# CLT for $\beta$-ensembles at high-temperature, and for integrable systems: a
transfer operator approach.

G. Mazzuca111Department of Mathematics, Tulane University, New Orleans,
Louisiana, USA.
email<EMAIL_ADDRESS>R. Memin222IMT, Toulouse, France.
email<EMAIL_ADDRESS>
###### Abstract

In this paper, we prove a polynomial Central Limit Theorem for several
integrable models, and for the $\beta$-ensembles at high-temperature with
polynomial potential. Furthermore, we connect the mean values, the variances
and the correlations of the moments of the Lax matrices of these integrable
systems with the ones of the $\beta$-ensembles. Moreover, we show that the
local functions’ space-correlations decay exponentially fast for the
considered integrable systems. For these models, we also established a
Berry–Esseen type bound.

###### Contents

  1. 1 Introduction
    1. 1.1 The Toda chain: an introductory case
    2. 1.2 Main results of the paper
  2. 2 Application
    1. 2.1 The Toda lattice, and the real $\beta$-ensemble at high-temperature
      1. 2.1.1 The Toda chain’s currents
    2. 2.2 The exponential Toda lattice, and the Laguerre $\beta$-ensemble at high-temperature
    3. 2.3 The Volterra lattice, and the antisymmetric $\beta$-ensemble at high-temperature
    4. 2.4 The defocusing Ablowitz-Ladik lattice, and the Circular $\beta$-ensemble at high-temperature
    5. 2.5 The defocusing Schur flow, and the Jacobi $\beta$-ensemble at high-temperature
    6. 2.6 The Itoh–Narita–Bogoyavleskii lattices
  3. 3 Transfer Operator Method
  4. 4 Nagaev–Guivarc’h theory
  5. 5 Proof Of Theorem 1.5-1.9
  6. 6 Technical Results
  7. 7 Conclusion and Outlooks
  8. A Proof of Lemma 1.3
  9. B Proof of Lemma 6.6

##  1 Introduction

In this paper, we establish a Central Limit Theorem for eigenvalue
fluctuations of the Lax matrices of several integrable models, and for the
classical $\beta$-ensembles of random matrix theory in the so-called high-
temperature regime. Motivated by the recent observation that both topics are
closely related, we connect, for the considered integrable systems, the
limiting variance in the CLT to the one of the associated $\beta$-ensemble in
the high-temperature regime, by establishing a relation between their free
energies. Furthermore, we establish exponential decay of space correlation for
the considered integrable models and a Berry–Esseen type bound.

We start by describing the case of the Toda chain, which will serve both as a
motivation for the results and as a way to introduce the key objects of the
present work.

###  1.1 The Toda chain: an introductory case

The classical Toda chain (or Toda lattice) [68] is the dynamical system
described by the following Hamiltonian:

$H_{T}({\bf p},{\bf
q}):=\frac{1}{2}\sum_{j=1}^{N}{p_{j}^{2}}+\sum_{j=1}^{N}V_{T}(q_{j+1}-q_{j})\
,\quad V_{T}(x)=e^{-x}+x-1\,,$ (1.1)

with periodic boundary conditions
$\,q_{j+N}=q_{j}+\Omega\quad\forall\,j\in\mathbb{Z}$, where $\Omega>0$ is
fixed. Its equations of motion take the form

$\dot{q}_{j}=\dfrac{\partial H_{T}}{\partial
p_{j}}=p_{j},\quad\dot{p}_{j}=-\dfrac{\partial H_{T}}{\partial
q_{j}}=V_{T}^{\prime}(q_{j+1}-q_{j})-V_{T}^{\prime}(q_{j}-q_{j-1}),\;\;j=1,\dots,N\,.$
(1.2)

It is well known that the Toda chain is an integrable system [68, 39]. One way
to prove it is to put the Toda equations in Lax pair form. This was obtained
by Flaschka [26], and Manakov [50] through the following non-canonical change
of coordinates:

$a_{j}:=-p_{j}\,,\qquad b_{j}:=e^{\frac{1}{2}(q_{j}-q_{j+1})}\equiv
e^{-\frac{1}{2}r_{j}},\qquad 1\leq j\leq N\,,$ (1.3)

where $r_{j}=q_{j+1}-q_{j}$ is the relative distance between particles $j$ and
$j+1$.

Defining the Lax operator $L$ as the periodic Jacobi matrix [70]

$L:=\left(\begin{array}[]{ccccc}a_{1}&b_{1}&0&\ldots&b_{N}\\\
b_{1}&a_{2}&b_{2}&\ddots&\vdots\\\ 0&b_{2}&a_{3}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ b_{N}&\ldots&0&b_{N-1}&a_{N}\\\
\end{array}\right)\,,$ (1.4)

and the antisymmetric matrix $B$

$B:=\frac{1}{2}\left(\begin{array}[]{ccccc}0&b_{1}&0&\ldots&-b_{N}\\\
-b_{1}&0&b_{2}&\ddots&\vdots\\\ 0&-b_{2}&0&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ b_{N}&\ldots&0&-b_{N-1}&0\\\
\end{array}\right)\,,$ (1.5)

a straightforward calculation shows that the equations of motions (1.2) are
equivalent to

$\dot{L}=\left[B;L\right]\,,$ (1.6)

where $[B;L]=BL-LB$ is the commutator of two matrices. This form implies that
the eigenvalues of $L$ are conserved, and therefore
$\mathrm{Tr}\left(L^{k}\right)$, $k=1,\ldots,N$ are constants of motions for
the Toda lattice, so the system is integrable [8]. We call these quantities
conserved fields. Also, note that the sum of the streches
$\sum_{j=1}^{N}r_{j}=\Omega$ is conserved along the dynamic.

Let $\alpha>0$ and $V:{\mathbb{R}}\to{\mathbb{R}}$ be a Lebesgue measurable
function such that for some $C\in{\mathbb{R}}$,
$\lim_{x\to\pm\infty}\frac{V(x)}{x^{2}}>0$ and $V(x)>C$ for all
$x\in{\mathbb{R}}$. We introduce, after [64], the probability measure on
${\mathbb{R}}^{N}\times{\mathbb{R}}^{N}_{+}$

${\rm d}\mu_{N,\text{Toda}}^{\alpha,V}({\bf a},{\bf
b}):=\frac{1}{Z_{N}^{\text{Toda}}(\alpha,V)}\
\prod_{j=1}^{N}b_{j}^{2\alpha-1}\mathbbm{1}_{b_{j}>0}e^{-\mathrm{Tr}\left(V(L)\right)}{\rm
d}{\bf a}\,{\rm d}{\bf b}\,,$ (1.7)

where $L$ is a periodic tridiagonal matrix of the form (1.4) and

$\mathrm{Tr}\left(V(L)\right)=\sum_{j=1}^{N}V(\lambda_{j}(L))\,,$

denoting by $\lambda_{j}(L)$, $1\leq j\leq N$ the eigenvalues of $L$.

