the
factor $n$ in the first term is due to the $1/n$ scaling in the stochastic
differential equation of the process. This explains the difference with the
analogue (3.3) in dimension $n$.

From the comparison between TV, Hellinger, Kullback and $\chi^{2}$ stated in
Lemma A.1, we easily deduce that the previous convergence remains true upon
replacing $\chi^{2}$ by $\mathrm{TV}$, $\mathrm{Hellinger}$ or
$\mathrm{Kullback}$.

It remains to cover the upper bound for the Wasserstein distance. This
distance is more sensitive to contraction arguments: according to Lemma A.2,
one needs to control the Lipschitz norm of the “contraction map” at stake. It
happens that the spectral map, restricted to the set $\mathrm{Herm}_{n}$ of
$n\times n$ Hermitian matrices, is $1$-Lipschitz: more precisely, the
Hoffman–Wielandt inequality, see [43] and [45, Th. 6.3.5], asserts that for
any two such matrices $A$ and $B$, denoting
$\Lambda(A)=(\lambda_{i}(A))_{1\leq i\leq n}$ and
$\Lambda(B)=(\lambda_{i}(B))_{1\leq i\leq n}$ the ordered sequences of their
eigenvalues, we have

$\sum_{i=1}^{n}|\lambda_{i}(A)-\lambda_{i}(B)|^{2}\leq\sum_{i,j}|A_{i,j}-B_{i,j}|^{2}.$

Applying Lemma A.2, we thus deduce that

$\mathrm{Wasserstein}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})\leq\mathrm{Wasserstein}(\mathrm{Law}(H_{t}),\mathrm{GUE}_{n}).$
(5.11)

Following the Gaussian computations in the proof of Theorem 1.2, we obtain

$\mathrm{Wasserstein}^{2}(\mathrm{Law}(H_{t}),\mathrm{GUE}_{n})=|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$
(5.12)

Set $c_{n}:=\log(|x_{0}^{n}|)$. If $c_{n}\to\infty$ as $n\to\infty$ then for
all $\varepsilon\in(0,1)$ we find

$\mathrm{Wasserstein}(\mathrm{Law}(X_{(1+\varepsilon)c_{n}}^{n}),P_{n}^{\beta})\underset{n\to\infty}{\longrightarrow}0.$

This completes the proof of Theorem 1.5.

###  5.2. Symmetric case ($\beta=1$)

The method is similar to the case $\beta=2$. Let us focus only on the
differences. Let $\mathrm{Sym}_{n}$ be the set of $n\times n$ real symmetric
matrices, namely the set of $s\in\mathcal{M}_{n,n}(\mathbb{R})$ with
$s_{i,j}=s_{j,i}$ for all $1\leq i,j\leq n$. An element $s\in\mathrm{Sym}_{n}$
is parametrized by the $n+\frac{n^{2}-n}{2}=\frac{n(n+1)}{2}$ real variables
$(s_{i,j})_{1\leq i\leq j\leq n}$. We define, for $s\in\mathrm{Sym}_{n}$ and
$1\leq i\leq j\leq n$,

$\pi_{i,j}(s)=\begin{cases}s_{i,i}&\text{if $i=j$}\\\
\sqrt{2}\,s_{i,j}&\text{if $i<j$}\end{cases}.$ (5.13)

Note that

$\mathrm{Tr}(s^{2})=\sum_{i,j=1}^{n}s_{i,j}^{2}=\sum_{i=1}^{n}s_{i,i}^{2}+2\sum_{i<j}s_{i,j}^{2}=\sum_{1\leq
i\leq j\leq n}\pi_{i,j}(s)^{2}.$

We thus identify isometrically $\mathrm{Sym}_{n}$, endowed with the norm
$\sqrt{\mathrm{Tr}(h^{2})}$, with
$\mathbb{R}^{n}\times\mathbb{R}^{\frac{n^{2}-n}{2}}=\mathbb{R}^{\frac{n(n+1)}{2}}$
endowed with the Euclidean norm.

The Gaussian Orthogonal Ensemble $\mathrm{GOE}_{n}$ is the Gaussian law on
$\mathrm{Sym}_{n}$ with density

$s\in\mathrm{Sym}_{n}\mapsto\frac{\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(s^{2})}}{C_{n}}\quad\text{where}\quad
C_{n}:=\int_{\mathbb{R}^{\frac{n(n+1)}{2}}}\mathrm{e}^{-\frac{n}{2}\mathrm{Tr}(s^{2})}\prod_{1\leq
i\leq j\leq n}\mathrm{d}s_{i,j}.$ (5.14)

If $S$ is a random $n\times n$ real symmetric matrix then
$S\sim\mathrm{GOE}_{n}$ if and only if the $\frac{n(n+1)}{2}$ real random
variables $\pi_{i,j}(S)$, $1\leq i\leq j\leq n$, are independent Gaussian
random variables with

$\pi_{i,j}(S)\sim\mathcal{N}\Bigr{(}0,\frac{1}{n}\Bigr{)},\quad 1\leq i\leq
j\leq n.$ (5.15)

