# Sharp total variation rates of convergence
for fluctuations of linear statistics of $\beta$-ensembles

Jürgen ANGST  IRMAR, Université de Rennes 1<EMAIL_ADDRESS>,
Ronan HERRY  IRMAR, Université de Rennes 1<EMAIL_ADDRESS>https://orcid.org/0000-0001-6313-1372 ,  Dominique MALICET  LAMA, Université
Gustave Eiffel<EMAIL_ADDRESS>https://orcid.org/0000-0003-2768-0125 and  Guillaume POLY  IRMAR, Université
de Rennes 1<EMAIL_ADDRESS>
###### Abstract.

In this article, we revisit the question of fluctuations of linear statistics
of beta ensembles in the single cut and non-critical regime for general
potentials $V$ under mild regularity and growth assumptions. Our main
objective is to establish sharp quantitative Central Limit Theorems (CLT) for
strong distances, such as the total variation distance, which to the best of
our knowledge, is new for general potentials, even qualitatively. Namely,
setting $\mu_{V}$ the equilibrium measure, for a test function
$\xi\in\mathscr{C}^{14}$, we establish the convergence in total variation of
$X_{n}=\sum_{i=1}^{n}\xi(\lambda_{i})-n\langle\xi,\mu_{V}\rangle$ to an
explicit Gaussian variable at the sharp speed $1/n$. Under the same
assumptions, we also establish multivariate CLTs for vectors of linear
statistics in $p-$Wasserstein distances for any $p\geq 1$, with the optimal
rate $1/n$, a result which already in dimension one sharpens the speed of
convergence established in the recent contribution [26] as well as the
required regularity on the test functions. A second objective of this paper,
in a more qualitative direction, is to establish the so-called super-
convergence of linear statistics, that is to say the convergence of all
derivatives of the densities of $X_{n}$ uniformly on $\mathbb{R}$, provided
that $\xi\in\mathscr{C}^{\infty}(\mathbb{R})$ and is not too degenerated in
some sense.
As far as quantitative results are concerned, our approach shares similar
tools with the latter reference as we also rely on Stein’s method and
integration by part formalism provided by the Dyson generator $\mathsf{L}$.
Nevertheless, we adopt a different and innovative strategy which consists in
proving that, up to centering, the above linear statistics $X_{n}$ can be
written in the form $\mathsf{L}F_{n}+Z_{n}$ where $Z_{n}$ is some small
remainder and $\Gamma[X_{n},F_{n}]$ converges to some constant as direct
consequences of the law of large numbers. We emphasize that this approach
bypasses the costly requirement of the invertibility of the Markov operator
which is central in Malliavin-Stein’s method and seems robust enough to be
implemented for various models of Gibbs measures. On the qualitative side, the
proof of the super-convergence of densities, for its part, is a consequence of
the existence of negative moments for the carré du champ of linear statistics,
associated with integration by parts techniques.

###### Contents

  1. 1 Introduction and statement of the results
  2. 2 Stein’s method for Markov diffusive operators
  3. 3 The master equation for linear statistics
  4. 4 Proofs of the main results

##  1\. Introduction and statement of the results

###  1.1. Overview of our contributions

Given a parameter $\beta>0$ interpreted as an inverse temperature, the
_$\beta$ -ensemble_, with $n$ particles associated with a continuous potential
$V\colon\mathbb{R}\to\mathbb{R}$ such that
$\liminf_{x\to\pm\infty}V(x)-\log\lvert x\rvert>0$, is the Gibbs probability
distribution on $\mathbb{R}^{n}$ associated with the _energy_

$H_{n}(\lambda):=\sum_{i<j}\log\frac{1}{\lvert\lambda_{i}-\lambda_{j}\rvert}+n\sum_{i}V(\lambda_{i}).$

Namely, the $\beta$-ensemble is the probability measure

(1.1)
$\operatorname{\mathbf{P}}(\mathtt{d}\lambda)=\operatorname{\mathbf{P}}_{n,\beta}(\mathtt{d}\lambda):=\frac{1}{Z_{n,\beta}}\mathrm{e}^{-\beta
H_{n}(\lambda)}\mathtt{d}\lambda,$

where $Z_{n,\beta}:=\int\mathrm{e}^{-\beta H_{n}}\mathtt{d}\lambda$ is the
_partition function_ , and $\mathtt{d}\lambda$ stands for the Lebesgue measure
on $\mathbb{R}^{n}$. In this paper, we establish both qualitative and
quantitative Central Limit Theorems (CLT) for _linear statistics_ of
$\beta$-ensemble, which are random variables of the form

$X_{n}:=\sum_{i=1}^{n}\xi(\lambda_{i})-n\int\xi\mathtt{d}\mu_{V},\qquad
n\in\mathbb{N},$

for some non-constant test function $\xi\colon\mathbb{R}\to\mathbb{R}$
satisfying some mild regularity assumptions, where $\mu_{V}$ is the so-called
_equilibrium measure_ , and where the random vector
$(\lambda_{1},\dots,\lambda_{n})$ is drawn from the Gibbs measure
$\operatorname{\mathbf{P}}$.

Our first main results quantifies precisely the fluctuations of such linear
statistics under mild assumptions on the potential $V$ and the test function
$\xi$. Whenever $V\in\mathscr{C}^{7}(\mathbb{R})$ is semi-convex, regular, and
that we are in the single-cut regime (see Assumption 1 below), we indeed
obtain a CLT for linear statistics, in both total variation and
$p-$Wasserstein distances ($1\leq p<\infty$), with the following rates:

  * •

(Theorem 1.2) If $\xi\in\mathscr{C}^{{14}}(\mathbb{R})$, we obtain the optimal
rate $\frac{1}{n}$.

  * •

(Theorem 1.1) If $\xi\in\mathscr{C}^{6}(\mathbb{R})$, we obtain the almost
optimal rate $\frac{1}{n^{1-\alpha}}$ for all $\alpha>0$.

  * •

(Theorem 1.3) If $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ with
$\gamma\in(0,1)$, we obtain an explicit rate of convergence depending on the
Hölder exponent $\gamma$.

Our bounds in $p-$Wasserstein distance are furthermore obtained in the
multivariate case. In fact, we obtain precise rates of convergence, with
explicit upper bounds involving $n$, $\beta$ and norms of derivatives of
$\xi$. This could allow for limit theorems with varying temperature and/or
with a test functions $\xi$ also depending on $n$.

Our next main result Theorem 1.4 is of qualitative nature and show that the
convergence in law of a linear statistic to a Gaussian automatically upgrades
to a very strong form of convergence. Namely, if the potential $V$ and the
test function $\xi$ are of class $\mathscr{C}^{\infty}$ and if the latter is
not degenerate is some sense, then $(X_{n})$ _super-converges_ to a Gaussian
(see Section 1.3.3 below for definition of the super-convergence). Informally
it means that the density of $(X_{n})$ converges in the
$\mathscr{C}^{\infty}$-topology to that of a Gaussian. In particular, it
entails that the convergence of linear statistics holds in relative entropy.

###  1.2. Motivations

The $\beta$-ensembles, also known as _$1d$ -log gas_, are related to random
matrix theory. We refer to the monographs [33, 20] for details on the subject,
as well as [36] for a broader introduction to Coulomb gases. Indeed, in the
case $\beta\in\\{1,2,4\\}$ and $V(x)=x^{2}$, the corresponding
$\beta$-ensemble describes the joint law of the spectrum of an $n\times n$
random matrix whose density is proportional to $\exp(-\frac{\beta
n}{4}\operatorname{Tr}V(M))\mathtt{d}M$, where $\mathrm{d}M$ is the Haar
measure on the sets of symmetric, hermitian, or symplectic matrices
respectively. The observation that certain $\beta$-ensembles relate to
eigenvalues of Gaussian ensembles goes back at least to [17].

The spectral macroscopic properties of large Gaussian matrices first observed
by [41], actually extend to more general $\beta$-ensembles, and it is by now
well-understood, see for instance [33, Thm. 11.1.2], that

$\frac{1}{n}\sum_{i=1}^{n}\xi(\lambda_{i})\xrightarrow[n\to\infty]{a.s}\int\xi\mathtt{d}\mu_{V},$

where the non-random _equilibrium measure_ $\mu_{V}$ is the unique minimizer
of the _mean-field energy_

$\mathcal{I}_{V}(\mu):=\int V\mathtt{d}\mu-\frac{1}{2}\iint\log\lvert
x-y\rvert\mu(\mathtt{d}x)\mu(\mathtt{d}y).$

In this article, our goal is to study the fluctuations associated with the
above “law of large numbers”, with different motivations that we detail below.

####  1.2.1. Sharp quantitative Central Limit Theorems

The seminal work of [24] shows that, in the matrix case and under suitable
assumptions on the potential $V$ and the test function $\xi$, the fluctuations
of $\sum_{i=1}^{n}\xi(\lambda_{i})$ around the value
$n\int\xi\mathtt{d}\mu_{V}$ are Gaussian. The quantitative aspect of this
convergence has attracted a lot of attention in the last decade: $i)$ in [13],
total variation bounds are provided in the matrix case via second order
Poincaré inequalities; $ii)$ in [3], still in the matrix case, near optimal
rates of convergence are derived for Kolmogorov distance; $iii)$ for general
$\beta$-ensembles, polynomial rates of convergence are obtained for
$2-$Wasserstein distance in [26] using a variant of Stein’s method due to
Meckes. However, in the last reference, the rates obtained are at most
$O(n^{-2/3+\varepsilon})$ for general potential and of near optimal order
$O(n^{-1+\varepsilon})$ only in the quadratic case. Our primary motivation in
this work is to derive total variation estimates, which is a much stronger
distance than the Wasserstein ones, but also to get optimal rate of
convergence of order $O(n^{-1})$. We also provide sharp rates of convergence
in Wasserstein metric in both univariate and multivariate settings, and our
findings require less regularity on the test functions than previous results
on this question, see [26, 5].

####  1.2.2. Implementing a robust Stein’s approach for $\beta$-ensembles

Historically, Stein’s method [38] has proved its remarkable efficientcy to
obtain quantitative CLTs in strong probabilistic metrics such as total
variation or relative entropy. Despite numerous applications to various
probabilistic models such as:  (a) Erdös–Rényi random graphs [35];  (b)
infinite-dimensional Gaussian fields [32];  (c) Poisson point processes [34];
(d) free probability [19];  to the best of our knowledge, Stein’s method to
study $\beta$-ensemble has only been implemented in [26, 21], through a
variant due to Meckes [30], which in this context consists in approaching
linear statistics by infinite sums of approximate eigenvectors of the Dyson
generator. This demanding task requires strong regularity assumptions on the
test function $\xi$ and prevents one to both get optimal rates and to handle
total variation metric. A second motivation for this works is to develop a
Stein’s method for $\beta$-ensembles, sufficiently robust and general to be
possibly applied to other models in statistical physics involving Gibbs
measures.