The measure $\mu_{N,\text{Toda}}^{\alpha,V}$ is called Generalized Gibbs
Ensemble (GGE) of the Toda chain with pressure parameter $\alpha>0$ and
potential $V$. Because both quantities $\mathrm{Tr}\left(V(L)\right)$ and

$\prod_{j=1}^{N}b_{j}=\exp\left\\{-\frac{1}{2}\sum_{j=1}^{N}r_{j}\right\\}$

are constants of motion, $\mu_{N,T}^{\alpha,V}$ is invariant under the Toda
flow, this is a consequence of a more general result, see [37, 33].

Note that the conditions on $V(x)$ are enough to ensure that the normalizing
constant $Z^{\text{Toda}}_{N}(\alpha,V)$ is finite. It is called the partition
function of the system, and it reads

$Z^{\text{Toda}}_{N}(\alpha,V)=\int_{{\mathbb{R}}^{N}\times{\mathbb{R}}_{+}^{N}}\prod_{j=1}^{N}b_{j}^{2\alpha-1}\mathbbm{1}_{b_{j}>0}e^{-\mathrm{Tr}\left(V(L)\right)}{\rm
d}{\bf a}\,{\rm d}{\bf b}\,.$

In the framework of generalized hydrodynamics, it is expected that for generic
initial data, the long-time distribution of the Toda chain converges towards a
GGE whose pressure $\alpha$ and potential $V$ are prescribed by the initial
condition. Formally, it is expected that for sufficiently regular observables
$f:{\mathbb{R}}^{N}\times{\mathbb{R}}_{+}^{N}\to{\mathbb{R}}$ we have as
$t\to+\infty$

$\frac{1}{t}\int_{0}^{t}f({\bf a}(s)),{\bf b}(s)){\rm
d}s\to{\mathbb{E}}_{N}^{\alpha,V}[f({\bf a},{\bf b})]\,,$

where the expectation is taken under $\mu^{\alpha,V}_{N,T}$, and where
$\alpha$, $V$ depend on $({\bf a}(0),{\bf b}(0))$. From there, understanding
the GGE with arbitrary potential and pressure parameter is key to
understanding the Toda lattice’s long-time behaviour.

Under $\mu^{\alpha,V}_{N,\text{Toda}}$, $L$ is a random matrix whose law is
invariant under the Toda flow. It turns out, as was observed by Spohn in [64],
that its spectral properties can be linked with the ones of the real
$\beta$-ensemble in the high-temperature regime.

Fix $N\in{\mathbb{N}}$, $\beta>0$ and $V:{\mathbb{R}}\to{\mathbb{R}}$ Lebesgue
measurable such that for some $\delta>0$, $c\in{\mathbb{R}}$, we have for all
$x\in{\mathbb{R}}$

$V(x)\geq(N\beta+1+\delta)\log|x|+c\,.$

The real $\beta$-ensemble of size $N$ with potential $V$ is the probability
measure on ${\mathbb{R}}^{N}$ given by

$d\mathbf{P}^{\beta,V}_{N}(\lambda_{1},\ldots,\lambda_{N})=\frac{1}{\mathbf{Z}^{\beta,V}_{N}}\prod_{1\leq
i<j\leq
N}|\lambda_{i}-\lambda_{j}|^{\beta}e^{-\sum_{j=1}^{N}V(\lambda_{j})}{\rm
d}\boldsymbol{\lambda}\,.$ (1.8)

When $\beta\in\\{1,2,4\\}$ and $V$ is quadratic, this measure is the joint law
of the (unordered) spectrum of respectively the GOE, GUE and GSE (see [6,
Section 2.5] for example), giving matrix representation for the
$\beta$-ensemble in those specific cases. In [19], Dumitriu and Edelman gave a
tridiagonal matrix model for the $\beta$-ensemble of size $N$ with quadratic
potential for any choice of $\boldsymbol{\beta>0}$:

Let $T$ be the following random tridiagonal matrix with independent
coefficients (up to the symmetry):

  * •

The diagonal entries $T_{i,i}$ are standard Gaussians

  * •

For $1\leq i\leq N-1$, the off-diagonal entry $T_{i,i+1}=T_{i+1,i}$ is
distributed as $\frac{1}{\sqrt{2}}\chi_{\beta(N-i)}$ ,

Where for $a>0$, the $\chi_{a}$ law is given by the density

$f_{a}(x)=\frac{2^{1-a/2}}{\Gamma(a/2)}x^{a-1}e^{-x^{2}/2}\,.$

Then, [19, Theorem 2.12] asserts that the distribution of the eigenvalues of
$T$ is given by the $\beta$-ensemble of size $N$ with potential
$V(x)=x^{2}/2$.

With this last point point in mind, let

$T:=\left(\begin{array}[]{ccccc}a_{1}&b_{1}&0&\ldots&0\\\
b_{1}&a_{2}&b_{2}&\ddots&\vdots\\\ 0&b_{2}&a_{3}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ 0&\ldots&0&b_{N-1}&a_{N}\\\
\end{array}\right)\,$ (1.9)

be the (nonperiodic) tridiagonal matrix whose entries are distributed
according to the measure

${\rm
d}\mu^{\beta,V}_{N,\text{beta}}=\frac{1}{Z^{\text{beta}}_{N}(\beta,V)}\prod_{j=1}^{N-1}b_{j}^{\beta(N-j)-1}\mathbbm{1}_{b_{j}>0}e^{-\mathrm{Tr}\left(V(T)\right)}{\rm
d}{\bf a}{\rm d}{\bf b}\,,$ (1.10)

with ${\bf a}=(a_{1},\ldots,a_{N})$ and ${\bf b}=(b_{1},\ldots,b_{N-1})$.
Then, the eigenvalues of $T$ are distributed according to the $\beta$-ensemble
(1.8) with potential $V$, the case $V=\frac{x^{2}}{2}$ being the case where
$T$ has independent entries described above.