The law $\mathrm{GOE}_{n}$ is the unique invariant law of the real symmetric
matrix OU process ${(S_{t})}_{t\geq 0}$ on $\mathrm{Sym}_{n}$ solution of the
stochastic differential equation

$S_{0}=s_{0}\in\mathrm{Sym}_{n},\quad\mathrm{d}S_{t}=\sqrt{\frac{2}{n}}\mathrm{d}B_{t}-S_{t}\mathrm{d}t$
(5.16)

where $B={(B_{t})}_{t\geq 0}$ is a Brownian motion on $\mathrm{Sym}_{n}$, in
the sense that the stochastic processes $(\pi_{i,j}(B_{t}))_{t\geq 0}$, $1\leq
i\leq j\leq n$, are independent standard one-dimensional BM. The coordinates
stochastic processes ${(\pi_{i,j}(S_{t}))}_{t\geq 0}$, $1\leq i\leq j\leq n$,
are independent real OU processes.

For any $s$ in $\mathrm{Sym}_{n}$, we denote by $\Lambda(s)$ the vector of the
eigenvalues of $s$ ordered in non-decreasing order. Lemma 5.2 below is the
real symmetric analogue of Lemma 5.1.

######  Lemma 5.2 (From matrix OU to DOU).

The image of $\mathrm{GOE}_{n}$ by the map $\Lambda$ is the Coulomb gas
$P_{n}^{\beta}$ given by (1.6) with $\beta=1$. Moreover the stochastic process
$X^{n}={(X^{n}_{t})}_{t\geq 0}={(\Lambda(S_{t}))}_{t\geq 0}$ is well-defined
and solves the stochastic differential equation (1.3) with $\beta=1$ and
$x_{0}^{n}=\Lambda(s_{0})$.

As for the case $\beta=2$, the idea now is that the DOU process is sandwiched
between a real OU process and a matrix OU process.

By similar computations to the case $\beta=2$, the analogue of (5.8) becomes

$\chi^{2}(\mathrm{Law}(H_{t})\mid\mathrm{GOE}_{n})=-1+\frac{1}{(1-\mathrm{e}^{-4t})^{\frac{(n(n+1))^{2}}{8}}}\exp\left(n|h_{0}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right).$
(5.17)

This allows to deduce the upper bound for TV, Hellinger, Kullback and
$\chi^{2}$. Regarding the Wasserstein distance, the analogue of (5.12) reads

$\mathrm{Wasserstein}^{2}(\mathrm{Law}(S_{t}),\mathrm{GOE}_{n})=|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$
(5.18)

If $\lim_{n\to\infty}\log(|x_{0}^{n}|)=\infty$ then we deduce the asserted
result, concluding the proof of Theorem 1.5.

###  5.3. Proof of Corollary 1.6

Let $\beta\in\\{1,2\\}$. Recall the definitions of $a_{n}$ and $c_{n}$ from
the statement. Take $x_{0}^{n,i}=a_{n}$ for all $i$, and note that
$\pi(x_{0}^{n})=na_{n}$. Given our assumptions on $a_{n}$, Corollary 1.4
yields for this particular choice of initial condition and for any
$\varepsilon\in(0,1)$

$\lim_{n\to\infty}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=\max.$

On the other hand, in the proof of Theorem 1.5 we saw that

$\chi^{2}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq-1+\frac{1}{(1-\mathrm{e}^{-4t})^{b_{n}/2}}\exp\left(n|x_{0}^{n}|^{2}\frac{\mathrm{e}^{-2t}}{1+\mathrm{e}^{-2t}}\right),$

where $b_{n}=n^{2}$ for $\beta=2$ and $b_{n}=(n(n+1)/2)^{2}$ for $\beta=1$.
Since $|x_{0}^{n}|\leq\sqrt{n}a_{n}$ for all $x_{0}^{n}\in[-a_{n},a_{n}]^{n}$,
and given the comparison between TV, Hellinger, Kullback and $\chi^{2}$ stated
in Lemma A.1 we obtain for
$\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger},\mathrm{Kullback},\chi^{2}\\}$
and for all $\varepsilon\in(0,1)$

$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0,$

thus concluding the proof of Corollary 1.6 regarding theses distances.
Concerning Wasserstein, the proof of Theorem 1.5 shows that for any
$x_{0}^{n}\in[-a_{n},a_{n}]^{n}$ we have

$\displaystyle\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}^{n}),P_{n}^{\beta})$
$\displaystyle\leq|x_{0}^{n}|^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}$
$\displaystyle\leq
na_{n}^{2}\mathrm{e}^{-2t}+2-\mathrm{e}^{-2t}-2\sqrt{1-\mathrm{e}^{-2t}}.$

If $\sqrt{n}a_{n}\to\infty$, then for $c_{n}=\log(\sqrt{n}a_{n})$ we deduce
that for all $\varepsilon\in(0,1)$

$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}})\mid
P_{n}^{\beta})=0.$

##  6\. Cutoff phenomenon for the DOU in TV and Hellinger

In this section, we prove Theorem 1.7 and Corollary 1.8 for the TV and
Hellinger distances. We only consider the case $\beta\geq 1$, although the
arguments could be adapted _mutatis mutandis_ to cover the case $\beta=0$:
note that the result of Theorem 1.7 and Corollary 1.8 for $\beta=0$ can be
deduced from Theorem 1.2. At the end of this section, we also provide the
proof of Theorem 1.10.