####  1.2.3. Super-convergence phenomenon

For sums of independent and identically distributed random variables, due to
the convolution structure and provided the common law has some small initial
regularity, the regularity in fact improves along the classical CLT. In
particular, it is possible to reinforce the classical convergence in law to
the get the $\mathscr{C}^{\infty}$ convergence of densities, see [28]. This
particular behavior also appears in the free CLT [8], from which the
terminology of _super-convergence_ proceeds. Recently, three of the authors
have revisited this regularization phenomenon on Wiener chaoses [23] and for
quadratic forms [22]. In the setting of $\beta$-ensembles, despite the linear
nature of the statistics in consideration, no convolution structure arises due
to the interaction between the particles. Establishing better-than-expected
limit theorems in this dependent setting serves as the third motivation of
this article.

###  1.3. Statement of the main results

Let us now precise our assumptions, fix our notations and state formally our
main results.

####  1.3.1. Assumption and notations

In the rest of the paper, we will always assume that the potential $V$
satisfies the following conditions. Note that the latter are classical, in
particular they coincide with the assumptions in [26].

######  Assumption 1.

The potential $V$ and associated equilibrium measure $\mu_{V}$ are such that

  * •

(Smoothness) $V\in\mathscr{C}^{7}(\mathbb{R})$.

  * •

(Single-cut) The equilibrium measure $\mu_{V}$ is supported on the interval
$\Sigma_{V}:=[-1,1]$.

  * •

(Semi-convexity) $\inf_{\mathbb{R}}V^{\prime\prime}>-\infty$.

  * •

(Regularity) The measure $\mu_{V}$ has a positive density with respect to the
semi-circle law.

Recall that the measure $\mu_{V}$ is here the unique minimizer of the mean-
field energy $\mathcal{I}_{V}$. In particular, it satisfies the Euler–Lagrange
equation, for some constant $C_{V}\in\mathbb{R}$:

$V(x)-\int\log(|x-y|)\mu_{V}(\mathtt{d}y)=C_{V},\qquad\forall x\in\Sigma_{V}.$

Under the above assumptions, the equilibrium measure $\mu_{V}$ admits a
density with respect to the semi-circular law
$\mu_{sc}(\mathtt{d}x):=\frac{2}{\pi}\sqrt{1-x^{2}}\mathds{1}_{[-1,1]}(x)\mathtt{d}x$,
namely setting $\rho(\mathtt{d}x):=\frac{\mathtt{d}x}{\pi\sqrt{1-x^{2}}}$

$\mu_{V}(\mathtt{d}x)=S(x)\mu_{sc}(\mathtt{d}x),\quad\text{where}\quad
S(x):=\frac{1}{2}\int_{-1}^{1}\frac{V^{\prime}(x)-V^{\prime}(y)}{x-y}\rho(\mathtt{d}y).$

As already mentioned, our main objective is to provide (multivariate)
qualitative and quantitative CLTs for linear statistics of $\beta$-ensembles.
We consider test functions
$\xi_{1},\dots,\xi_{d}\colon\mathbb{R}\to\mathbb{R}$ and we define the random
vector $X=(X_{1},\dots,X_{d})$ by

$X_{k}:=\sum_{i=1}^{n}\xi_{k}(\lambda_{i})-n\int\xi_{k}\mathtt{d}\mu_{V},\quad
1\leq k\leq d.$

For such a vector $X$, as $n$ goes to infinity, following Equations $(1.10)$
and $(1.11)$ in [26], the candidate limiting mean $m=(m_{i})_{1\leq i\leq d}$
and covariance matrix $C=(c_{i,j})_{1\leq i,j,d}$ are given by

$m_{i}:=\left(\frac{1}{2}-\frac{1}{\beta}\right)\left[\frac{\xi_{i}(-1)+\xi_{i}(1)}{2}-\int_{\Sigma_{V}}\xi_{i}(x)\rho(\mathtt{d}x)-\frac{1}{2}\int_{\Sigma_{V}^{2}}\frac{S^{\prime}(x)}{S(x)}\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\rho(\mathtt{d}y)\mu_{sc}(\mathtt{d}x)\right],$
$c_{ij}:=\frac{1}{2\beta}\int\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\frac{\xi_{j}(x)-\xi_{j}(y)}{x-y}(1-xy)\rho(\mathtt{d}x)\rho(\mathtt{d}y).$

The above matrix $C$ is well-defined as soon as the $\xi_{i}$’s are
$1/2$-Hölder continuous. We shall see in Section 4.1.1 below that $C$ is
indeed a covariance matrix as soon as

(1.2) $1,\xi_{1},\dots,\xi_{d}\ \text{are linearly independent}.$

For an open set $U\subset\mathbb{R}$, we write $\mathscr{C}^{r}(U)$ for the
space of $r$ times continuously differentiable functions on $U$ with bounded
derivatives. We endow it with the Banach norm

$\lVert\xi\rVert_{\mathscr{C}^{r}(U)}:=\max_{r^{\prime}\leq r}\sup_{x\in
U}\lvert\xi^{(r^{\prime})}(x)\rvert.$

When $\xi=(\xi_{1},\dots,\xi_{d})$, we also write

$\lVert\xi\rVert_{\mathscr{C}^{r}(U)}=\sum_{k=1}^{d}\lVert\xi_{k}\rVert_{\mathscr{C}^{r}(U)}.$

We write indifferently $\lvert\cdot\rvert$ for the absolute value of a real
number, the Euclidean norm of a vector, or the Euclidean norm of a square
matrix, also known as its Hilbert–Schmidt norm. For a square symmetric matrix
$A$, we write $\lVert A\rVert_{op}$ for its _operator norm_ , that is its
largest singular value. Given two random variables $X$ and $Y$, the _$p-$
Wasserstein distance_ is

$\mathbf{W}_{p}(X,Y):=\inf\left\\{\operatorname{\mathbf{E}}\left[\lvert
X^{\prime}-Y^{\prime}\rvert^{p}\right]:X^{\prime}\overset{\operatorname{\mathbf{law}}}{=}X,Y^{\prime}\overset{\operatorname{\mathbf{law}}}{=}Y\right\\}.$

The Wasserstein distances only depends on $X$ and $Y$ through their respective
laws but we favor a probabilistic notation more suited for the approximation
theorems we establish. The $p-$Wasserstein distance induces a topology
corresponding to convergence in law together with convergence of the $p$-th
moment [39, Thm. 6.9]. We also work with the total variation distance of real-
valued random variables

$\begin{split}\mathbf{TV}(X,Y)&:=\sup\left\\{\operatorname{\mathbf{P}}[X\in
B]-\operatorname{\mathbf{P}}[Y\in B]:B\ \text{Borel}\right\\}.\end{split}$

Let us finally define

$\displaystyle A_{\beta}:=\left\lparen\frac{1}{\beta}\lVert
C^{1/2}\rVert_{op}\lvert C^{-1}\rvert\lVert
N\rVert_{L^{p}}+\left\lvert\frac{1}{2}-\frac{1}{\beta}\right\rvert\lVert
C^{1/2}\rVert_{op}\right\rparen;$ $\displaystyle
a_{\beta}:=\left\lparen\frac{1}{\beta}\frac{1}{\sigma^{2}}+\left\lvert\frac{1}{2}-\frac{1}{\beta}\right\rvert\frac{\sqrt{\pi}}{2}\right\rparen.$

####  1.3.2. Quantitative CLT in total variation and Wasserstein distances

We can now state our main results concerning the quantitative behavior of the
fluctuations of linear statistics of $\beta$-ensembles. The following bounds
are optimal or nearly optimal depending on the regularity of the test
functions.

######  Theorem 1.1.

Let us fix $1\leq p<\infty$ and $\alpha>0$. There exists $K_{p,\alpha}>0$ such
that for all test functions
$\xi_{1},\dots,\xi_{d}\in\mathscr{C}^{{6}}(\mathbb{R})$ satisfying 1.2, we
have, with $N$ a standard Gaussian in $\mathbb{R}^{d}$

$\mathbf{W}_{p}(X,C^{1/2}N+m)\leq
K_{p,\alpha}A_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1-\alpha}}.$

In the univariate case $d=1$, with $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, we
also have

$\mathbf{TV}(X,\sigma N+m)\leq
K_{1,\alpha}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1-\alpha}}.$

Assuming more regularity on the test functions, we obtain an optimal rate of
convergence.

######  Theorem 1.2.

For $1\leq p<\infty$ and $\alpha>0$, there exists $K_{p,\alpha}>0$ such that
for all test functions
$\xi_{1},\dots,\xi_{d}\in\mathscr{C}^{{14}}(\mathbb{R})$ satisfying 1.2, we
have, with $N$ a standard Gaussian in $\mathbb{R}^{d}$

$\mathbf{W}_{p}(X,C^{1/2}N+m)\leq
K_{p}A_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{7}(\mathbb{R})}^{2}}{n}.$

In the univariate case $d=1$, with $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, we
also have

$\mathbf{TV}(X,\sigma N+m)\leq
K_{1}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}+\lVert\xi\rVert_{\mathscr{C}^{7}(\mathbb{R})}^{2}}{n}.$

We also prove a theorem for functions of lower regularity. For
$\gamma\in(0,1)$, we write $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$,
provided $\xi\in\mathscr{C}^{1}(\mathbb{R})$ and it satisfies

$\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}:=\lVert\xi\rVert_{\mathscr{C}^{1}(\mathbb{R})}+\sup_{x\neq
y}\frac{\lvert\xi^{\prime}(x)-\xi^{\prime}(y)\rvert}{\lvert
x-y\rvert^{\gamma}}<\infty.$

######  Theorem 1.3.

Let $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ for some $\gamma\in(0,1)$.
Setting $\sigma:=\sqrt{c_{11}}$ and $m=m_{1}$, and for any
$a<\frac{\gamma}{6+\gamma}$, we have

$\mathbf{TV}(X,\sigma N+m)\leq
K_{a}a_{\beta}\frac{\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}}{n^{a}}.$

####  1.3.3. Super-convergence

We now turn to our qualitative results and the reinforcement of the mode of
convergence is the smooth case. We say that a sequence $(X_{n})$ _super-
converges_ to a non-degenerate Gaussian $N\sim\mathcal{N}(m,\sigma^{2})$ with
density $\varphi_{m,\sigma^{2}}$ provided that, for all $r\in\mathbb{N}$,
there exists $n_{0}\in\mathbb{N}$ such that for $n\geq n_{0}$, the law of
$X_{n}$ admits a density $\varphi_{n}\in\mathscr{C}^{r}(\mathbb{R})$, and

$\limsup_{n\to\infty}\lVert\varphi_{n}-\varphi_{m,\sigma^{2}}\rVert_{\mathscr{C}^{r}(\mathbb{R})}=0.$

Regarding linear statistics, we obtain that convergence in law can easily be
upgraded to super-convergence. For $\alpha>0$, we say that
$\xi\in\mathscr{C}^{1}(\mathbb{R})$ is _$\alpha$ -regular_ provided that

$\mathsf{Leb}(x\in\mathbb{R}^{d}:|\xi^{\prime}(x)|\leq\epsilon)\lesssim\epsilon^{\alpha}.$

It is in particular true for any polynomial function $\xi$.