The parameter $\beta$ is allowed to be $N$-dependent and can be interpreted as
an inverse temperature. In this paper, we call the high-temperature regime the
framework where $\beta=\frac{2\alpha}{N}$ for some fixed $\alpha>0$. We then
define the high temperature version of $\mu^{\beta,V}_{N,\text{beta}}$ as

${\rm d}\mu^{\alpha,V}_{N,\text{HT}}:={\rm
d}\mu^{2\alpha/N,V}_{N,\text{beta}}=\frac{1}{Z^{\text{HT}}_{N}(\alpha,V)}\prod_{j=1}^{N-1}b_{j}^{2\alpha\frac{N-j}{N}-1}\mathbbm{1}_{b_{j}>0}e^{-\mathrm{Tr}\left(V(T)\right)}{\rm
d}{\bf a}{\rm d}{\bf b}\,.$ (1.11)

We can now sketch the link between the Toda chain distributed according to a
GGE and the high temperature real $\beta$-ensemble. Consider the GGE (1.7)
with quadratic potential $V(x)=x^{2}/2$. Then, under
$\mu^{\alpha,x^{2}/2}_{N,\text{Toda}}$, the entries of $L$ become independent
(up to the symmetry) and their law is given by

  * •

the diagonal entries $L_{i,i}$ are standard Gaussian

  * •

the off diagonal entries $L_{i,i+1}=L_{i+1,i}$, $1\leq i\leq N$ (where the
indices are taken modulo $N$) are distributed as
$\frac{1}{\sqrt{2}}\chi_{2\alpha}$.

With the choice $\beta=2\alpha/N$ in (1.8), the entries near the top left
corner of Dumitriu-Edelman representation (i.e. the matrix $T$ distributed
according to $\mu^{\alpha,x^{2}/2}_{N,\text{HT}}$) resemble the ones of the
top left corner of $L$ when $({\bf a},{\bf b})$ is distributed according to
$\mu^{\alpha,x^{2}/2}_{N,\text{Toda}}$. One can then hope to pull this link
between both models for more general choices of $V$. As already mentioned,
this fundamental remark was made by Spohn in [64] where the author was able to
derive the limit of the empirical measure of eigenvalues of the Toda Lax
matrix $L$ under the GGE in terms of the limiting measure of the
$\beta$-ensemble in the high-temperature regime, both with polynomial
potential $V$. Namely, denoting the empirical measure of eigenvalues of any
$N\times N$ complex matrix $A$

$\nu_{N}(A)=\frac{1}{N}\sum_{j=1}^{N}\delta_{\lambda_{j}(A)},$ (1.12)

Spohn argued that with $({\bf a},{\bf b})$ distributed according to
$\mu^{\alpha,V}_{N,\text{Toda}}$, the empirical measure $\nu_{N}(L)$ converges
almost surely as $N\to\infty$ towards some deterministic measure
$\nu^{V}_{\alpha}$, which is given in terms of the limiting empirical measure
$\mu^{V}_{\alpha}$ of the high temperature $\beta$-ensemble,
$\beta=\frac{2\alpha}{N}$, by

$\nu^{V}_{\alpha}=\partial_{\alpha}\left(\alpha\mu^{V}_{\alpha}\right)\,,$
(1.13)

where the last equality is understood in the weak sense, i.e. tested against
bounded continuous functions. This was proved for the quadratic potential in
[51] and generalized to potentials with polynomial growth in [36], where the
authors established large deviations for the empirical measure of the Toda Lax
matrix and related the associated rate function with the one of the
$\beta$-ensemble at high temperature.

In the present paper, as an application of Theorem 1.5, we push the comparison
between both models further by linking the fluctuations of the empirical
measure of the Toda chain to the ones of the high temperature
$\beta$-ensemble.

Namely, we show that taking a polynomial potential $V$, considering

  * case 1:

the periodic tridiagonal matrix $L$ distributed according to
$\mu_{N,\text{Toda}}^{\alpha,V}$, eq. (1.7)

  * case 2:

the tridiagonal matrix $T$ distributed according to
$\mu^{\alpha,V}_{N,\text{HT}}$, eq. (1.11),

we have for some $\sigma^{2}_{\text{Toda}}(\alpha,V,g)>0$ and
$\sigma^{2}_{\text{HT}}(\alpha,V,g)>0$ the following convergences in
distribution:

in case 1,

$\sqrt{N}\left(\int_{{\mathbb{R}}}g{\rm
d}\nu_{N}(L)-\lim_{N\to+\infty}\int_{{\mathbb{R}}}g{\rm
d}\nu_{N}(L)\right)\xrightarrow{N\to\infty}\mathcal{N}(0,\sigma^{2}_{\text{Toda}}(\alpha,V,g))$

where $\nu_{N}(L)$ (1.12) is the empirical measure of $L$.

in case 2,

$\sqrt{N}\left(\int_{{\mathbb{R}}}g{\rm
d}\nu_{N}(T)-\lim_{N\to+\infty}\int_{{\mathbb{R}}}g{\rm
d}\nu_{N}(T)\right)\xrightarrow{N\to\infty}\mathcal{N}(0,\sigma^{2}_{\text{HT}}(\alpha,V,g))$

where $\nu_{N}(T)$ (1.12) is the empirical measure of $T$.

Furthermore, we establish that both limiting variances are linked through

$\sigma^{2}_{\text{Toda}}(\alpha,V,g)=\partial_{\alpha}\left(\alpha\sigma^{2}_{\text{HT}}(\alpha,V,g)\right)\,,$
(1.14)

which is the analogue of 1.13 at the level of fluctuations of the empirical
measures of eigenvalues.

This result is particularly relevant in the context of the so-called
Generalized Hydrodynamics, a recent physical theory that allows computing the
correlation functions for classical integrable models, for an introduction to
the subject see [18, 66]. According to this theory, one of the main
ingredients to compute the correlation functions for the integrable model at
hand is to be able to calculate the correlation functions for the conserved
fields at time $0$. We are able to access these quantities as a consequence of
Theorem 1.5. We show how to do it in Section 2. We also mention the recent
work [52], where the authors made molecular dynamics simulations of the
correlation functions of the Toda lattice, and they compared them with the
predictions of linear Generalized Hydrodynamics, showing an astonishing
agreement.

The relation between the Toda chain and the high-temperature regime of the
real $\beta$-ensemble is not a unique case — it is an instance of a broader
picture linking two types of models:

  * Type 1)

discrete space, continuous time integrable system with nearest neighbour
interaction endowed with some GGE

  * Type 2)

high temperature $\beta$-ensemble on some given curve of the complex plane.

Indeed, an analogue result to the one of Spohn for the Toda lattice holds true
also for several other pairs integrable system/$\beta$-ensemble at high-
temperature matrix model as it was described in [33, 35, 53, 66, 65].