###  6.1. Proof of Theorem 1.7 in TV and Hellinger

By the comparison between TV and Hellinger stated in Lemma A.1, it suffices to
prove the result for the TV distance, so we concentrate on this distance until
the end of this section. Our proof is based on the exponential decay of the
relative entropy at an explicit rate given by the optimal logarithmic Sobolev
constant. However, this requires the relative entropy of the initial condition
to be _finite_. Consequently, we proceed in three steps. First, given an
arbitrary initial condition $x_{0}^{n}\in\overline{D}_{n}$, we build an
absolutely continuous probability measure $\mu_{x_{0}^{n}}$ on $D_{n}$ that
approximates $\delta_{x_{0}^{n}}$ and whose relative entropy is not too large.
Second, we derive a decay estimate starting from this regularized initial
condition. Third, we control the total variation distance between the two
processes starting respectively from $\delta_{x_{0}^{n}}$ and
$\mu_{x_{0}^{n}}$.

####  6.1.1. Regularization

In order to have a finite relative entropy at time $0$, we first regularize
the initial condition by smearing out each particle in a ball of radius
bounded below by $n^{-(\kappa+1)}$, for some $\kappa>0$. Let us first
introduce the regularization at scale $\eta$ of a Dirac distribution
$\delta_{z}$, $z\in\mathbb{R}$ by

$\delta_{z}^{(\eta)}(\mathrm{d}u)=\mathrm{Uniform}([z,z+\eta])(\mathrm{d}u)=\eta^{-1}\mathbf{1}_{[z,z+\eta]}\mathrm{d}u.$

Given $x\in\overline{D}_{n}$ and $\kappa>0$, we define a regularized version
of $\delta_{x}$ at scale $n^{-\kappa}$, that we denote $\mu_{x}$, by setting

$\mu_{x}=\otimes_{i=1}^{n}\delta_{x_{i}+3i\eta}^{(\eta)},$ (6.1)

where $\eta:=n^{-(\kappa+1)}$. The parameters have been tuned in such a way
that, independently of the choice of $x\in\overline{D}_{n}$, the following
properties hold. The supports of the Dirac masses
$\delta_{x_{i}+3i\eta}^{(\eta)}$, $i\in\\{1,\ldots,n\\}$, lie at distance at
least $\eta$ from each other. The volume of the support of $\mu_{x}$ is equal
to $\eta^{n}$, and therefore the relative entropy of $\mu_{x}$ with respect to
the Lebesgue measure is not too large. Finally, provided $X_{0}^{n}=x$ and
$Y_{0}^{n}$ is distributed according to $\mu_{x}$, almost surely
$|X_{0}^{n}-Y_{0}^{n}|_{\infty}\leq(3n+1)\eta$.

####  6.1.2. Convergence of the regularized process to equilibrium

######  Lemma 6.1 (Convergence of regularized process).

Let $(Y_{t}^{n})_{t\geq 0}$ be a DOU process solution of (1.3), $\beta\geq 1$,
and let $P_{n}^{\beta}$ be its invariant law. Assume that
$\mathrm{Law}(Y^{n}_{0})$ is the regularized measure $\mu_{x_{0}^{n}}$ in
(6.1) associated to some initial condition $x_{0}^{n}\in\overline{D}_{n}$.
Then there exists a constant $C>0$, only depending on $\kappa$, such that for
all $t\geq 0$, all $n\geq 2$ and all $x_{0}^{n}\in\overline{D}_{n}$

$\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid P_{n}^{\beta})\leq
C(n|x_{0}^{n}|^{2}+n^{2}\log(n))\mathrm{e}^{-2t}.$

###### Proof of Lemma 6.1.

By Lemma B.2 and since $\mathrm{Law}(Y^{n}_{0})=\mu_{x_{0}^{n}}$, for all
$t\geq 0$, there holds

$\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid
P_{n}^{\beta})\leq\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})\mathrm{e}^{-2t}.$ (6.2)

Now we have

$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})=\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\log\frac{\mathrm{d}\mu_{x_{0}^{n}}}{\mathrm{d}P_{n}^{\beta}}\right].$

Recall the definition of $S$ in (1.15). As $P_{n}^{\beta}$ has density
$\frac{\mathrm{e}^{-E}}{C_{n}^{\beta}}$, we may re-write this as

$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid
P_{n}^{\beta})=S(\mu_{x_{0}^{n}})+\mathbb{E}_{\mu_{x_{0}^{n}}}[E]+\log
C_{n}^{\beta}.$ (6.3)