######  Theorem 1.4.

Let $\xi\in\mathscr{C}^{\infty}(\mathbb{R})$ be $\alpha$-regular such that
$(X_{n})$ converges in law to a Gaussian variable
$N\sim\mathcal{N}(m,\sigma^{2})$. Then, $(X_{n})$ super-converges to $N$.

###  1.4. Comparison with existing results

####  1.4.1. Gaussian fluctuations for linear statistics

The mathematical study of fluctuations of linear statistics around their
equilibrium starts with the seminal work [24]. His method consists in writing
the Laplace transform of a linear statistics as the ratio of a partition
function associated with a perturbed potential by that associated with the
initial potential. Interestingly, the perturbation involves the so-called
master operator, noted $\Theta_{V}$ in the sequel, see section 3.1 for precise
definition. This operators plays also a central role in our approach. The
method developed in [24] has been continued in [25, 37, 9, 10] to obtain
qualitative CLT with increasing levels of generality, allowing, for instance,
results in the multi-cut regime. All the above results require the potential
$V$ to be analytic. Using the same approach on the Laplace transform, [5]
proves a CLT when $V\in\mathscr{C}^{5}(\mathbb{R})$ and
$\xi\in\mathscr{C}^{3}_{c}(\mathbb{R})$. To the best of our knowledge, these
are the best results available regarding the regularity of $V$ and $\xi$. In
contrast, we obtain a CLT for $V\in\mathscr{C}^{7}(\mathbb{R})$ but allowing
to lower the regularity to $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$. Our
results are also limited to the single-cut case. We stress that in the multi-
cut regime, non-Gaussian fluctuations are known, and additional compatibility
conditions on $\xi$ are required to ensure a CLT. We also mention [6] that
derives a CLT for linear statistics at all meso-scales, that is for linear
statistics where the particles are rescaled by a factor $n^{-\alpha}$ for
$\alpha\in(0,1)$. Even if our main result allows to recover CLT at some meso-
scale, we cannot reach their full range with our method.

####  1.4.2. Stein’s method and $\beta$-ensemble

To the best of our knowledge, the reference [26] is the first to implement
Stein’s method in the setting of general $\beta$-ensembles and to provide
quantitative bounds in $2-$Wasserstein distance which are near optimal for
$V(x)=x^{2}$ but sub-optimal for more general potentials. Their method works
for linear statistics that are close to be eigenvalues of the Dyson generator,
and technically completing their requires, among others, the tour de force of
diagonalizing the so-called master operator. Relying on the concept of
exchangeable pairs which plays an important role in Stein’s method (see e.g.
[30]), quantitative CLTs for linear statistics of Haar distributed random
variables on compact classical groups are established in [16, 15] and later
on, in the context of circular $\beta$-ensembles, in [40] where exchangeable
pairs are built through the $n$-dimensional circular Dyson Brownian motion.
Finally, the contribution [21] which builds upon the techniques given in [26]
to provide quantitative (near optimal) CLTs in the high temperature regime.

Let us also mention the seminal article [13]. There, the author introduced a
variation around Stein’s method, based on _second-order Poincaré inequality_
in order to study fluctuations of eigenvalues of matrices with random
coefficients, possibly not identically distributed, whose distributions admit
suitable densities. It is somewhat complicated to compare the results there
with ours, simply because for general potentials $V$ we cannot a priori
interpret $\beta$-ensembles as spectrum of random matrices with independent
coefficients. Besides, [13, Thm. 4.2] obtains convergence to a Gaussian of
random variables of the form $\operatorname{Tr}A_{n}^{p_{n}}$ where $(A_{n})$
are some random matrices as above, and $p_{n}=o(\log n)$. Note that, in
comparison, in our context, it should be possible to get CLTs for
$\xi_{n}(x)=x^{p_{n}}$ when $p_{n}$ grows polynomially although the models are
pretty different and only coincide for $V(x)=x^{2}$.

We would like to point out that despite the aforementioned articles and ours
being inspired by the philosophy of Stein’s method, they strongly differ in
the way Stein’s method is implemented. Indeed, [26] relies on ideas coined in
[30] using exchangeable pairs while our builds on a novel refinement of the
well established Malliavin Stein’s approach, which allows to handle random
variables which are not in the image of the underlying Markov generator.
Indeed, whenever $X$ belongs to the image of $\mathsf{L}$ one can directly
provide a Stein’s kernel by setting
$\tau(x)=\mathbb{E}\left[\Gamma\left[X,-L^{-1}X\right]\right]$ which fulfils
the equation $\mathbb{E}\left[\phi^{\prime}(X)\tau(X)-X\phi(X)\right]=0$. The
great advantage of this point of view is to control the total variation
distance (among others) by $\mathbb{E}\left[|1-\tau(X)|\right]$ but it
requires the costly assumption that $\mathsf{L}$ is invertible which is not
true in general. To overcome this obstacle, we notice that it is enough to
show that $X$ is near $\text{Im}(Z)$ in the following sense: $X=LF+Z$ for $Z$
small in probability. This simple remark, which is new to the best of our
knowledge, increases considerably the applicability of the above Malliavin-
Stein’s approach and still enables to provide total variation bounds but is a
priori not sufficient to build a Stein’s kernel. While this being noticed, our
strategy then consists in proving that any linear statistics is close to the
range of $\mathsf{L}$, a step which proceeds from the invertibility of the
master operator. As a result, our findings thus refine those of [26] by
providing optimal rates of convergence for stronger metrics such as the total
variation distance and we believe that our approach could also be successfully
implemented in the framework of circular $\beta$-ensembles and Haar
distributed matrices on compact groups.

####  1.4.3. Super-convergence

To the best of our knowledge, in the context of random matrices or
$\beta$-ensembles, the question of establishing convergence in metrics
stronger than total variation or Wasserstein distances has not been considered
yet. We deploy here ideas that are classical in the framework of Malliavin
calculus and stochastic analysis and which merely consists in establishing
negative moments for the carré-du-champ operator applied to the considered
linear statistics. Then, relying on integration by parts techniques, we are
able to prove strong forms of convergence for the densities of the underlying
sequences of random variables.

###  1.5. Outline of the proofs an plan of the paper

Let us give here more details on our strategy of proof and on the plan of the
paper.

####  1.5.1. The generator of the Dyson Brownian motion

The overall strategy behind our quantitative and qualitative estimates
leverage the characterization of the $\beta$-ensemble as the unique invariant
distribution of the Dyson Brownian motion. Namely, consider the _generator of
the Dyson Brownian motion_

(1.3) $\mathsf{L}=\mathsf{L}_{n,\beta,V}=:=\Delta-\beta\nabla
H\cdot\nabla=\sum_{i=1}^{n}\partial_{\lambda_{i}}^{2}-\beta
n\sum_{i=1}^{n}V^{\prime}(\lambda_{i})\partial_{\lambda_{i}}+\frac{\beta}{2n}\sum_{i\neq
j}\frac{\partial_{\lambda_{i}}-\partial_{\lambda_{j}}}{\lambda_{i}-\lambda_{j}}.$

The operator $\mathsf{L}$ is the diffusive Markov generator canonically
associated with the $\beta$-ensemble. We have indeed the following
characterization of the $\beta$-ensemble Gibbs measure
$\operatorname{\mathbf{P}}$:

$\tilde{\operatorname{\mathbf{P}}}=\operatorname{\mathbf{P}}\Leftrightarrow\left\lparen\tilde{\operatorname{\mathbf{E}}}\mathsf{L}F=0,\
\forall
F\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n})\right\rparen\Leftrightarrow\left\lparen\tilde{\operatorname{\mathbf{E}}}\left[F\mathsf{L}G\right]=\tilde{\operatorname{\mathbf{E}}}[G\mathsf{L}F],\
\forall F,G\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n})\right\rparen.$

The differential structure induced by $\mathsf{L}$ on $\mathbb{R}^{n}$,
technically called a _Dirichlet structure_ , comes with a _carré du champ_
operator

$\Gamma[F,G]:=\nabla F\cdot\nabla G,\qquad
F,\,G\in\mathscr{C}^{1}(\mathbb{R}^{n}).$

We have the following integration by parts formula

(1.4)
$\operatorname{\mathbf{E}}\left[\Gamma[F,G]\right]=-\operatorname{\mathbf{E}}\left[F\mathsf{L}G\right],\qquad
F,\,G\in\mathscr{C}^{\infty}_{c}(\mathbb{R}^{n}).$

####  1.5.2. $\Gamma$-Stein’s method for $\beta$-ensemble

At an informal level, the data of a diffusive Markov generator generally
combines well with Stein’s method to provide quantitative bounds for normal
approximation. Provided $\ker\mathsf{L}$ is limited to constant functions,
$\mathsf{L}$ is invertible on mean-zero functions, and the celebrated
_$\Gamma$ -Stein_ approach [32, 34, 1, 27], or _Malliavin–Stein_ approach in
the setting of Gaussian fields or Poisson point processes, yields that the
variance of $\Gamma[X,-\mathsf{L}^{-1}X]$ controls the Gaussian fluctuations
of $X$, in total variation or Wasserstein distance. However, in the case
$\beta$-ensembles, the operator $\mathsf{L}$ is in general _not invertible_.

To overcome this difficulty, we amend the classical $\Gamma$-Stein approach.
Intuitively, whenever $X=\mathsf{L}F$ for some $F$, $\mathsf{L}^{-1}X$ makes
sense despite the non-invertibility of $\mathsf{L}$. In this case, it is
natural to expect that $\Gamma[X,-\mathsf{L}^{-1}X]=\Gamma[X,-F]$ controls the
Gaussian fluctuations of $X$. In the next Section 2 and more precisely in
Theorem 2.1 below, we formalize this intuition by proving quantitative bounds
for the normal approximation, in Wasserstein distance and total variation, of
random variables of the form $X=\mathsf{L}F+Z$. Provided, $Z$ is small and
that $\Gamma[X,-F]$ is close to a constant $\sigma^{2}$, then $X$ is close to
a Gaussian with variance $\sigma^{2}$. This theorem should not come as a
surprise to Stein’s method aficionados and its proof is rather
straightforward. However, this observation has, to the best of our knowledge,
never been remarked and this “almost invertibility” decomposition is precisely
what allows us to provide an efficient proof. We stress that our bounds hold
for generic abstract diffusive Markov generators and we believe it could prove
useful in other statistical physics models where the Gibbs measure has an
explicit density.