In the present manuscript, we extend the aforementioned association. We
establish a similar outcome analogous to the one delineated for the
fluctuations of the Toda lattice and the real $\beta$-ensemble at high
temperature. This extension is applicable to other pairs of integrable systems
and $\beta$-ensembles at high temperatures, as presented in Table 1. The
correspondence between the systems of this table was already described in [33]
at the level of empirical measures under particular choices of potentials
defining the GGEs and their associated $\beta$-ensemble. We introduce the
systems at stake in Section 2.

Integrable System (Type 1) |  $\beta$-ensemble at high-temperature  (Type 2)
---|---
Toda lattice | Real
Defocusing Ablowitz-Ladik lattice | Circular
Exponential Toda lattice | Laguerre
Defocusing Schur flow | Jacobi
Volterra lattice | Antisymmetric
Table 1: Integrable systems and random matrix ensembles

In the next section, we model the present picture by introducing two types of
matrices and of distribution for their entries: the first type models are the
Lax matrix of integrable systems, distributed according to a GGE, and the
second type models the matrix representation of the associated
$\beta$-ensemble in the high-temperature regime. We then state the main
results of this paper.

###  1.2 Main results of the paper

We consider the following matrix models.

  * •

Type 1-i) Periodic Jacobi matrices, which are periodic tridiagonal matrix of
the form

$\left(\begin{array}[]{ccccc}a_{1}&b_{1}&0&\ldots&b_{N}\\\
b_{1}&a_{2}&b_{2}&\ddots&\vdots\\\ 0&b_{2}&a_{3}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ b_{N}&\ldots&0&b_{N-1}&a_{N}\\\
\end{array}\right)\,,$ (1.15)

for ${\bf a}=(a_{1},\ldots,a_{N})\in{\mathbb{R}}^{N}$, ${\bf
b}=(b_{1},\ldots,b_{N})\in{\mathbb{R}}^{N}_{+}$.

  * •

Type 1-ii) Periodic positive definite Jacobi matrices, which are periodic
tridiagonal matrix $L$ of the form

$L=BB^{\intercal}\,,$ (1.16)

where $B\in\text{Mat}({\mathbb{R}},N)$ read

$B=\left(\begin{array}[]{ccccc}a_{1}&b_{1}&0&\ldots&0\\\
0&a_{2}&b_{2}&\ddots&\vdots\\\ 0&0&a_{3}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ b_{N}&\ldots&0&0&a_{N}\\\
\end{array}\right)\,,$ (1.17)

for ${\bf a}=(a_{1},\ldots,a_{N})\in{\mathbb{R}}^{N}_{+}$, ${\bf
b}=(b_{1},\ldots,b_{N})\in{\mathbb{R}}^{N}_{+}$.

  * •

Type 1-iii) Antisymmetric tridiagonal periodic matrices:

$\left(\begin{array}[]{ccccc}0&a_{1}&0&\ldots&-a_{N}\\\
-a_{1}&0&a_{2}&\ddots&\vdots\\\ 0&-a_{2}&0&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&a_{N-1}\\\ a_{N}&\ldots&0&-a_{N-1}&0\\\
\end{array}\right)\,,$ (1.18)

for ${\bf a}=(a_{1},\ldots,a_{N})\in{\mathbb{R}}^{N}_{+}$.

  * •

Type 1-iv) Periodic CMV (after Cantero, Moral and Velazquez) matrices, which
are $2N\times 2N$ unitary matrices given by

${\mathcal{E}}={\mathcal{L}}{\mathcal{M}}\,,$ (1.19)

where we define ${\mathcal{L}}$ and ${\mathcal{M}}$ in the following way. Let
${\bf a}=(a_{1},\ldots a_{2N})$ be complex numbers of the unit disk
${\mathbb{D}}$. Define the $2\times 2$ unitary matrix $\Xi_{j}$

$\Xi_{j}=\begin{pmatrix}{\overline{a}}_{j}&\rho_{j}\\\
\rho_{j}&-a_{j}\end{pmatrix}\,,\quad
j=1,\dots,2N\,,\quad\rho_{j}=\sqrt{1-|a_{j}|^{2}}\,.$ (1.20)

Then, ${\mathcal{L}}$ and ${\mathcal{M}}$ are the $2N\times 2N$ matrices

${\mathcal{L}}=\begin{pmatrix}\Xi_{1}\\\ &\Xi_{3}\\\ &&\ddots\\\
&&&\Xi_{2N-1}\end{pmatrix}\,,\qquad{\mathcal{M}}=\begin{pmatrix}-a_{2N}&&&&&\rho_{2N}\\\
&\Xi_{2}\\\ &&&\ddots\\\ &&&&\Xi_{2N-2}\\\
\rho_{2N}&&&&&{\overline{a}}_{2N}\end{pmatrix}\,.$ (1.21)

The matrix ${\mathcal{E}}$ is a unitary pentadiagonal periodic matrix.

  * •

Type 1-v) Two diagonals periodic matrices given by

$\displaystyle\begin{pmatrix}0&a_{1}&0&\cdots&\leavevmode\hbox to21.11pt{\vbox
to9.32pt{\pgfpicture\makeatletter\hbox{\hskip 10.55295pt\lower-4.66222pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.55295pt}{-2.28223pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$b_{N-r+1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&0&0\\\
0&0&a_{2}&\cdots&0&b_{N-r+2}&0&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\\\
0&0&0&\cdots&a_{r-1}&0&0&b_{N}\\\
b_{1}&0&\cdots&\cdots&0&a_{r}&0&\leavevmode\hbox to5pt{\vbox
to6.44pt{\pgfpicture\makeatletter\hbox{\hskip 2.5pt\lower-3.22221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
0&b_{2}&0\cdots&&\ddots&\ddots&\ddots&\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0&0&a_{N-1}\\\
a_{N}&0&\cdots&b_{N-r}&\cdots&0&0&0\\\ \end{pmatrix}\leavevmode\hbox
to72.18pt{\vbox to23.73pt{\pgfpicture\makeatletter\hbox{\hskip
1.19998pt\lower-1.59998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}{}{}{}}{}{{}}{}{{}{}}{{}}{}
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to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (1.22)

Where ${\bf a},{\bf b}\in{\mathbb{R}}^{N}_{+}$. In applications, we consider
either $a_{1}=a_{2}=\ldots=a_{N}=1$ or $b_{1}=b_{2}=\ldots=b_{N}=1$.