Recall the partition function $C_{*n}^{\beta}=n!C_{n}^{\beta}$ from Subsection
2.2. It is proved in [12], using explicit expressions involving Gamma
functions via a Selberg integral, that for some constant $C>0$

$\log C_{n}^{\beta}\leq\log C_{*n}^{\beta}\leq Cn^{2}.$ (6.4)

Next, we claim that $S(\mu_{x_{0}^{n}})\leq n\log(n^{1+\kappa})$. Indeed since
$\mu_{x_{0}^{n}}$ is a product measure, the tensorization property of entropy
recalled in Lemma A.4 gives

$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid\mathrm{d}x)=\sum_{i=1}^{n}\mathrm{Kullback}(\delta_{0}^{(\eta)}\mid\mathrm{d}x).$

Moreover an immediate computation yields
$\mathrm{Kullback}(\delta_{0}^{(\eta)}\mid\mathrm{d}x)=\log(\eta^{-1})$ so
that, given the definition of $\eta$, we get

$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid\mathrm{d}x)=n\log(n^{\kappa+1}).$ (6.5)

We turn to the estimation of the term $\mathbb{E}_{\mu_{x_{0}^{n}}}[E].$ The
confinement term can be easily bounded:

$\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\frac{n}{2}\sum_{i=1}^{n}{x_{i}^{2}}\right]\leq(n|x_{0}^{n}|^{2}+n^{2}\eta^{2}).$

Let us now estimate the logarithmic energy of $\mu_{x_{0}^{n}}$. Using the
fact that the logarithmic function is increasing, together with the fact the
supports of $\delta_{x_{i}+3i\eta}^{(\eta)}$ lie at distance at least $\eta$
from each other, we notice that for any $i>j$ there holds

$\displaystyle\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\log|x_{i}-x_{j}|\right]$
$\displaystyle=\iint\log|x-y|\delta_{x_{i}+3i\eta}^{(\eta)}(\mathrm{d}x)\delta_{x_{j}+3j\eta}^{(\eta)}(\mathrm{d}y)$
$\displaystyle\geq\iint\log|x-y|\delta_{3\eta}^{(\eta)}(\mathrm{d}x)\delta_{0}^{(\eta)}(\mathrm{d}y)$
$\displaystyle\geq\log\eta.$

It follows that the initial logarithmic energy cannot be much larger than
$n^{2}\log n$:

$\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}\right]\leq\frac{n(n-1)}{2}\log
n^{\kappa+1}.$

This implies that there exists a constant $C>0$, only depending on $\kappa$,
such that for all $n\geq 2$

$\mathbb{E}_{\mu_{x_{0}^{n}}}[E]=\mathbb{E}_{\mu_{x_{0}^{n}}}\left[\frac{n}{2}\sum_{i=1}^{n}|x_{i}|^{2}+{\beta}\sum_{i>j}\log\frac{1}{|x_{i}-x_{j}|}\right]\leq
C\big{(}n|x_{0}^{n}|^{2}+n^{2}\log n\big{)}.$ (6.6)

Inserting (6.4), (6.5) and (6.6) into (6.3) we obtain (for a different
constant $C>0$)

$\mathrm{Kullback}(\mu_{x_{0}^{n}}\mid P_{n}^{\beta})\leq
C\big{(}n|x_{0}^{n}|^{2}+n^{2}\log n\big{)}.$

This bound, combined with (6.2), concludes the proof of Lemma 6.1. ∎

####  6.1.3. Convergence to the regularized process in total variation
distance

Let $(X_{t}^{n})_{t\geq 0}$ and $(Y_{t}^{n})_{t\geq 0}$ be two DOU processes
with $X_{0}^{n}=x_{0}^{n}$ and $\mathrm{Law}(Y_{0}^{n})=\mu_{x_{0}^{n}}$,
where the measure $\mu_{x_{0}^{n}}$ is defined in (6.1). Below we prove that,
as soon as the parameter $\kappa$ is large enough, the total variation
distance between $\mathrm{Law}(X_{t}^{n})$ and $\mathrm{Law}(Y_{t}^{n})$ tends
to $0$, for any fixed $t>0$.

Note that at time $0$, almost surely, there holds $X_{0}^{n,i}\leq
Y_{0}^{n,i}$, for every $i\in\\{1,\ldots,n\\}$. We now introduce a coupling of
the processes $(X_{t}^{n})_{t\geq 0}$ and $(Y_{t}^{n})_{t\geq 0}$ that
preserves this ordering at all times. Consider two independent standard BM
$B^{n}$ and $W^{n}$ in $\mathbb{R}^{n}$. Let $X^{n}$ be the solution of (1.3)
driven by $B^{n}$, and let $Y^{n}$ be the solution of