####  1.5.3. The master equation

The next Section 3 of the paper consists in checking that the Dyson generator
is indeed in line with the global strategy developed in Section 2. In order to
apply our abstract bound, we need to show that our linear statistics $X$ is of
the form $\mathsf{L}F+Z$. With the notations of Section 3, using mostly
algebraic manipulations together with the minimality of $\mu_{V}$ with respect
to the mean-field energy $\mathcal{I}_{V}$, we show, in Theorem 3.8, that
whenever $F:=\sum_{i=1}^{n}f(\lambda_{i})$ is a linear statistic, setting
$m_{f}:=\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\langle
f^{\prime\prime}|\mu_{V}\rangle$ which is such that the term in parenthesis
below is asymptotically centered, we have

$\frac{1}{n\beta}\mathsf{L}F=\left(\sum_{i=1}^{n}(\Theta_{V}f^{\prime})(\lambda_{i})-n\int(\Theta_{V}f^{\prime})\mathtt{d}\mu_{V}-m_{f}\right)+\frac{Z}{\beta},$

where $Z$ is a quadratic (which should be thought as a remainder) term and
$\Theta_{V}$ is the so-called _master operator_. In particular, under
Assumption 1, $\Theta_{V}$ is invertible in a neighborhood of $[-1,1]$.
Namely, there exists an open neighborhood $U$ of $[-1,1]$, such that given
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, we can find
$\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$ and $c_{\xi}\in\mathbb{R}$ satisfying

$(\Theta_{V}\psi)(x)=\xi(x)+c_{\xi},\qquad x\in U.$

Under Assumption 1, by [11] the $(\lambda_{i})$ enjoy a strong rigidity
property, precisely recalled in Theorem 3.3. Informally, the probability that
the $\lambda_{i}$’s deviate from the grid given by the $i/n$-th quantiles of
$\mu_{V}$ is overwhelmingly small. Note that this rigidity phenomenon is also
used in [26]. Thus up to paying a loss factor, that is negligible compared to
our bound, we can assume that all the $\lambda_{i}$’s are localized in the
neighborhood $U$. By the master equation, choosing $f$ any primitive of
$\psi$, we are thus left with establishing the quantitative CLT for the random
variable

$X=\frac{1}{n\beta}\mathsf{L}F-\frac{Z}{\beta},$

which falls precisely under the scope of our Stein’s bound.

####  1.5.4. Controlling the carré du champ and the remainder

In Section 4, we concretely implement our strategy and complete the proofs of
our main results. We first establish the quantitative statements, namely
Theorem 1.1, Theorem 1.2 and Theorem 1.3. In view of our strategy, we show
that $(n\beta)^{-1}\Gamma[X,-F]$ is close to a constant and that
$(\beta)^{-1}Z$ is small. We control the term involving the carré du champ by
relying on a simple yet remarkable property of the carré du champ: it
preserves linear statistics. Precisely, we have that

$\Gamma\left[\sum\varphi(\lambda_{i}),\sum\chi(\lambda_{i})\right]=\sum\varphi^{\prime}(\lambda_{i})\chi^{\prime}(\lambda_{i}).$

It follows that

$\frac{1}{n}\Gamma[X,F]=\frac{1}{n}\sum\xi^{\prime}(\lambda_{i})f^{\prime}(\lambda_{i}).$

By the convergence to equilibrium, the above quantity converges to
$\int\xi^{\prime}f^{\prime}\mathtt{d}\mu_{V}$. Using the minimality property
of $\mu_{V}$, it can be shown that this last integral coincides with limiting
covariance given above. To derive quantitative bounds from there, we rely on a
quantitative convergence to equilibrium which also follows from the rigidity
estimate from [11].

Handling the quadratic remainder $Z$ is more tedious. We use Fourier inversion
to decompose $Z$ into a product of linear statistics. By the aforementioned
quantitative law, one of term goes to $0$ at a given rate while the other
stays bounded. This allows us to conclude for the proof Theorem 1.1. From
there, Theorem 1.2 follows by a bootstrap argument: we essentially redo the
same proof but instead of using the sub-optimal quantitative law of large
numbers provided by [11], we use the CLT we just established that gives us a
law of large numbers at speed $1/n$. To conclude for Theorem 1.3, we
approximate $\xi\in\mathscr{C}^{1,\gamma}(\mathbb{R})$ by a sequence of smooth
functional $\xi_{\varepsilon}$ while choosing $\varepsilon\simeq n^{a}$, for a
well-chosen $a$.

####  1.5.5. Super-convergence

The proof of our qualitative result Theorem 1.4 is finally given in Section
4.4. The derivation of the super-convergence relies on a lemma, classical in
Dirichlet forms / Malliavin calculus theory, stating that negative moments of
$\Gamma[X,X]$ control the Sobolev norms of the density of $X$. We rely again
on the fact that $\Gamma$ preserves linear statistics. In this precise case,

$\Gamma{[X,X]}=\sum(\xi^{\prime}(\lambda_{i}))^{2}.$

We obtain negative moments through a direct control on
$\operatorname{\mathbf{P}}\left[\sum(\xi^{\prime}(\lambda_{i}))^{2}\leq\varepsilon\right]$.

##  2\. Stein’s method for Markov diffusive operators

In order to carry out our program, we need to establish quantitative bounds in
total variation for random variables which are close in some sense to the
range of a given Markov diffusive operator $\mathsf{L}$. To the best of our
knowledge, the following estimates seem to be new in the well-studied area of
Malliavin–Stein’s method and are of independent interest. We stress that all
the results of this section are valid for any diffusive Markov operator
$\mathsf{L}$ associated with $\Gamma$ the so-called carré du champ and
$\operatorname{\mathbf{P}}$ the invariant measure for $\mathsf{L}$. We refer
to [2] for definitions in this abstract setting. Indeed, the only properties
used in our proofs are the chain rule and the integration by parts which hold
in full generality. The cornerstone of our method relies on the following
Theorem which will proved in Section 2.3 based on the content of Sections 2.1
and 2.2.

In case of vector-valued random variables $F$ and $G$, we extend our
definition to the _matrix-valued carré du champ_

$\Gamma[F,G]_{ij}:=\Gamma[F_{i},G_{j}],\qquad i,j=1,\dots,d.$

######  Theorem 2.1.

Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$,
$Z_{1},\dots,Z_{d}\in L^{p}$, and
$X:=(\mathsf{L}F_{1}+Z_{1},\dots,\mathsf{L}F_{d}+Z_{d})$. Let $\Sigma$ be a
positive symmetric matrix, $C=\Sigma^{2}$ and $N$ be standard Gaussian in
$\mathbb{R}^{d}$. Then, we have

(2.1) $\begin{split}\mathbf{W}_{p}(X,\Sigma
N)&\leq\lVert\Sigma\rVert_{op}\lVert
N\rVert_{L^{p}}\lVert\mathrm{id}-\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}\\\ &\leq\lVert\Sigma\rVert_{op}\lVert C^{-1}\rVert\lVert
N\rVert_{L^{p}}\lVert
C-\Gamma[X,-F]\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}.\end{split}$

Moreover, in the univariate case $d=1$, setting $\sigma=\Sigma_{11}$, we
obtain

(2.2) $\mathbf{TV}(X,\sigma
N)=\mathbf{TV}(\sigma^{-1}X,N)\leq\frac{2}{\sigma^{2}}\lVert\sigma-\Gamma[X,-F]\rVert_{L^{1}}+\frac{\sqrt{\pi}}{2}\lVert
Z\rVert_{L^{1}}.$

###  2.1. Reminders on Stein kernels

Let us first recall important results regarding Stein kernels. Here we work on
a general probability space $(\Omega,\mathfrak{W},\operatorname{\mathbf{P}})$.
Given a multivariate random variable $X=(X_{1},\dots,X_{d})$, we say that a
matrix-valued random variable $\tau$, measurable with respect to $X$, is a
_Stein kernel_ for $X$ provided

$\operatorname{\mathbf{E}}\left[X\cdot\nabla\varphi(X)\right]=\operatorname{\mathbf{E}}\left[\tau\cdot\nabla^{2}\varphi(X)\right],\qquad\varphi\in\mathscr{C}^{\infty}(\mathbb{R}^{d}).$

Heuristically, Stein kernels are relevant for normal approximation, since $X$
is a standard multivariate normal variable if and only if it admits
$\tau=\mathrm{id}$ as a Stein kernel. A key observation at the heart of
Stein’s method are the following quantitative normal approximation
inequalities.

######  Lemma 2.2 (Stein kernel bounds [27, 18, 31] ).

Let $X=(X_{1},\dots,X_{d})$, let $N$ be a standard normal variable on
$\mathbb{R}^{d}$, and $\tau$ be Stein kernel for $X$.

  1. (i)

Let $p\in[1,\infty)$. If $X\in L^{p}(\Omega)$, then

$\mathbf{W}_{p}(X,N)^{p}\leq\operatorname{\mathbf{E}}\left[\lvert
N\rvert^{p}\right]\operatorname{\mathbf{E}}\left[\lVert\tau-\mathrm{id}\rVert^{p}\right].$

  2. (ii)

In the univariate setting $d=1$, we have

$\mathbf{TV}(X,N)\leq
2\operatorname{\mathbf{E}}\left[\lvert\tau-1\rvert\right].$

###  2.2. Computations of Stein kernels on $\operatorname{Im}\mathsf{L}$

The celebrated Malliavin–Stein method [32] puts forward a strikingly efficient
way to compute a Stein kernel for sufficiently smooth functionals of an
infinite-dimensional Gaussian field.

######  Lemma 2.3.

Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$. Let
$X:=(\mathsf{L}F_{1},\dots,\mathsf{L}F_{d})$, and define the matrix-valued
random variable $\tau=(\tau_{ij})$, where

$\tau_{ij}:=\operatorname{\mathbf{E}}\left[\Gamma[X_{i},-F_{j}]\nonscript\>|\allowbreak\nonscript\>\mathopen{}X\right],\qquad
i,j=1,\dots,d.$

Then, $\tau$ is a Stein kernel for $X$.

###### Proof.

Take $\varphi\in\mathscr{C}^{\infty}(\mathbb{R}^{d})$. Then, using the
integration by parts, and then the chain rule, we get

$\operatorname{\mathbf{E}}\left[X\cdot\nabla\varphi(X)\right]=\sum_{i=1}^{l}\operatorname{\mathbf{E}}\left[\mathsf{L}F_{i}\partial_{i}\varphi(X)\right]=-\sum_{i,j}\operatorname{\mathbf{E}}\left[\partial_{ij}\varphi(X)\Gamma[F_{i},X_{j}]\right].$

This concludes the proof in view of the definition of a Stein kernel. ∎

Combining Lemmas 2.2 and 2.3, we obtain the following $\Gamma$-Stein bound.

######  Theorem 2.4.

Take $F_{1},\dots,F_{d}\in\operatorname{\mathbb{D}om}\mathsf{L}$, and
$X:=(\mathsf{L}F_{1},\dots,\mathsf{L}F_{d})$. Then

$\mathbf{W}_{p}(X,N)^{p}\leq\operatorname{\mathbf{E}}\left[\lvert
N\rvert^{p}\right]\operatorname{\mathbf{E}}\left[\lVert\operatorname{id}-\Gamma[X,-F]\rVert^{p}\right].$

Moreover, in the univariate case $d=1$, we find

$\mathbf{TV}(X,N)\leq 2\operatorname{\mathbf{E}}\left[\lvert
1-\Gamma[X,-F]\rvert\right].$

######  Remark 2.5.

The classical approach on the Wiener space relies on the invertibility of the
generator of the Ornstein–Uhlenbeck process. In this case, it is known that

$\tau_{ij}:=\operatorname{\mathbf{E}}\left[\Gamma[X_{i},-\mathsf{L}^{-1}X_{j}]\nonscript\>|\allowbreak\nonscript\>\mathopen{}X\right],\qquad
i,j=1,\dots,d,$

is a Stein kernel for $X\in L^{2}(\Omega)$ with
$\operatorname{\mathbf{E}}X=0$. This strategy has also been applied in the
setting of other diffusive Markov generators with discrete spectrum, see for
instance [1, 27]. This actually could be further generalized to any diffusive
Markov generators that is invertible away from constants. Note that, the
generator of the Dyson Brownian motion $\mathsf{L}$ is in general not
invertible, in particular it is a priori not true that a given linear
statistics belongs to the range of $\mathsf{L}$. Therefore, we need to extend
the above techniques for random variables not belonging to the range of
$\mathsf{L}$ which is precisely the object of the next section.