We also consider the non-periodic counterparts of the previous matrices. More
specifically:

  * •

Type 2-i) Jacobi matrices, which are symmetric tridiagonal matrices

$\begin{pmatrix}a_{1}&b_{1}&&&\\\ b_{1}&a_{2}&b_{2}&&\\\
&b_{2}&\ddots&\ddots&\\\ &&\ddots&\ddots&b_{N-1}\\\
&&&b_{N-1}&a_{N}\end{pmatrix},$ (1.23)

where ${\bf a}\in{\mathbb{R}}^{N}$ and ${\bf b}\in{\mathbb{R}}^{N-1}_{+}$.

  * •

Type 2-ii) Positive definite Jacobi matrices, tridiagonal matrix $T$ of the
form

$T=BB^{\intercal}\,,$ (1.24)

where $B\in\text{Mat}({\mathbb{R}},N)$ read

$B=\left(\begin{array}[]{ccccc}a_{1}&b_{1}&0&\ldots&0\\\
0&a_{2}&b_{2}&\ddots&\vdots\\\ 0&0&a_{3}&\ddots&0\\\
\vdots&\ddots&\ddots&\ddots&b_{N-1}\\\ 0&\ldots&0&0&a_{N}\\\
\end{array}\right)\,,$ (1.25)

for ${\bf a}=(a_{1},\ldots,a_{N})\in{\mathbb{R}}^{N}_{+}$, ${\bf
b}=(b_{1},\ldots,b_{N-1})\in{\mathbb{R}}^{N}_{+}$.

  * •

Type 2-iii) Tridiagonal Antisymmetric matrices:

$\left(\begin{array}[]{ccccc}0&a_{1}&0&&\\\ -a_{1}&0&a_{2}&\ddots&\\\
0&-a_{2}&0&\ddots&0\\\ &\ddots&\ddots&\ddots&a_{N-1}\\\ &&0&-a_{N-1}&0\\\
\end{array}\right)\,,$ (1.26)

for ${\bf a}\in{\mathbb{R}}^{N-1}_{+}$.

  * •

Type 2-iv) CMV matrices, $2N\times 2N$ unitary matrices of the form

${\mathfrak{E}}={\mathfrak{L}}{\mathfrak{M}}\,,$ (1.27)

where

${\mathfrak{L}}=\mbox{diag}\left(\Xi_{0},\Xi_{2},\Xi_{4},\ldots,\Xi_{2N}\right)\qquad\text{and}\qquad{\mathfrak{M}}=\mbox{diag}\left(\Xi_{1},\Xi_{3},\Xi_{5}\ldots,\Xi_{2N-1}\right)\,,$
(1.28)

and the blocks $\Xi_{j}$, $j=1,\dots,2N-1$ are defined as

$\Xi_{j}=\begin{pmatrix}{\overline{a}}_{j}&\rho_{j}\\\
\rho_{j}&-a_{j}\end{pmatrix}\,,\quad
j=1,\dots,2N-1\,,\quad\rho_{j}=\sqrt{1-|a_{j}|^{2}}\,.$ (1.29)

for $\mathbf{a}=(a_{1},\ldots,a_{2N})\in\mathbb{D}^{2N}$, while $\Xi_{0}=(1)$
and $\Xi_{2N}=({\overline{a}}_{2N})$ are $1\times 1$ matrices. The matrix
${\mathcal{E}}$ is a pentadiagonal matrix.

The periodic matrices (Type 1) that we consider are the Lax matrices of some
integrable models. These are particular dynamical systems that are Liouville
integrable, and their integrability is proved by obtaining a Lax pair $(L,A)$
[48] representation of the model, meaning that the equations of motions for
each of these systems are equivalent to the following linear system of
ordinary differential equations for some matrices $L,A$

$\dot{L}\equiv\frac{{\rm d}L}{{\rm d}t}=[L;A]=LA-AL\,.$ (1.30)

This formulation is useful since it implies that
$\\{\mathrm{Tr}\left(L^{k}\right)\\}_{k=1}^{N}$ are a system of independent
constants of motion $\left(\dfrac{{\rm d}}{{\rm
d}t}\mathrm{Tr}\left(L^{k}\right)=0)\right)$ for the system at hand, so the
system is integrable in classical sense. We call these quantities conserved
fields. For a comprehensive introduction to classical integrable systems, we
refer to [8].

We introduce each integrable model with its matrix representation in Section
2, but we anticipate that

  * •

the Toda lattice [68] has as Lax matrix a periodic Jacobi matrix (type $1$-i),

  * •

the Exponential Toda lattice [33] has a positive definite periodic Jacobi
matrix (type $1$-ii),

  * •

the Volterra lattice [33] has an antisymmetric periodic one (type $1$-iii),

  * •

the Ablowitz-Ladik lattice [2] and the Schur flow [31] have a periodic CMV one
(type $1$-iv),

  * •

the family of Itoh–Narita–Bogoyavleskii [10] lattices have a bidiagonal
periodic one (type $1$-v).

We endow the periodic matrices of type $\boldsymbol{1}$ with the so-called
Generalized Gibbs Ensemble of the corresponding dynamical system. An important
property of these measures is that they are invariant with respect to the
dynamics of the corresponding integrable system. These Generalized Gibbs
Ensembles are probability measures $X^{N}$, where $X$ is a convex subset of
${\mathbb{R}}^{d}$, $d=1$ or $2$ (with ${\mathbb{C}}$ identified with
${\mathbb{R}}^{2}$) depending on the considered model, of the form

$\mu^{(1)}_{N}=\frac{1}{Z^{(1)}_{N}(\alpha,\Re
P)}\left(\prod_{j=1}^{N}F(x_{j},\alpha)\right)e^{-\mathrm{Tr}\left(\Re
P(L)\right)}{\rm d}{\bf x}\,.$ (1.31)

Here $L$ is a matrix of type 1, and the integration variable ${\bf
x}=(x_{1},\ldots,x_{N})$ is to be understood as the entries of $L$. In our
applications, we will specify the set $X$ to be

  * •

model 1-i) ${\bf x}=({\bf a},{\bf
b})\in{\mathbb{R}}^{N}\times{\mathbb{R}}_{+}^{N}$,
$X={\mathbb{R}}\times{\mathbb{R}}_{+}$

  * •

model 1-ii) ${\bf x}=({\bf a},{\bf b})\in{\mathbb{R}}_{+}^{2N}$,
$X={\mathbb{R}}_{+}\times{\mathbb{R}}_{+}$

  * •

model 1-iii) ${\bf x}={\bf a}\in{\mathbb{R}}_{+}^{N}$, $X={\mathbb{R}}_{+}$

  * •

model 1-iv) ${\bf x}={\bf a}\in{\mathbb{D}}^{N}$, $X={\mathbb{D}}$

  * •

model 1-v) ${\bf x}={\bf a}$ (resp. ${\bf x}={\bf b}$)
$\in{\mathbb{R}}_{+}^{N}$ if $b_{1}=\ldots=b_{N}=1$ (resp. if
$a_{1}=\ldots=a_{N}=1$), $X={\mathbb{R}}_{+}$.