$\mathrm{d}Y_{t}^{n,i}=\sqrt{\frac{2}{n}}\Big{(}\mathbf{1}_{\\{Y_{t}^{n,i}\neq
X_{t}^{n,i}\\}}\mathrm{d}W^{i}_{t}+\mathbf{1}_{\\{Y_{t}^{n,i}=X_{t}^{n,i}\\}}\mathrm{d}B^{i}_{t}\Big{)}-Y_{t}^{n,i}\mathrm{d}t+\frac{\beta}{n}\sum_{j\neq
i}\frac{\mathrm{d}t}{Y_{t}^{n,i}-Y_{t}^{n,j}},\quad 1\leq i\leq n.$

We denote by $\mathbb{P}$ the probability measure under which these two
processes are coupled. Let us comment on the driving noise in the equation
satisfied by $Y^{n}$. When the $i$-th coordinates of $X^{n}$ and $Y^{n}$
equal, we take the same driving Brownian motion and the difference
$Y^{n,i}-X^{n,i}$ remains non-negative due to the convexity of $-\log$ defined
in (1.5), see the monotoncity result stated in Lemma B.4. On the other hand,
when these two coordinates differ, we take independent driving Brownian
motions in order for their difference to have non-zero quadratic variation
(this allows to increase their merging probability). Under this coupling, the
ordering of $X^{n}$ and $Y^{n}$ is thus preserved at all times, and if
$X_{s}^{n}=Y_{s}^{n}$ for some $s\geq 0$, then it remains true at all times
$t\geq s$. Note however that if $X_{s}^{n,i}=Y_{s}^{n,i}$, then this equality
does not remain true at all times except if all the coordinates match.

As in (A.7), the total variation distance between the laws of $X_{t}^{n}$ and
$Y_{t}^{n}$ may be bounded by

$\|\mathrm{Law}(Y_{t}^{n})-\mathrm{Law}(X_{t}^{n})\|_{\mathrm{TV}}\leq\mathbb{P}(X_{t}^{n}\neq
Y_{t}^{n}),$

for all $t\geq 0$. We wish to establish that for any given $t>0$,

$\lim_{n\to\infty}\mathbb{P}(X_{t}^{n}\neq Y_{t}^{n})=0.$

To do so, we work with the area between the two processes $X^{n}$ and $Y^{n}$,
defined by

$A^{n}_{t}:=\sum_{i=1}^{n}\big{(}Y^{n,i}_{t}-X^{n,i}_{t}\big{)}=\pi(Y^{n}_{t})-\pi(X^{n}_{t}),\quad
t\geq 0.$

As the two processes are ordered at any time, this is nothing but the
geometric area between the two discrete interfaces $i\mapsto X_{t}^{n,i}$ and
$i\mapsto Y_{t}^{n,i}$ associated to the configurations $X_{t}^{n}$ and
$Y_{t}^{n}$. We deduce that the merging time of the two processes coincide
with the hitting time of $0$ by this area, that we denote by
$\tau=\inf\\{t\geq 0:A_{t}^{n}=0\\}$.

The process $A^{n}$ has a very simple structure: it is a semimartingale that
behaves like an OU process with a randomly varying quadratic variation. Let
$N_{t}$ be the number of coordinates that do not coincide at time $t$, that is

$N_{t}:=\\#\big{\\{}i\in\\{1,\ldots,n\\}:X^{n,i}_{t}\neq
Y^{n,i}_{t}\big{\\}}.$

Then $A^{n}$ satisfies

$\mathrm{d}A^{n}_{t}=-A^{n}_{t}\mathrm{d}t+\mathrm{d}M_{t},$

where $M$ is a centered martingale with quadratic variation

$\mathrm{d}\langle M\rangle_{t}=\frac{2}{n}N_{t}\mathrm{d}t.$ (6.7)

Note that whenever $t<\tau$ we have

$\mathrm{d}\langle M\rangle_{t}\geq\frac{2}{n}.$

This _a priori_ lower bound on the quadratic variation of $M$, combined with
the Dubins–Schwarz theorem, allows to check that $\tau<\infty$ almost surely.
Note that in view of the coupling between $X_{t}^{n}$ and $Y_{t}^{n}$, we have
$X_{t}^{n}=Y_{t}^{n}$ for all $t\geq\tau$.

Recall the following informal fact: with large probability, a Brownian motion
starting from $a$ hits $b$ by a time of order $(a-b)^{2}$. For a continuous
martingale, this becomes: with large probability, a continuous martingale
starting from $a$ accumulates a quadratic variation of order $(a-b)^{2}$ up to
its first hitting time of $b$. Our next lemma states such a bound on the
supermartingale $A^{n}$.