###  2.3. Normal approximation away from $\operatorname{Im}\mathsf{L}$

Given $X$, finding $F$ such that $\mathsf{L}F=X$ is a demanding task. Indeed
as already mentioned, $\mathsf{L}$ may be not invertible. Nevertheless, as we
now demonstrate, the bounds obtained by the Stein’s method can be amended in
order to cover the case where $X$ is close to the range of $\mathsf{L}$ that
is to say of the form $X=\mathsf{L}F+Z$ with small perturbation $Z$.

###### Proof of Theorem 2.1.

We first prove the claim in the case $\Sigma=\mathrm{id}$, and then explain
how to handle the case of general non-degenerate covariance. For the upper
bound 2.1, by the triangle inequality, we have that

$\mathbf{W}_{p}(X,N)\leq\mathbf{W}_{p}(\mathsf{L}F,N)+\mathbf{W}_{p}(\mathsf{L}F,\mathsf{L}F+Z).$

We apply Lemma 2.3 on the first term on the right-hand side. For the second
term by choosing the coupling $(\mathsf{L}F,\mathsf{L}F+Z)$ in the
minimization problem defining $\mathbf{W}_{p}$, we see that

$\mathbf{W}_{p}(\mathsf{L}F,\mathsf{L}F+Z)^{p}\leq\operatorname{\mathbf{E}}\left[\lVert\mathsf{L}F-\mathsf{L}F+Z\rVert^{p}\right]=\operatorname{\mathbf{E}}\left[\lVert
Z\rVert^{p}\right].$

This concludes the proof of 2.1.

We now prove 2.2. Since, by the chain rule an integration by parts, we have

$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\Gamma[X,-F]\right]=\operatorname{\mathbf{E}}\left[\varphi(X)\mathsf{L}F\right]=\operatorname{\mathbf{E}}\left[\varphi(X)X\right]-\operatorname{\mathbf{E}}\left[\varphi(X)Z\right],$

we find that

$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)-X\varphi(X)\right]=\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)(1-\Gamma[X,-F])\right]-\operatorname{\mathbf{E}}\left[\varphi(X)Z\right].$

Recalling that, by [32, Thm. 3.3.1],

$\mathbf{TV}(X,N)\leq\sup\left\\{\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\right]-\operatorname{\mathbf{E}}\left[X\varphi(X)\right]:\lVert\varphi\rVert_{\infty}\leq\sqrt{\frac{\pi}{2}},\,\lVert\varphi^{\prime}\rVert_{\infty}\leq
2\right\\},$

this concludes the proof in the case where $\Sigma=\mathrm{id}$. Now we take
an general positive, hence invertible, covariance matrix $\Sigma$. On the one
hand, by the stability of Wasserstein distances under the Lipschitz map
$\mathbb{R}^{n}\ni v\mapsto\Sigma v$, we find that

$\mathbf{W}_{p}(X,\Sigma
N)\leq\lVert\Sigma\rVert_{op}\mathbf{W}_{p}(\Sigma^{-1}X,N).$

Recall that $\lVert\cdot\rVert_{op}$ is the operator norm with respect to the
Euclidean norm on $\mathbb{R}^{n}$. On the other hand, a routine computation
yields

$\Gamma[\Sigma^{-1}X,-\Sigma^{-1}F]=\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}.$

Thus by 2.1, we get

(2.3) $\begin{split}\mathbf{W}_{p}(X,\Sigma
N)&\leq\lVert\Sigma\rVert_{op}\lVert
N\rVert_{L^{p}}\lVert\mathrm{id}-\Sigma^{-1}\Gamma[X,-F]\Sigma^{-1}\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}\\\ &\leq\lVert\Sigma\rVert_{op}\lVert C^{-1}\rVert\lVert
N\rVert_{L^{p}}\lVert
C-\Gamma[X,-F]\rVert_{L^{p}}+\lVert\Sigma\rVert_{op}\lVert
Z\rVert_{L^{p}}.\end{split}$

Similarly, in the univariate case $d=1$, we obtain the bound 2.2. ∎

##  3\. The master equation for linear statistics

We now apply the content of the previous section to the specific operator
$\mathsf{L}$ which is the generator of the Dyson Brownian motion which was
given in equation (1.3).

###  3.1. Notations and introduction of relevant operators

We show here that linear statistics associated with $\beta$-ensembles can
indeed be expressed in the form $\mathsf{L}F+Z$ thus making them good
candidates to apply our Theorem 2.1. For a more concise exposition of our
results and proofs, we adopt the following notations related to the empirical
measure:

$\displaystyle\nu_{n}:=\sum_{i=1}^{n}\delta_{\lambda_{i}},\ \text{and}\quad\
\bar{\nu}_{n}:=\nu_{n}-n\mu_{V};$ $\displaystyle\mu_{n}:=\frac{\nu_{n}}{n},\
\text{and}\quad\ \bar{\mu}_{n}:=\mu_{n}-\mu_{V}.$

We will write $\langle f|\mu\rangle$ for the integration of a function $f$
against a measure $\mu$. We consider the following operators, acting on
functions $f\in\mathscr{C}^{2}(\mathbb{R})$:

$\displaystyle
T_{V}(f)(x):=\int\frac{f(x)-f(y)}{x-y}\mathtt{d}\mu_{V}(y)=\int\int_{0}^{1}f^{\prime}((1-u)x+uy)\mathtt{d}u\mu_{V}(\mathtt{d}y);$
$\displaystyle\Theta_{V}(f):=-V^{\prime}f+T_{V}(f);$ $\displaystyle
T_{n}(f)(x):=\int\int_{0}^{1}f^{\prime\prime}((1-u)x+uy)\mathtt{d}u\mu_{n}(\mathtt{d}y)=\frac{1}{n}\sum_{i=1}^{n}\frac{f^{\prime}(x)-f^{\prime}(\lambda_{i})}{x-\lambda_{i}}1_{x\neq\lambda_{i}}+f^{\prime\prime}(\lambda_{i})1_{x=\lambda_{i}};$
$\displaystyle\Theta_{n}(f):=-V^{\prime}f^{\prime}+T_{n}(f).$

To some extend, $T_{n}$ and $\Theta_{n}$ are the empirical counterparts of
$T_{V}$ and $\Theta_{V}$. Since $\mu_{V}$ is compactly supported and since $f$
is chosen $\mathscr{C}^{2}$, the integrals above are all convergent. This
shows in particular that the operators $T_{V}$ and $\Theta_{V}$ are well-
defined. We recall an _invertibility result_ for $\Theta_{V}$ taken from [5].

######  Lemma 3.1 ([5, Lem. 3.3]).

There exists $\delta>0$, such that if $U:=(-1-\delta,1+\delta)$, for all
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, there exist a constant $c_{\xi}$ and
$\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$ such that

(3.1) $\displaystyle(\Theta_{V}\psi)(x)=\xi(x)+c_{\xi},\qquad x\in U,$ (3.2)
$\displaystyle\lVert\psi\rVert_{\mathscr{C}^{5}(\mathbb{R})}\leq
C\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}.$

For the rest of the paper, we fix the margin $\delta>0$ and the associated
neighborhood $U$ given by Lemma 3.1 above.

######  Remark 3.2.

  1. (a)

A slight variant of this invertibility result can be found in [4, Lem. 3.2].

  2. (b)

We do not assume [5, (H3)], however as noticed in this last reference, this
assumption is not used in the proof of [5, Lem. 3.3].

  3. (c)

Since we are in the single-cut case, with no singular density, with the
notations of [5], we have here $\mathsf{k}=0$ and $\mathsf{m}=0$.

###  3.2. The rigidity estimates and its consequences

Many ingredients in our proof proceed from a rigidity result from [11] which
is also used in [26].

######  Theorem 3.3 ([11, Thm 2.4]).

Let $\rho_{0}:=-1$, and $\rho_{j}$ be the $j/n$-quantile of $\mu_{V}$ for
$j\in\\{1,\cdots,n\\}$. Set $\hat{\jmath}:=\min(j,n-j+1)$ and write
$(\lambda_{(j)})$ for the increasingly-ordered vector of $\lambda_{j}$’s. For
any $\varepsilon>0$, there exist $c_{\varepsilon}>0$ and $N_{\varepsilon}>0$
such that for all $n\geq N_{\varepsilon}$

(3.3) $\operatorname{\mathbf{P}}\left[\exists
j\in[1,n]:\lvert\lambda_{(j)}-\rho_{j}\rvert>\hat{\jmath}^{-\frac{1}{3}}n^{-\frac{2}{3}+\varepsilon}\right]\leq\exp(-n^{c_{\varepsilon}}).$

As anticipated, we repeatedly use in our proofs, the following corollary on
the control of the outliers. Philosophically, it allows us to work on
$U=(-1-\delta,1+\delta)$ and to discard the rest up to accepting a negligible
loss. The next result follows directly from the Theorem 3.3 and fact that the
locations $(\rho_{j})_{1\leq j\leq n}$ belong to $[-1,1]$.

######  Lemma 3.4.

There exists $C>0$ and $c>0$ such that for $n$ large enough

(3.4) $\operatorname{\mathbf{P}}\left[\max_{1\leq i\leq n}|\lambda_{i}|\geq
1+\delta\right]\leq C\exp\left\lparen-n^{c}\right\rparen.$

######  Remark 3.5.

Similar results are numerous in the literature: even with weaker assumptions
on the potential $V$ than the ones in [11] for which the rigidity is not yet
known, one can derive better rate of convergence, such as exponential decay.
Such exponential decay follows, for instance, from the large deviations
principle for the extreme positions, as in [9, Prop. 2.1]. Under strong
assumptions on the potential $V$, the reference [29, Lem. 4] offers an
exponential decay with an explicit dependence on $\delta$, which, of course we
do not need since our margin $\delta$ is fixed. See also [12, Thm. 1.12] for
similar a similar result under weaker assumptions in dimension greater than
two. As the rigidity estimate given by Theorem 3.3 is in fact used to quantify
the speed of convergence to equilibrium in the next Lemma 3.6 and since the
associated bound 3.4 is sufficient for our purpose, we have decided to use the
rigidity also to control the outliers.

As proved in [26], Theorem 3.3 provides a polynomial speed of convergence to
equilibrium. Namely, they obtain the following lemma that we reproduce below
for completeness.