For our applications, we consider polynomial potentials: $\Re P$ stands for
the real part of a polynomial $P\in{\mathbb{C}}[X]$. In all models except type
1-iv, we take directly $P\in{\mathbb{R}}[X]$. Model 1-iv is the only one with
complex entries, so in this case we consider just $\Re P$. Furthermore, $P$
has to be chosen in such a way that the measure (1.31) is normalizable, i.e.
such that the associated partition function converges. Note that with this
choice of potential, the trace $\mathrm{Tr}\left(\Re
P(L)\right)=\Re\mathrm{Tr}\left(P(L)\right)$ is simply expressed either in
terms of the diagonal entries of powers of $L$, or in terms of the eigenvalues
of the matrix $L$.

We recover the GGE of the Toda lattice (1.7) with polynomial potential taking
for $x=(a,b)\in{\mathbb{R}}\times{\mathbb{R}}_{+}$

$F(x,\alpha)=F(a,b,\alpha)=b^{2\alpha-1}\,.$

In this case, taking $P\in{\mathbb{R}}[X]$ polynomial with even degree $n\geq
2$ and positive leading coefficient ensures the convergence of the partition
function $Z^{\text{Toda}}_{N}(\alpha,P)$ of the model.

The non-periodic matrices of type $\boldsymbol{2}$ are related to the
classical $\beta$-ensembles. More specifically

  * •

The Real $\beta$-ensemble has a Jacobi matrix representation [20] (type 2-i)

  * •

The Laguerre $\beta$-ensemble has a positive definite Jacobi matrix
representation [20] (type 2-ii)

  * •

The Antisymmetric $\beta$-ensemble has a tridiagonal antisymmetric matrix
representation [21] (type 2-iii)

  * •

The Circular and Jacobi $\beta$-ensemble have a CMV matrix representation [45]
(type 2-iv)

We introduce properly those ensembles in Section 2. As in (1.11) for the case
of the real $\beta$-ensemble, we consider these ensembles in the high-
temperature regime, meaning that the parameter $\beta$ scales with the matrix
size $N$ as $\beta=\frac{2\alpha}{N}$, for some $\alpha\in{\mathbb{R}}_{+}$
independent of $N$.

In all the cases we consider, the joint density of the entries of the matrix
representation of the $\beta$-ensembles with polynomial potential can be
expressed as

$\mu^{(2)}_{N}=\frac{1}{Z^{(2)}_{N}(\alpha,\Re
P)}\left(\prod_{j=1}^{N-1}F\left(x_{j},\alpha\left(1-\frac{j}{N}\right)\right)\right)R(x_{N})e^{-\mathrm{Tr}\left(\Re
P(T)\right)}{\rm d}{\bf x}\,.$ (1.32)

As before, this probability measure is the entry-wise law of $T$; furthermore,
in our applications, the space is the same as in the corresponding type 1
model:

  * •

model 2-i) ${\bf x}=({\bf a},{\bf
b})\in{\mathbb{R}}^{N}\times{\mathbb{R}}_{+}^{N}$

  * •

model 2-ii) ${\bf x}=({\bf a},{\bf b})\in{\mathbb{R}}_{+}^{2N}$

  * •

model 2-iii ${\bf x}={\bf a}\in{\mathbb{R}}^{N}_{+}$

  * •

model 2-iv) ${\bf x}={\bf a}\in{\mathbb{D}}^{2N}$

In the previous equation, $R$ is a distribution defining a probability measure
supported on a subset of $X$: we denote, with an abuse of notation, $R(x){\rm
d}x$ the integration against $R$. Thus, for some $X^{\prime}\subset X$,
$\int_{X^{\prime}}R(x){\rm d}x=1$ and for any $\varphi\in C^{\infty}_{c}(X)$
with support disjoint from $X^{\prime}$, $\int_{X}\varphi(x)R(x){\rm d}x=0$.

In this notation, the type 2 models are defined on the same space as the type
1. As an example, considering $x=(a,b)\in{\mathbb{R}}\times{\mathbb{R}}_{+}$

$F(x,\alpha)=F(a,b,\alpha)=b^{2\alpha-1}\,,\qquad
R(x)=R(a,b)=\delta_{0}(b)\,,$

we recover the tridiagonal representation of the real $\beta$-ensemble in the
high-temperature regime (1.11) with polynomial potential. As for the matrices
of type 1, $P\in{\mathbb{R}}[X]$ except for the model 2-iv associated with CMV
matrices, which is the only model of type 2 with complex entries. We remark
that the function $F$ specified for the real $\beta$-ensemble at high
temperature is the same as the one for the Toda lattice (1.7).

As we already mentioned, we focus on the fluctuations (or linear statistics)
around the equilibrium measure of these general models, where we choose the
functions $F$, $P$ and the distribution $R$ so that the partition functions

$\begin{split}&Z^{(1)}_{N}(\alpha,\Re
P)=\int_{X^{N}}\left(\prod_{j=1}^{N}F(x_{j},\alpha)\right)e^{-\mathrm{Tr}\left(\Re
P(L)\right)}{\rm d}{\bf x}\,\\\ &Z^{(2)}_{N}(\alpha,\Re
P)=\int_{X^{N}}\left(\prod_{j=1}^{N-1}F\left(x_{j},\alpha\left(1-\frac{j}{N}\right)\right)\right)R(x_{N})e^{-\mathrm{Tr}\left(\Re
P(T)\right)}{\rm d}{\bf x}\end{split}$ (1.33)

are finite for all $N$. Recall that $X$ is a convex subset of
${\mathbb{R}}^{d}$. In our applications, $d=1$ or $2$, and ${\mathbb{C}}$ is
identified with ${\mathbb{R}}^{2}$.