######  Lemma 6.2.

Let $a>b\geq 0$. Let $\tau_{b}=\inf\\{t>0:A_{t}=b\\}<\infty$ almost surely.
Then, for all $u\geq 1$,

$\mathbb{P}(\langle A\rangle_{\tau_{b}}\geq(a-b)^{2}u\mid A_{0}=a)\leq
4u^{-1/2}.$

###### Proof.

Without loss of generality one can assume that $A_{0}=a$ almost surely.
By Itô’s formula, for all $\lambda\geq 0$, the process

$S_{t}=\exp\Bigr{(}-\lambda A_{t}-\frac{\lambda^{2}}{2}\langle
A\rangle_{t}\Bigr{)},$

defines a submartingale (taking its values in $[0,1]$). Doob’s stopping
theorem yields

$\mathbb{E}[\mathrm{e}^{-\frac{\lambda^{2}}{2}\langle
A\rangle_{\tau_{b}}}]=\mathrm{e}^{\lambda
b}\mathbb{E}[S_{\tau_{b}}]\geq\mathrm{e}^{\lambda
b}\mathbb{E}[S_{0}]=\mathrm{e}^{-\lambda(a-b)}.$

On the other hand, for $\lambda=2(a-b)^{-1}u^{-1/2}$, there holds

$\displaystyle\mathbb{E}[\mathrm{e}^{-\frac{\lambda^{2}}{2}\langle
A\rangle_{\tau_{b}}}]$ $\displaystyle\leq\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}<(a-b)^{2}u\big{)}+\mathrm{e}^{-\frac{\lambda^{2}}{2}(a-b)^{2}u}\,\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}$ $\displaystyle\leq
1-(1-\mathrm{e}^{-\frac{\lambda^{2}}{2}(a-b)^{2}u})\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}$ $\displaystyle\leq
1-\frac{1}{2}\mathbb{P}\big{(}\langle
A\rangle_{\tau_{b}}\geq(a-b)^{2}u\big{)}.$

Consequently one deduces that

$\mathbb{P}(\langle A\rangle_{\tau_{b}}\geq(a-b)^{2}u)\leq
2(1-\mathrm{e}^{-\lambda(a-b)})\leq 4u^{-1/2}.$

∎

We are now ready to prove the following lemma:

######  Lemma 6.3.

If $\kappa>\frac{3}{2}$, then for every sequence of times ${(t_{n})}_{n}$ with
$\varliminf_{n\to\infty}t_{n}>0$, we have

$\lim_{n\to\infty}\sup_{x_{0}^{n}\in\overline{D}_{n}}\|\mathrm{Law}(Y_{t_{n}}^{n})-\mathrm{Law}(X_{t_{n}}^{n})\|_{\mathrm{TV}}=0.$

###### Proof of Lemma 6.3.

Let ${(t_{n})}_{n}$ be a sequence of times such that
$\varliminf_{n\to\infty}t_{n}>0$. In view of the definition of
$\mu_{{x_{0}^{n}}}$ and $\eta$, the initial area satisfies almost surely

$A_{0}^{n}\leq 4n^{1-\kappa}.$

According to Lemma 6.2, with a probability that goes to $1$, one has

$\langle A^{n}\rangle_{\tau}-\langle A^{n}\rangle_{0}<16n^{2-2\kappa}\log n.$

On the other hand, by (6.7), we have the following control on the quadratic
variation:

$\langle A\rangle_{\tau}-\langle A\rangle_{0}\geq\frac{2}{n}\tau.$

One deduces that, with a probability that goes to $1$,

$\tau\leq\frac{16}{2}n^{3-2\kappa}\log n,$

and this quantity goes to $0$ as $n\to\infty$, whenever $\kappa>\frac{3}{2}$.
Therefore for $\kappa>\frac{3}{2}$, there holds

$\lim_{n\to\infty}\sup_{x_{0}^{n}\in\overline{D}_{n}}\mathbb{P}(X_{t_{n}}^{n}\neq
Y_{t_{n}}^{n})=0,$

thus concluding the proof of Lemma 6.3. ∎

###### Proof of Theorem 1.7 in TV and Hellinger.

Let $\kappa>\frac{3}{2}$ and fix some initial condition
$x_{0}^{n}\in\overline{D}_{n}$. By the triangle inequality for $\mathrm{TV}$,
there holds

$\|\mathrm{Law}(X_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}\leq\|\mathrm{Law}(Y_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}+\|\mathrm{Law}(X_{t}^{n})-\mathrm{Law}(Y_{t}^{n})\|_{\mathrm{TV}}.$
(6.8)

Taking $t=t_{n}(1+\varepsilon)$ with
$t_{n}=\log(\sqrt{n}|x_{0}^{n}|)\vee\log(n)$, one deduces from Lemma 6.1 and
the Pinsker inequality stated in Lemma A.1 that the first term in the right-
hand side of (6.8) vanishes as $n$ tends to infinity. Meanwhile Lemma 6.3
guaranties that the second term tends to $0$ as $n$ tends to infinity. We also
conclude using the comparison between TV and Hellinger (see Lemma A.1) that

$\lim_{n\to\infty}\mathrm{Hellinger}(\mathrm{Law}(X_{t_{n}}^{n}),P_{n}^{\beta})=0.$

∎

###  6.2. Proof of Corollary 1.8 in TV and Hellinger

###### Proof of Corollary 1.8 in TV and Hellinger.

By Lemma A.1 and the triangle inequality for $\mathrm{TV}$, we have

$\displaystyle\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\|\mathrm{Law}(X_{t}^{n})-P_{n}^{\beta}\|_{\mathrm{TV}}$
$\displaystyle\leq\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\|\mathrm{Law}(Y_{t}^{n})-\mathrm{Law}(X_{t}^{n})\|_{\mathrm{TV}}$
$\displaystyle\quad+\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\sqrt{2\,\mathrm{Kullback}(\mathrm{Law}(Y_{t}^{n})\mid
P_{n}^{\beta})}.$