######  Lemma 3.6 ([26, Lemma 5.3]).

Let $\alpha>0$ and $p\geq 1$. Then, there exists $K_{p,\alpha}>0$ such that
for any bounded and Lipschitz function $f$ we have

(3.5) $\lVert\langle
f|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-\alpha}}K_{p,\alpha}(\lVert
f\rVert_{\infty}+\lVert f^{\prime}\rVert_{\infty}).$

######  Remark 3.7.

As we shall see below, our findings allow to upgrade the rate of convergence
given in Lemma 3.6 to get an optimal rate $O(n^{-1})$ for regular enough test
functions $f$.

###  3.3. The master equation

In relation to the Dyson generator $\mathsf{L}$, the master operator
$\Theta_{V}$ allows us to derive the following _master equation_ , which is at
the heart of our argument.

######  Theorem 3.8.

Consider a test function $f\in\mathscr{C}^{2}(\mathbb{R})$ and define
$F:=\langle f|\bar{\nu}_{n}\rangle$. Then, we have the following decomposition

(3.6) $\frac{\mathsf{L}F}{n}=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle+\beta\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}-T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.$

###### Proof.

Recalling the definitions of $\mathsf{L}$ and $T_{n}$, we get that

(3.7) $\begin{split}\frac{\mathsf{L}F}{n}&=\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\nu_{n}\rangle+\frac{\beta}{2n}\sum_{i\neq
j}^{n}\frac{f^{\prime}(\lambda_{i})-f^{\prime}(\lambda_{j})}{\lambda_{i}-\lambda_{j}}\\\
&=\frac{2-\beta}{2}\langle f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\nu_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle\\\ &=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle-n\beta\langle
V^{\prime}f^{\prime}|\mu_{V}\rangle.\end{split}$

Recall that the Euler–Lagrange equation for the minimality of $\mu_{V}$ with
respect to the energy reads, for some $C_{V}\in\mathbb{R}$:

$V(x)-\int\log(|x-y|)\mu_{V}(\mathtt{d}y)=C_{V},\qquad x\in\Sigma_{V}.$

Differentiating with respect to $x$ leads to

$V^{\prime}(x)-\int\frac{1}{x-y}\mu_{V}(\mathtt{d}y)=0,\qquad x\in\Sigma_{V}.$

From there, multiplying by $f^{\prime}(x)$, integrating with respect to
$\mu_{V}(\mathtt{d}x)$, and symmetrizing yields

$\langle
V^{\prime}f^{\prime}|\mu_{V}\rangle=\frac{1}{2}\int\frac{f^{\prime}(x)-f^{\prime}(y)}{x-y}\mu_{V}(\mathtt{d}x)\mu_{V}(\mathtt{d}y)=\frac{1}{2}\langle
T_{V}f^{\prime}|\mu_{V}\rangle.$

Substituting the latter in 3.7, one gets

(3.8) $\begin{split}\frac{\mathsf{L}F}{n}&=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|\nu_{n}\rangle-\frac{\beta}{2}\langle
T_{V}f^{\prime}|n\mu_{V}\rangle\\\ &=\frac{2-\beta}{2}\langle
f^{\prime\prime}|\mu_{n}\rangle-\beta\langle
V^{\prime}f^{\prime}|\bar{\nu}_{n}\rangle-\frac{\beta}{2}\langle
T_{V}f^{\prime}|n\mu_{V}\rangle\\\ &+\frac{\beta}{2}\langle
T_{n}f^{\prime}-T_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f^{\prime}|n\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.\end{split}$

By Fubini Theorem, we have that $\langle
T_{n}f^{\prime}|n\mu_{V}\rangle=\langle T_{V}f^{\prime}|\nu_{n}\rangle$. This
gives

$\langle T_{n}f^{\prime}|n\mu_{V}\rangle-\langle
T_{V}f^{\prime}|n\mu_{V}\rangle=\langle T_{V}f^{\prime}|\bar{\nu}_{n}\rangle.$

Plugging this equality in 3.8 leads to the announced result. ∎

##  4\. Proofs of the main results

We now complete the proofs of our main results. Regarding quantitative normal
approximation for linear statistics of $\beta$-ensembles, we follow the
strategy described in Sections 2 and 3, applying the general quantitative
bounds given by Theorem 2.1.

###  4.1. Proof of Theorem 1.1

We first give the proof of Theorem 1.1, which gives a near optimal rate of
convergence $O(n^{-1+\alpha})$ for all $\alpha>0$.

####  4.1.1. Handling the covariance

Let us recall the form of the limit covariance matrix $C$ of the linear
statistics and let us first prove that it is invertible. Following [26, §4.1,
in particular Eq. (4.14)], we have

$c_{ij}=\frac{1}{2\beta}\int\frac{\xi_{i}(x)-\xi_{i}(y)}{x-y}\frac{\xi_{j}(x)-\xi_{j}(y)}{x-y}(1-xy)\rho(\mathtt{d}x)\rho(\mathtt{d}y)=:\langle\xi_{i},\xi_{j}\rangle_{\mathscr{H}^{1/2}}.$

The bilinear symmetric form
$\langle\xi_{i},\xi_{j}\rangle_{\mathscr{H}^{1/2}}$ is not a scalar product,
since $\langle g,g\rangle_{\mathscr{H}^{1/2}}=0$ if and only if $g$ is
constant. Nevertheless, the matrix $C:=(c_{ij})$ is a Gram matrix with respect
to the semi-scalar product $\langle\cdot,\cdot\rangle_{\mathscr{H}^{1/2}}$.
Thus under the freeness condition 1.2, the matrix $C$ is symmetric definite
positive, and we write $\Sigma$ for its unique positive symmetric square root.

####  4.1.2. Preparatory computations

We now use the master equation (3.6) to decompose the linear statistics in a
suitable form to apply Theorem 2.1.

Reduction to the case $\mathsf{L}F+Z$. Let us consider
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, $\psi\in\mathscr{C}^{5}_{c}(\mathbb{R})$
associated with $\xi$ through Lemma 3.1 and let $f$ be any primitive of
$\psi$. We then have

$\Theta_{V}f^{\prime}=\xi+c_{\xi}+\left\lparen\Theta_{V}f^{\prime}-\xi-
c_{\xi}\right\rparen 1_{\mathbb{R}\setminus U}.$

Since $\langle 1|\bar{\nu}_{n}\rangle=0$, and that
$\operatorname{supp}\mu_{V}\subset U$, we find that

$\langle\xi|\bar{\nu}_{n}\rangle=\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle+\sum_{i=1}^{n}1_{\lvert\lambda_{i}\rvert>1+\delta}(\Theta_{V}f^{\prime}-\xi-
c_{\xi}).$

By Lemma 3.4, we have that

$\left\lVert\sum_{i=1}^{n}1_{\lvert\lambda_{i}\rvert>1+\delta}(\Theta_{V}f^{\prime}-\xi-
c_{\xi})\right\rVert_{L^{p}}=n\mathrm{e}^{-n^{c}}O\left(\lVert\xi\rVert_{\infty}+\lVert\Theta_{V}f^{\prime}\rVert_{\infty}+1\right).$

From the explicit expression of $\Theta_{V}$ and 3.2, we find that

$\lVert\Theta_{V}f^{\prime}\rVert_{\infty}\leq c(\lVert
f^{\prime}\rVert_{\infty}+\lVert f^{\prime\prime}\rVert_{\infty})\leq
c\lVert\psi\rVert_{\mathscr{C}^{5}(\mathbb{R})}\leq
c\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})}.$

Combining those two estimates, and since $\langle 1|\bar{\nu}_{n}\rangle=0$,
it is sufficient to establish our bound for linear statistics of the form
$\langle\Theta_{V}f^{\prime}|\bar{\nu}_{n}\rangle$, where
$f^{\prime}=\psi\in\mathscr{C}_{c}^{5}(\mathbb{R})$ is associated with
$\xi\in\mathscr{C}^{6}(\mathbb{R})$ by Lemma 3.1. Thus, we now study

$X:=\left\lparen\langle\Theta_{V}f_{1}^{\prime}|\bar{\nu}_{n}\rangle,\dots,\langle\Theta_{V}f_{d}^{\prime}|\bar{\nu}_{n}\rangle\right\rparen.$

By Theorem 3.8, we can then decompose $X$ under the form

$X=m+\frac{1}{n}\mathsf{L}F+Z,$

where $m=(m_{i})_{1\leq i\leq d}$ with $m_{i}:=(1/2-1/\beta)\langle
f_{i}^{\prime\prime}|\mu_{V}\rangle$ where $F_{i}:=\frac{1}{\beta}\langle
f_{i}|\bar{\nu}_{n}\rangle$, and

(4.1) $Z_{i}:=\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\langle
f_{i}^{\prime\prime}|\mu_{n}\rangle-m_{i}-\frac{1}{2}\langle
T_{n}f_{i}^{\prime}-T_{V}f_{i}^{\prime}|\bar{\nu}_{n}\rangle.$

In the next section, we apply Theorem 2.1 to provide quantitative bounds.

Computation of $\Gamma[X,-F]$. We repeatedly use the simple yet remarkable
observation that linear statistics are stable under the action of the carré du
champ, namely

(4.2)
$\Gamma\big{[}\langle\varphi|\nu_{n}\rangle,\langle\psi|\nu_{n}\rangle\big{]}=\langle\varphi^{\prime}\psi^{\prime}|\nu_{n}\rangle,\qquad\varphi,\psi\in\mathscr{C}^{1}(\mathbb{R}).$

Thus, we find that for $1\leq i,j\leq d$

(4.3)
$\Gamma\left[X_{i},-\frac{F_{j}}{n}\right]=\frac{1}{\beta}\langle(\Theta_{V}f_{i}^{\prime})^{\prime}f_{j}^{\prime}|\mu_{n}\rangle.$

Heuristics. As a consequence of the convergence to equilibrium,
$\mu_{n}\to\mu_{V}$, we see that the sum of the first two terms on the right-
hand side in 4.1 converges to zero. Similarly, by Equation 4.3, the term
$\Gamma[X_{i},-F_{j}/n]$ converges to
$\frac{1}{\beta}\langle\xi_{i}^{\prime}f_{j}^{\prime}|\mu_{V}\rangle$ which
will be the limit covariance of
$(\langle\xi_{i},\bar{\nu_{n}}\rangle,\langle\xi_{j},\bar{\nu_{n}}\rangle)$.
Relying on the expression of the limit variance given in [26] and by
uniqueness one must therefore have

$c_{ij}=\lim_{n}\text{Cov}(\langle\xi_{i},\bar{\nu_{n}}\rangle,\langle\xi_{j},\bar{\nu_{n}}\rangle)=\frac{1}{\beta}\langle\xi_{i}^{\prime}f_{j}^{\prime}|\mu_{V}\rangle.$

However, the determination of the asymptotics of all the terms involving the
quadratic term $\langle(T_{n}-T_{v})f_{i}^{\prime}|\bar{\nu}_{n}\rangle$ is
more delicate. Indeed, the same heuristic on the convergence to equilibrium
shows that this term is of order $o(n)$ whereas one would expect $o(1)$. We
handle this remainder in two steps:

  1. (i)

We split the quadratic term in a product of linear terms, using Fourier
inversion. Indeed since $f_{i}^{\prime}\in\mathscr{C}_{c}^{5}(\mathbb{R})$ is
is in the image of the Fourier transform.