Specifically, we focus on analysing the fluctuations

$\int_{\mathbb{C}}Q{\rm d}\nu_{N}-\lim_{N\to\infty}\int_{\mathbb{C}}Q{\rm
d}\nu_{N}\,,$ (1.34)

for polynomial test functions $Q$, where $\nu_{N}$ is the empirical measure of
eigenvalues of the considered matrix $M$:

$\nu_{N}=\frac{1}{N}\sum_{j=1}^{N}\delta_{\lambda_{j}(M)}\,,$ (1.35)

with $\lambda_{j}(M)$ denoting the eigenvalues of $M$. In this equation,
$\delta_{x}$ denotes the Dirac delta function centred at $x$. Note that the
existence of the limit

$\lim_{N\to\infty}\int_{\mathbb{C}}Q{\rm d}\nu_{N}$ (1.36)

is not straightforward: we establish it in Theorem 1.5, see Remark 1.6.

Our paper utilizes a transfer operator technique to analyze the random
variable (1.34) for polynomial potentials $\Re P$.

The study of statistical spectral properties of Lax matrices was initiated by
Spohn in [64], establishing the link between the Lax matrix of the Toda chain
distributed according to any of the GGEs, and the real $\beta$-ensemble at
high temperature, as sketched in Section 1.1. See also[51]. In [64, 63, 65],
the author studies space-time correlation functions for the Toda lattice and,
leveraging the theory of Generalized Hydrodynamics [18], he argues that they
have a ballistic behaviour, characterized by symmetrically located peaks that
travel in opposite directions at a constant speed and decay as $t^{-1}$ when
$t$ approaches infinity. To achieve this result, Spohn computed the density of
states of the Toda Lax matrix by connecting the Generalized Gibbs Ensemble of
the Toda lattice to the real $\beta$-ensemble in the high-temperature regime
[4]. Later, one of the authors and T. Grava connected the Generalized Gibbs
ensemble for the Ablowitz-Ladik lattice and the Schur flow to the Circular
$\beta$-ensemble and the Jacobi $\beta$-ensemble, respectively, in the high-
temperature regime [38, 28], as did H. Spohn independently [35, 65]. For all
these models, a large deviation principle for their mean density of states was
developed in [53, 36]. Furthermore, in [33], the authors established
connections between the classical Gibbs ensemble for the Exponential Toda
lattice and the Volterra lattice with the Laguerre ensemble [20] and the
Antisymmetric $\beta$-ensemble [21], respectively.

In addition to integrable systems, our study also involves the classical
$\beta$-ensembles. Specifically, we investigate the random variable (1.34) for
these ensembles in the high-temperature regime. The study of these quantities
was initiated by Johansson in [41], where the author derived a central limit
theorem (CLT) for the Gaussian Unitary Ensemble with a polynomial potential.
This result was subsequently generalized to other models and different values
of $\beta$ in [12, 61, 22], and more recently in [13], where the authors also
obtained a rigidity result for the eigenvalues of the $\beta$-ensembles. See
also [46] for a refinement of the CLT for the circular $\beta$-ensemble at a
mesoscopic scale. Other relevant works include [56], where a CLT was derived
for the $\beta$-ensemble in the high-temperature regime with a quadratic
potential, [38], where a CLT for the Circular $\beta$-ensemble at high
temperatures was obtained using a normal approximation method, and the recent
paper [23], where the authors derived a CLT for the real $\beta$-ensemble in
the high-temperature regime for general confining potentials. We also mention
the work [7], where Coulomb gases in dimensions ${\mathtt{d}}\geq 2$ were
investigated, and local laws were studied for various temperature regimes.
Finally, we mention the recent works [42] and [15, 16], where the authors
established super-exponential bounds for the convergence of moments of the
CUE, COE, and CSE to a Gaussian vector.

##### Statement of the results.

We come to precise statements of the main results of the present paper. To do
that, we introduce the concept of circular functions.

######  Definition 1.1.

Let $N\in{\mathbb{N}}$ and $s\geq 1$. Write $N=sM+\ell$ where $0\leq\ell<s$.
We say that a function $Y:X^{N}\to{\mathbb{C}}$ is $s$-circular if there exist
two functions $y\,:\,X^{2s}\to{\mathbb{C}}$, and
$\widetilde{y}\,:\,X^{2s+\ell}\to{\mathbb{C}}$ such that

$Y({\bf x})=\sum_{j=1}^{M-1}y({\bf x}_{j},{\bf x}_{j+1})+\widetilde{y}({\bf
x}_{1},{\bf x}_{M},x_{sM+1},\ldots,x_{N})\,,$ (1.37)

where ${\bf x}_{j}=(x_{(j-1)s+1},\ldots,x_{js})$ and $s$ is independent of $N$
for $N$ big enough. We call $y$ a seed of $Y$, $\widetilde{y}$ a weed of $Y$,
and $s$ the circular index of $Y$. When we are not interested in the specific
circular index of the function $Y$, we just say that $Y$ is circular.

######  Remark 1.2.

We notice that

  * •

The seed of a function is not unique, indeed $\hat{y}({\bf x}_{j},{\bf
x}_{j+1},{\bf x}_{j+2},{\bf x}_{j+3})=y({\bf x}_{j},{\bf x}_{j+1})+y({\bf
x}_{j+1},{\bf x}_{j+2})+y({\bf x}_{j+2},{\bf x}_{j+3})$ is another seed;

  * •

the sum of a $s_{1}$-circular function $Y_{1}$ and a $s_{2}$-circular function
$Y_{2}$ is a $s_{3}$-circular function $Y_{3}$, where $s_{3}$ is the smallest
common multiple between $s_{1},s_{2}$. In this situation, we say that
$y_{1},y_{2}$ are compatible seeds of $Y_{1},Y_{2}$ if

    * –

$y_{1}+y_{2}$ is a seed of $Y_{3}$;

    * –

the circular indexes of $y_{1},y_{2}$ is $s_{3}$;

    * –

$y_{1}$ is a seed of $Y_{1}$, and $y_{2}$ is a seed of $Y_{2}$

Furthermore, the following lemma holds true

######  Lemma 1.3.

Consider any type 1-2 matrix $M$. Then for any polynomial
$P\in{\mathbb{C}}[X]$, $\mathrm{Tr}\left(P(M)\right)$ is circular.

We detail the proof of this Lemma in Appendix A.

Given this definition and this lemma, we can proceed by stating our main
result. First, we state the assumptions we make for the definition of our
measures of interest $\mu^{(1)}_{N}$ (1.31) and $\mu^{(2)}_{N}$ (1.32). Let
$M$ be a matrix of type $1$ or of type $2$, endowed with the distribution
$\mu^{(1)}_{N}$ or $\mu^{(2)}_{N}$ on $X^{N}$, $X\subset{\mathbb{R}}^{d}$,
$d=1$ or $2$.