Take $t=(1+\varepsilon)c_{n}$ with $c_{n}=\log(na_{n})$. Lemmas 6.1 and 6.3,
combined with the assumption made on $(a_{n})$, show that the two terms on the
right-hand side vanish as $n\to\infty$. Using Lemma A.1, the same result holds
for $\mathrm{Hellinger}$.
On the other hand, take $x_{0}^{n,i}=a_{n}$ for all $i$ and note that
$\pi(x_{0}^{n})=na_{n}$ goes to $+\infty$ as $n\to\infty$. By Corollary 1.4 we
find

$\lim_{n\to\infty}\sup_{x^{n}_{0}\in[-a_{n},a_{n}]^{n}}\mathrm{dist}(\mathrm{Law}(X^{n}_{(1-\varepsilon)c_{n}})\mid
P_{n}^{\beta})=1\;$

whenever $\mathrm{dist}\in\\{\mathrm{TV},\mathrm{Hellinger}\\}$. ∎

###  6.3. Proof of Theorem 1.10

###### Proof of Theorem 1.10.

_Lower bound._ The contraction property provided by Lemma A.2 gives

$\mathrm{Kullback}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\geq\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1}).$

By Theorem 1.3 $P_{n}\circ\pi^{-1}=\mathcal{N}(0,1)$ and $Y=\pi(X^{n})$ is an
OU process weak solution of $Y_{0}=\pi(X^{n}_{0})$ and
$\mathrm{d}Y_{t}=\sqrt{2}\mathrm{d}B_{t}-Y_{t}\mathrm{d}t$. In particular for
all $t\geq 0$, $\mathrm{Law}(Y_{t})$ is a mixture of Gaussian laws in the
sense that for any measurable test function $g$ with polynomial growth,

$\mathbb{E}_{\mathrm{Law}(Y_{t})}[g]=\mathbb{E}[g(Y_{t})]=\mathbb{E}[G_{t}(Y_{0})]\quad\text{where}\quad
G_{t}(y)=\mathbb{E}_{\mathcal{N}(y\mathrm{e}^{-t},1-\mathrm{e}^{-2t})}[g].$

Now we use (again) the variational formula used in the proof of Lemma A.2 to
get

$\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1})=\sup_{g}\\{\mathbb{E}_{\mathrm{Law}(\pi(X_{t}^{n}))}[g]-\log\mathbb{E}_{\mathcal{N}(0,1)}[\mathrm{e}^{g}]\\},$

and taking for $g$ the linear function defined by $g(x)=\lambda x$ for all
$x\in\mathbb{R}$ and for some $\lambda\neq 0$ yields

$\mathrm{Kullback}(\mathrm{Law}(\pi(X_{t}^{n}))\mid
P_{n}^{\beta}\circ\pi^{-1})\geq\lambda\mathrm{e}^{-t}\sum_{i=1}^{n}\int
x\mu_{i}(\mathrm{d}x)-\frac{\lambda^{2}}{2}.$

Finally, by using the assumption on first moment and taking $\lambda$ small
enough we get, for all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(\pi(X_{(1-\varepsilon)\log(n)}^{n})\mid
P_{n}^{\beta}\circ\pi^{-1})=+\infty,$

_Upper bound._ From Lemma B.2 we have, for all $t\geq 0$,

$\mathrm{Kullback}(\mathrm{Law}(X_{t}^{n})\mid
P_{n}^{\beta})\leq\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid
P_{n}^{\beta})\mathrm{e}^{-2t}.$

Arguing like in the proof of Lemma 6.1 and using the contraction property of
$\mathrm{Kullback}$ provided by Lemma A.2 for the map $\Psi$ defined in
(1.17), we can write the following decomposition

$\displaystyle\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid P_{n}^{\beta})$
$\displaystyle\leq\mathrm{Kullback}(\otimes_{i=1}^{n}\mu_{i}\mid
P_{*n}^{\beta})$
$\displaystyle=S(\otimes_{i=1}^{n}\mu_{i})+\mathbb{E}_{\otimes_{i=1}^{n}\mu_{i}}[E]+\log
C_{*n}^{\beta}$ $\displaystyle\leq\sum_{i=1}^{n}S(\mu_{i})+\sum_{i\neq
j}\iint\Phi\mathrm{d}\mu_{i}\otimes\mathrm{d}\mu_{j}+Cn^{2}.$

Combining (6.4) with the assumptions on the $\mu_{i}$’s yields for some
constant $C>0$

$\mathrm{Kullback}(\mathrm{Law}(X_{0}^{n})\mid P_{n}^{\beta})\leq Cn^{2}$

and it follows finally that for all $\varepsilon\in(0,1)$,

$\lim_{n\to\infty}\mathrm{Kullback}(\mathrm{Law}(X_{(1+\varepsilon)\log(n)})\mid
P_{n}^{\beta})=0.$