  2. (ii)

We control each of the linear term as before using convergence to equilibrium.
This is where we need the finer quantitative estimates recalled in Lemma 3.6.

####  4.1.3. Quantitative control and completion of the proof

Le us now quantify the remainders mentioned in the last Section.

Splitting the remainder by Fourier inversion. For simplicity, we omit the
indices here and write $f$ for $f_{i}$, $1\leq i\leq d$. Since
$f^{\prime}\in\mathscr{C}^{5}_{c}(\mathbb{R})$, we have

(4.4) $\lvert\widehat{f^{\prime\prime}}(t)\rvert\leq 2\frac{\lVert
f^{(6)}\rVert_{\infty}}{(1+|t|)^{4}},\qquad t\in\mathbb{R}.$

In particular, $\widehat{f^{\prime\prime}}\in L^{1}(\mathbb{R})$. By the
Fourier inversion Theorem, one finds

(4.5)
$\begin{split}\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle&=n\iint\int_{0}^{1}f^{\prime\prime}((1-u)x+uy)(\mu_{n}-\mu_{V})(\mathtt{d}x)(\mu_{n}-\mu_{V})(\mathtt{d}y)\\\
&=n\int_{\mathbb{R}}\int_{0}^{1}\widehat{f^{\prime\prime}}(t)\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathtt{d}t.\end{split}$

By Lemma 3.6 applied to the exponential functions and by Hölder’s inequality,
we have

$n\lVert\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}^{2}(1+\lvert
t\rvert+t^{2}).$

Thus, reporting in 4.5, and up to changing the constants from line to line, we
find that

(4.6)
$\lVert\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}\int_{\mathbb{R}}\lvert\widehat{f^{\prime\prime}}(t)\rvert\left\lparen
1+\lvert
t\rvert+t^{2}\right\rparen\mathtt{d}t\leq\frac{1}{n^{1-2\alpha}}K_{p,\alpha}\lVert
f^{(6)}\rVert_{\infty}.$

Completion of the proof. We can finally complete the proof Theorem 1.1. Again,
to simplify the expressions, we omit the indices $1\leq i\leq d.$ here. Recall
the definition of the term $Z$ given by Equation 4.1. By Lemma 3.6 and the
above Equation (4.6), we find that

$\lVert
Z\rVert_{L^{p}}\leq\left\lparen\frac{1}{2}-\frac{1}{\beta}\right\rparen\lVert\langle
f^{\prime\prime}|\mu_{n}-\mu_{V}\rangle\rVert_{L^{p}}+\lVert\langle(T_{n}-T_{V})f^{\prime}|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq\frac{1}{n^{1-\alpha}}K_{p,\alpha}\lVert
f^{\prime}\rVert_{\mathscr{C}^{5}(\mathbb{R})}.$

Introducing the random matrix

$C_{n}(i,j):=\langle\xi^{\prime}_{i}f^{\prime}_{i}|\mu_{n}\rangle,$

we have by Equation 4.3

$\left\|\frac{1}{\beta}C-\Gamma[X,-F/n]\right\|_{L^{p}}=\frac{1}{\beta}\lVert
C-C_{n}\rVert.$

Invoking Lemma 3.6 to control $C-C_{n}$ yields, again with constants which may
change from line to line,

(4.7) $\begin{split}\lVert C-C_{n}\rVert_{L^{p}}&\leq
K_{p,\alpha}\frac{1}{n^{1-\alpha}}\left\lparen\lVert\xi^{\prime}\rVert_{\infty}\lVert
f^{\prime}\rVert_{\infty}+\lVert\xi^{\prime}\rVert_{\infty}\lVert
f^{\prime\prime}\rVert_{\infty}+\lVert\xi^{\prime\prime}\rVert_{\infty}\lVert
f^{\prime}\rVert_{\infty}\right\rparen\\\
&\leq\frac{K_{p,\alpha}}{n^{1-\alpha}}\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}\lVert
f\rVert_{\mathscr{C}^{2}(\mathbb{R})}.\end{split}$

Since by Lemma 3.1, we have $\lVert f\rVert_{\mathscr{C}^{2}(\mathbb{R})}\leq
K\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}$, all the bounds above can thus
be rewritten only in term of the norm
$\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}$. Injecting these estimates in
the general upper bound given by Theorem 2.1, we arrive at the announced
result.

###  4.2. Proof of Theorem 1.2

We now give the proof of Theorem 1.2, which gives an optimal rate of
convergence $O(n^{-1})$ in the case where the test function $\xi$ is
sufficiently regular. We work here in dimension $d=1$ for simplicity, but the
proof will extend verbatim to higher dimensions.

The strategy used in order to obtain the optimal rate of convergence consists
in bootstrapping the argument used in the previous proof of Theorem 1.1.
Except that instead of invoking Lemma 3.6, we use here a better law of large
numbers provided by Theorem 1.1 itself. Namely, for
$\xi\in\mathscr{C}^{6}(\mathbb{R})$, taking $\alpha:=1/2$ and using the
$p$-Wasserstein bound for any $p\geq 1$ given by Theorem 1.1, we have that

$\lVert\langle\xi|\bar{\nu}_{n}\rangle\rVert_{L^{p}}\leq
K_{p}\left(\frac{\lVert\xi\rVert_{\mathscr{C}^{6}(\mathbb{R})})+\lVert\xi\rVert_{\mathscr{C}^{2}(\mathbb{R})}^{2}}{n^{1/2}}\right)+\lVert\sigma
N+m\rVert_{L^{p}},$

where $N$ is a standard Gaussian and both the mean $m=m_{1}$ and the standard
deviation $\sigma=\sqrt{c_{11}}$ are controlled by
$\|\xi\|_{\mathscr{C}^{1}(\mathbb{R})}$, see Section 1.3.1 where their
explicit expressions are given. This shows that provided we can control the
$6$-th derivative, we can remove the polynomial loss in the quantitative law
of large numbers given by Lemma 3.6. At the level of the upper bound 4.6, this
implies that we have a $t^{12}$ appearing in the integral, which leads to an
upper bound controlled by $\lVert\xi\rVert_{\mathscr{C}^{14}(\mathbb{R})}$.
Similarly, in the upper bound 4.7, the right hand side now yields a control of
the form $\lVert\xi^{\prime}\rVert_{\mathscr{C}^{6}(\mathbb{R})}\lVert
f^{\prime}\rVert_{\mathscr{C}^{6}(\mathbb{R})}$ and concludes the proof.

As a result and as already mentioned in Remark 3.7, this argument implies that
any polynomial speed of convergence on the law of large numbers can be
upgraded to the optimal speed $\frac{1}{n}$, the price to pay is to impose
stronger regularity assumption on the test functions.

###  4.3. Proof of Theorem 1.3

We now show how Theorem 1.3 dealing with functions with Hölder derivative
easily follows from Theorem 1.1 together with Lemma 3.6.

Take $\eta$ a smooth probability density supported on $[-1,1]$ with finite
first moment. Defining
$\eta_{\varepsilon}:=\frac{1}{\varepsilon}\eta(\frac{\bullet}{\varepsilon})$,
and $\xi_{\varepsilon}:=\xi\ast\eta_{\varepsilon}$, we see by direct
computations that

$\displaystyle\lVert\xi-\xi_{\varepsilon}\rVert_{\infty}\leq\lVert\xi^{\prime}\rVert_{\infty}\varepsilon\int\lvert
y\rvert\eta(y),$ (4.8)
$\displaystyle\lVert\xi^{\prime}-\xi_{\varepsilon}^{\prime}\rVert_{\infty}\leq\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}\,\varepsilon^{\gamma}\int\lvert
y\rvert^{\gamma}\eta(y).$

Thus by Lemma 3.6, for any $\alpha>0$, we have

$\lVert\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle\rVert_{1}\leq
K_{\alpha}\varepsilon^{\gamma}n^{\alpha}\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}.$

Now we use the master equation (3.6) for the linear statistics
$\xi_{\varepsilon}$ and we write:

$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\Theta_{v}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle,$

where $\Theta_{v}f_{\varepsilon}^{\prime}=\xi_{\varepsilon}$ on the
neighbourhood $U$. Mimicking the arguments developed in Section 4.1.2, we thus
have:

$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\xi_{\varepsilon}|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right).$

As a result, we get

$\frac{\mathsf{L}F_{\varepsilon}}{n}=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}\rangle+\beta\langle\xi|\bar{\nu}_{n}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right)+\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle.$

Hence, we may decompose the linear statistics in the form
$\langle\xi|\bar{\nu}_{n}\rangle=m_{\varepsilon}+\frac{\mathsf{L}F_{\varepsilon}}{n}+Z_{\varepsilon}$
as in the proof of Theorem 1.1 with this time

$\begin{array}[]{ll}\displaystyle{m_{\varepsilon}:=\left(\frac{1}{2}-\frac{1}{\beta}\right)\langle
f_{\varepsilon}^{\prime\prime}|\mu_{V}\rangle},\\\ \\\
\displaystyle{Z_{\varepsilon}:=\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}-\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle+O\left(ne^{-n^{c}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}\right)+\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle.}\end{array}$

Gathering the bounds given in the proof of Theorem 1.1 and the above Equation
(4.3), we then obtain the following controls

$\displaystyle\left\|\frac{2-\beta}{2}\langle
f_{\varepsilon}^{\prime\prime}|\mu_{n}-\mu_{V}\rangle+\frac{\beta}{2}\langle
T_{n}f_{\varepsilon}^{\prime}-T_{V}f_{\varepsilon}^{\prime}|\bar{\nu}_{n}\rangle\right\|_{1}\leq\frac{K_{\alpha}}{n^{1-2\alpha}}\left(\|\xi_{\varepsilon}\|_{\mathscr{C}^{6}}+\|\xi_{\varepsilon}\|_{\mathscr{C}^{2}}^{2}\right),$
$\displaystyle\|\langle\xi-\xi_{\varepsilon}|\bar{\nu}_{n}\rangle\|_{1}\leq
K_{\alpha}\varepsilon^{\gamma}n^{\alpha}\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})},$
$\displaystyle\lVert
C_{\varepsilon}-C_{n,\varepsilon}\rVert_{L^{p}}\leq\frac{K}{n^{1-\alpha}}\|\xi_{\varepsilon}\|_{\mathscr{C}^{2}}^{2},$

where $C_{\varepsilon}$ denote the limit variance associated with the linear
statistic $\xi_{\varepsilon}$ and
$C_{n,\varepsilon}:=\langle\xi^{\prime}_{\varepsilon}f^{\prime}_{\varepsilon}|\mu_{n}\rangle$
is the empirical analogue. By Equation (2.2) in Theorem 2.1, combined with the
three above bounds we may write for some constant $K_{\alpha}$

(4.9)
$\mathbf{TV}\left(\langle\xi|\bar{\nu}_{n}\rangle,m_{\varepsilon}+\sigma_{\varepsilon}N\right)\leq
K_{\alpha}\left(\frac{1}{\varepsilon^{6}n^{1-2\alpha}}+\varepsilon^{\gamma}n^{\alpha}\right)\lVert\xi\rVert_{\mathscr{C}^{1,\gamma}(\mathbb{R})}.$

In order to complete the proof, one needs to show that $m_{\varepsilon}$ and
$\sigma_{\varepsilon}$ converge towards $m$ and $\sigma$ and to estimate
$mathbf{TV}\left(m_{\varepsilon}+\sigma_{\varepsilon}N,m+\sigma N\right)$ as
$\varepsilon$ goes to zero. First of all, relying on the explicit expressions
of the limit means and variances, we have indeed as $\varepsilon$ goes to zero

$|m_{\varepsilon}-m|=O\left(\|\xi_{\varepsilon}-\xi\|_{\mathscr{C}^{1}}\right)=O\left(\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}\right),\quad\text{as
well
as}\quad|\sigma_{\varepsilon}-\sigma|=O\left(\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}\right).$

Finally, using for instance Theorem 1.3 in [14], we derive that

$\mathbf{TV}\left(m_{\varepsilon}+\sigma_{\varepsilon}N,m+\sigma N\right)\leq
K\varepsilon^{\gamma}\|\xi\|_{\mathscr{C}^{1,\gamma}}.$

Optimizing in the parameters in Equation (4.9), since $\alpha$ can be chosen
arbitrarily small, we get the announced bound of $O(n^{-a})$ for any
$a<\frac{\gamma}{\gamma+6}$.

###  4.4. Proof of Theorem 1.4

We finally give the proof of Theorem 1.4 stating the super-convergence of
linear statistics of $\beta-$ensembles. If $X$ denotes our linear statistics,
the proof is based on the fact that $\Gamma[X,X]$ admit negative moments. This
fact, combined with classical integration by parts techniques indeed ensure
the convergence of densities.

####  4.4.1. Control of the negative moments of $\Gamma$

Let us first establish the following lemma.

######  Lemma 4.1.

Take $\xi\in\mathscr{C}^{1}(\mathbb{R})$ such that there exist $\alpha>0$ such
that

$\mathsf{Leb}\left\lparen
x\in\mathbb{R}:\lvert\xi^{\prime}(x)\rvert\leq\varepsilon\right\rparen\lesssim\varepsilon^{\alpha},\qquad\varepsilon>0.$

Then, there exist $\gamma>0$, such that for $n$ large enough

(4.10)
$\operatorname{\mathbf{P}}\left[\frac{1}{n}\sum_{i=1}^{n}\xi^{\prime}(\lambda_{i})^{2}\leq\varepsilon\right]\lesssim\varepsilon^{\gamma
n},\qquad\varepsilon>0.$

###### Proof of Lemma 4.1.

To simplify the expressions in this proof, we set $g:=(\xi^{\prime})^{2}$ and
$m:=\langle g|\mu_{V}\rangle$. We distinguish here two regimes depending on
the magnitude of $\varepsilon$.

The large deviation regime. Take $\varepsilon<\frac{1}{2}m$, then by the large
deviations principle for $\beta$-ensembles [36, Thm. 2.3] (originally proved
in [7]), there exist $c=c_{g}>0$ such that, for $n$ large enough

$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\operatorname{\mathbf{P}}\left[\lvert\langle
g|\bar{\mu}_{n}\rangle\rvert\geq\frac{m}{2}\right]\leq\exp\left\lparen-
cn^{2}\right\rparen.$

In this case, in the regime $\varepsilon\geq\mathrm{e}^{-kn}$ where $k$ is
some constant to be fixed later, we have

$\mathrm{e}^{-cn^{2}}=(\mathrm{e}^{-kn})^{n\frac{c}{k}}\leq\varepsilon^{n\frac{c}{k}}.$

The free energy regime. Let us work in the regime
$\varepsilon\leq\mathrm{e}^{-kn}$, where we recall that we have the freedom to
choose $k$. By the explicit form of the density for the $\beta$-ensemble, we
find that

$\begin{split}\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]&\leq\operatorname{\mathbf{P}}\left[g(\lambda_{i})\leq\varepsilon
n,\,\forall i\in[1,n]\right]\\\
&=\frac{1}{Z_{n,\beta}}\int_{\mathbb{R}^{n}}\prod_{i=1}^{n}\mathbf{1}_{\left\\{g(\lambda_{i})\leq
n\varepsilon\right\\}}\mathrm{e}^{-\beta
H_{n}(\lambda_{1},\dots,\lambda_{n})}\mathtt{d}\lambda_{1}\dots\mathtt{d}\lambda_{n}.\end{split}$

Thus applying Cauchy–Schwarz inequality and using the $\alpha-$regular
condition on $\xi$ hence $g$, we obtain

$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\frac{Z_{n,2\beta}^{1/2}}{Z_{n,\beta}}(n\varepsilon)^{n\alpha/2}.$

By the large deviations principle, see for example [36, Thm. 2.3], we know
that

$\frac{1}{n^{2}}\log
Z_{n,\beta}\xrightarrow[n\to\infty]{}-\frac{\beta}{2}\mathcal{I}_{V}(\mu_{V}).$

In particular, there exists a constant $a(\beta)$ such that

$\frac{Z_{n,2\beta}^{1/2}}{Z_{n,\beta}}\leq\mathrm{e}^{n^{2}a(\beta)}.$

As a result, we have

$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq(n\varepsilon)^{n\alpha/2}\mathrm{e}^{a(\beta)n^{2}}.$

We can assume that $a(\beta)\geq 0$ otherwise the claim is trivial. In this
regime, we have

$\mathrm{e}^{a(\beta)n^{2}}\leq\varepsilon^{-a(\beta)n/k},\qquad
n^{n\alpha/2}\leq\varepsilon^{-\alpha/(2k)\log n}.$

It follows that

$\operatorname{\mathbf{P}}\left[\langle
g|\mu_{n}\rangle\leq\varepsilon\right]\leq\varepsilon^{n(\alpha/2-a(\beta)/k)-\alpha/(2k)\log
n}.$

Choosing $k$ large enough so that $\alpha/2-a(\beta)/kr>0$ then yields and
upper bound for the probability of order $\varepsilon^{cn}$, for some $c>0$. ∎

Recall that if $X=\sum_{i=1}^{n}\xi(\lambda_{i})$ is a linear statistics, then
we have $\Gamma[X,X]=\sum_{i=1}^{n}\xi^{\prime}(\lambda_{i})^{2}$. As a
result, given an exponent $\alpha>0$, we can write

$\operatorname{\mathbf{E}}\left[\frac{1}{\Gamma[X,X]^{\alpha}}\right]=\int_{\mathbb{R}^{+}}\operatorname{\mathbf{P}}\left[\frac{1}{\Gamma[X,X]^{\alpha}}>t\right]dt=\int_{\mathbb{R}^{+}}\operatorname{\mathbf{P}}\left[\Gamma[X,X]<\frac{1}{t^{1/\alpha}}\right]dt.$

As a result, the last Lemma 4.1 indeed ensures that $\Gamma[X,X]$ admits
negative moments.

####  4.4.2. Regularity and super convergence

The proof of Theorem 1.4 now follows from a well-known integration by parts
procedure see [22, §2.1], or [23, §3.2.2], for details. We only recall the
first part of the proof to highlight that, in the setting of $\beta$-ensemble,
the correct quantities to control are the negative moments of $\Gamma[X,X]/n$,
and not those of $\Gamma[X,X]$, as it is the case in [22, 23]. Take
$\varphi\in\mathscr{C}^{1}_{c}(\mathbb{R})$ and write

(4.11)
$\operatorname{\mathbf{E}}\left[\varphi^{\prime}(X)\right]=\operatorname{\mathbf{E}}\left[\varphi(X)\left\lparen\frac{\Gamma[X,\Gamma[X,X]]}{\Gamma[X,X]^{2}}-\frac{\mathsf{L}X}{\Gamma[X,X]}\right\rparen\right].$

By Equation 4.2, we find that

$\Gamma[X,X]=\langle(\xi^{\prime})^{2}|\nu_{n}\rangle,\qquad\Gamma[X,\Gamma[X,X]]=2\langle(\xi^{\prime})^{2}\xi^{\prime\prime}|\nu_{n}\rangle.$

As a result, as $n$ goes to infinity, we have

$\frac{\Gamma[X,\Gamma[X,X]]}{\Gamma[X,X]^{2}}\to 0.$

Now in view of the master equation 3.6, we have the decomposition

(4.12)
$\begin{split}\Gamma\left[\frac{\mathsf{L}X}{n},\frac{X}{n}\right]&=\frac{2-\beta}{2}\frac{1}{n}\langle\xi^{(3)}\xi^{\prime}|\mu_{n}\rangle+\frac{1}{\beta}\langle(\Theta_{V}\xi^{\prime})^{\prime}\xi^{\prime}|\mu_{n}\rangle\\\
&-\frac{1}{2\beta}\Gamma\left[\langle(T_{V}-T_{n})\xi^{\prime}|\nu_{n}\rangle,\frac{X}{n}\right].\end{split}$

Using again the Fourier splitting, by the chain rule, dominated convergence,
and 4.2, we find that

$\begin{split}R_{n}&:=\Gamma\left[\langle(T_{n}-T_{V})\xi^{\prime}|\bar{\nu}_{n}\rangle,\langle\psi|\mu_{n}\rangle\right]\\\
&=\mathrm{i}\int_{\mathbb{R}}\widehat{\xi^{\prime\prime}}(t)t\int_{0}^{1}u\langle\mathrm{e}^{\mathrm{i}u\bullet}\psi^{\prime}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathrm{d}t\\\
&+\mathrm{i}\int_{\mathbb{R}}\widehat{\xi^{\prime\prime}}(t)t\int_{0}^{1}(1-u)\langle\mathrm{e}^{\mathrm{i}t(1-u)\bullet}\psi^{\prime}|\mu_{n}-\mu_{V}\rangle\langle\mathrm{e}^{\mathrm{i}tu\bullet}|\mu_{n}-\mu_{V}\rangle\mathtt{d}u\mathrm{d}t.\end{split}$

By Lemma 3.6, we then obtain that $R_{n}\to 0$ in $L^{2}$. By Lemma 2.3, this
shows that $\mathsf{L}F/n$ converges to a Gaussian with respect to
$\mathbf{W}_{2}$. In particular, its variance is of order $1$. Thus re-
writing, the last term in 4.11 as the product

$\frac{\mathsf{L}X}{n}\frac{n}{\Gamma[X,X]},$

the need to control the negative moments of $\Gamma[X,X]/n$ is more obvious.
It implies in particular that for some constant $C>0$ and any
$\phi\in\mathscr{C}^{1}$ we have
$\left|\mathbb{E}\left[\phi^{\prime}(X)\right]\right|\leq C\|\phi\|_{\infty}$.
This argument can be iterated and for any $p>1$ one would get similarly for
$n$ large enough that $\left|\mathbb{E}\left[\phi^{(p)}(X)\right]\right|\leq
C\|\phi\|_{\infty}$. The latter combined with the convergence in law proved in
Theorem 1.1 entails the announced super-convergence, see [23] for more
details.

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