∎

##  7\. Cutoff phenomenon for the DOU in Wasserstein

###  7.1. Proofs of Theorem 1.7 and Corollary 1.8 in Wasserstein

Let ${(X_{t})}_{t\geq 0}$ be the DOU process. By Lemma B.2, for all $t\geq 0$
and all initial conditions $X_{0}\in\overline{D}_{n}$,

$\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{t}),P_{n}^{\beta})\leq\mathrm{e}^{-2t}\mathrm{Wasserstein}^{2}(\mathrm{Law}(X_{0}),P_{n}^{\beta}).$

Suppose now that $\mathrm{Law}(X^{n}_{0})=\delta_{x^{n}_{0}}$. Then the
triangle inequality for the Wasserstein distance gives

$\mathrm{Wasserstein}^{2}(\delta_{x^{n}_{0}},P_{n}^{\beta})=\int\left|x^{n}_{0}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)\leq
2|x^{n}_{0}|^{2}+2\int\left|x\right|^{2}P_{n}^{\beta}(\mathrm{d}x).$

By Theorem 1.3, the mean at equilibrium of $|X_{t}^{n}|^{2}$ equals
$1+\frac{\beta}{2}(n-1)$ and therefore

$\int\left|x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=1+\frac{\beta}{2}(n-1).$

We thus get

$\mathrm{Wasserstein}^{2}(\mathrm{Law}(X^{n}_{t}),P_{n}^{\beta})\leq
2(|x^{n}_{0}|^{2}+1+\frac{\beta}{2}(n-1))\mathrm{e}^{-2t}.$

Set $c_{n}:=\log(|x_{0}^{n}|)\vee\log(\sqrt{n})$. For any
$\varepsilon\in(0,1)$, we have

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}}),P_{n}^{\beta})=0$

and this concludes the proof of Theorem 1.7 in the Wasserstein distance.
Regarding the proof of Corollary 1.8, if $x_{0}^{n}\in[-a_{n},a_{n}]^{n}$ then
$|x_{0}^{n}|\leq\sqrt{n}a_{n}$. Therefore if $\inf_{n}a_{n}>0$, setting
$c_{n}=\log(\sqrt{n}a_{n})$ we find, as required,

$\lim_{n\to\infty}\sup_{x_{0}^{n}\in[-a_{n},a_{n}]^{n}}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)c_{n}}),P_{n}^{\beta})=0.$

###  7.2. Proof of Theorem 1.9

This is an adaptation of the previous proof. We compute

$\displaystyle\mathrm{Wasserstein}^{2}(\delta_{x_{0}^{n}},P_{n}^{\beta})$
$\displaystyle=\int\left|x^{n}_{0}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)$
$\displaystyle\leq
2\left|x^{n}_{0}-\rho_{n}\right|^{2}+2\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x),$

where $\rho_{n}\in D_{n}$ is the vector of the quantiles of order $1/n$ of the
semi-circle law as in (1.14). The rigidity estimates established in [17, Th.
2.4] justify that

$\lim_{n\to\infty}\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=0.$

If $|x^{n}_{0}-\rho_{n}|$ diverges with $n$, we deduce that for all
$\varepsilon\in(0,1)$, with $t_{n}=\log(|x^{n}_{0}-\rho_{n}|)$,

$\lim_{n\to\infty}\mathrm{Wasserstein}(\mathrm{Law}(X^{n}_{(1+\varepsilon)t_{n}}),P_{n}^{\beta})=0.$

On the other hand, if $|x^{n}_{0}-\rho_{n}|$ converges to some limit $\alpha$
then we easily get, for any $t\geq 0$,

$\varlimsup_{n\to\infty}\mathrm{Wasserstein}^{2}(\mathrm{Law}(X^{n}_{t}),P_{n}^{\beta})\leq\alpha^{2}\mathrm{e}^{-2t}.$

######  Remark 7.1 (High-dimensional phenomena).

With $X_{n}\sim P_{n}^{\beta}$, in the bias-variance decomposition

$\int\left|\rho_{n}-x\right|^{2}P_{n}^{\beta}(\mathrm{d}x)=|\mathbb{E}X_{n}-\rho_{n}|^{2}+\mathbb{E}(|X_{n}-\mathbb{E}X_{n}|^{2}),$

the second term of the right hand side is a variance term that measures the
concentration of the log-concave random vector $X_{n}$ around its mean
$\mathbb{E}X_{n}$, while the first term in the right hand side is a bias term
that measures the distance of the mean $\mathbb{E}X_{n}$ to the mean-field
limit $\rho_{n}$. Note also that
$\mathbb{E}(|X_{n}-\mathbb{E}X_{n}|^{2})=\mathbb{E}(|X_{n}|^{2})-|\mathbb{E}X_{n}|^{2}=1+\frac{\beta}{2}(n-1)-|\mathbb{E}X_{n}|^{2}$,