Throughout the paper, we will consider polynomial potentials with the
following properties, depending on the considered model.

######  Hypotheses 1.1.

Models 1-i and 2-i: $P\in{\mathbb{R}}[X]$, with degree $d\geq 2$ and positive
leading coefficient

Models 1-ii and 2-ii: $P\in{\mathbb{R}}[X]$, with degree $d\geq 1$ and
positive leading coefficient

Models 1-iii and 2-iii: $P\in{\mathbb{R}}[X]$, with $P$ of the form
$P(x)=(-1)^{d}a_{d}x^{2d}+Q(x)$ with $a_{d}>0$ and $Q\in{\mathbb{R}}[X]$ with
degree smaller than $2d-1$.

Models 1-iv and 2-iv: $P\in{\mathbb{C}}[X]$

Model 1-v: $P\in{\mathbb{R}}[X]$ with degree $j(r+1)$ for some $j\geq 1$ (with
$r$ appearing in the definition of the matrix model) and positive leading
coefficient.

######  Hypotheses 1.2.

To prove our main result, we assume the following:

  1. HP 1.

$X$ is a convex subset of ${\mathbb{R}}^{d}$, $d=1$ or $2$ (${\mathbb{C}}$
being identified with ${\mathbb{R}}^{2}$).

  2. HP 2.

$F(x,\alpha)\,:\,X\times(0,\infty)\to{\mathbb{R}}_{+}$ is such that for any
$\alpha>0$ $F(\cdot,\alpha)\in C^{1}(X)$, and for any $x\in X,F(x,\cdot)\in
C^{\infty}((0,+\infty))$ ;

  3. HP 3.

The seed $W$ and weed ${\widetilde{W}}$ of $\mathrm{Tr}\left(\Re P(M)\right)$
are bounded away from $-\infty$

  4. HP 4.

for all $\alpha\in(0,\infty)$, $F(\cdot,\alpha)\in L^{1}(X)$, and
$[c,d]\subset(0,+\infty)$ one can find $g_{c,d}\in L^{2}(X)$ such that for all
$c\leq\alpha\leq d$, $\left|\sqrt{F(\cdot,\alpha)}\right|,\
\left|\partial_{\alpha}\sqrt{F(\cdot,\alpha)}\right|\leq g_{c,d}$; moreover,
there exist a ${\mathtt{c}}\in{\mathbb{N}}$ and $\varepsilon_{0}$ such that
for all $\varepsilon<\varepsilon_{0}$ there exists a compact set
$\mathcal{O}_{\varepsilon}\subseteq X$ and $d_{1},d_{2},d_{3}>0$, depending on
$\varepsilon$, such that

     * •

$||F(\cdot,\alpha)||_{1}=d_{1}\alpha^{-{\mathtt{c}}}(1+o(1))$ as $\alpha\to 0$

     * •

$||F(\cdot,\alpha)||_{1,\mathcal{O}_{\varepsilon}}=\int_{\mathcal{O}_{\varepsilon}}|F(x,\alpha)|{\rm
d}x=d_{2}\alpha^{-{\mathtt{c}}}(1+o(1))$ as $\alpha\to 0$

     * •

$||F(\cdot,\alpha)||_{1,\mathcal{O}^{c}_{\varepsilon}}\leq d_{3}$ here
$\mathcal{O}_{\varepsilon}^{c}=X\setminus\mathcal{O}_{\varepsilon}$

     * •

There exists a continuous function $w\,:\,X^{k}\to{\mathbb{R}}$ such that for
${\bf x},{\bf y}\in\mathcal{O}_{\varepsilon}$

$W({\bf x},{\bf y})=w({\bf x})+w({\bf y})+o(\varepsilon)\,.$ (1.38)

     * •

If $\varepsilon_{1}<\varepsilon_{2}$ then
$\mathcal{O}_{\varepsilon_{1}}\subseteq\mathcal{O}_{\varepsilon_{2}}$

  5. HP 5.

$R$ is a distribution defining a probability measure on a subset of $X$.

Here $L^{p}(X,{\mathcal{B}})$ is the usual $L^{p}$ space.

######  Remark 1.4.

The definition of circular function and seeds was introduced in this context
in [30].

Let $s\in{\mathbb{N}}$ and $t\in{\mathbb{R}}$. Then, for $P$, $F$ satisfying
the previous assumptions, both

$\left(\prod_{j=1}^{N}F(x_{j},\alpha)\right)e^{-\mathrm{Tr}\big{(}\Re
P(L)+it\Re L^{s}\big{)}}$

and

$\left(\prod_{j=1}^{N-1}F\left(x_{j},\alpha\left(1-\frac{j}{N}\right)\right)\right)e^{-\mathrm{Tr}\big{(}\Re
P(L)+it\Re L^{s}}\big{)}R(x_{N})$

are integrable on $X^{N}$. We denote by

$Z^{(1)}_{N}(\alpha,\Re P+it\Re z^{s})\text{ and }Z^{(2)}_{N}(\alpha,\Re
P+it\Re z^{s})$

the associated integrals.

Under these assumptions, we prove our main theorem:

######  Theorem 1.5.

Under Hypotheses 1.1-1.2. Consider $\mu^{(1)}_{N},\,\mu^{(2)}_{N}$
(1.31)-(1.32), and let $s\in{\mathbb{N}}$, $W,h$ be compatible seeds of
$\mathrm{Tr}\left(\Re P(L)\right)$ and $\mathrm{Tr}\left(\Re L^{s}\right)$,
and assume that $|h|^{a}e^{-W}$ is bounded, for $1\leq a\leq 3$. Then, there
exist two continuous functions

$\displaystyle A(x)\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}\,,$ (1.39)
$\displaystyle\sigma^{2}(x)\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}_{+}\,,$
(1.40)

such that under $\mu^{(1)}_{N}$ (1.31)

$\left(\mathrm{Tr}\left(\Re L^{s}\right)-NA(\alpha)\right)/\sqrt{N}$

converges to a Gaussian distribution ${\mathcal{N}}(0,\sigma^{2}(\alpha))$ as
$N$ tends to infinity. Similarly, under $\mu^{(2)}_{N}$ (1.32),

$\left(\mathrm{Tr}\left(\Re T^{s}\right)-N\int_{0}^{1}A(\alpha x){\rm
d}x\right)/\sqrt{N}$

converges to a Gaussian distribution
${\mathcal{N}}(0,\int_{0}^{1}\sigma^{2}(\alpha x){\rm d}x)$ as $N$ tends to
infinity. Furthermore, defining the free energies
${\mathcal{F}}^{(1)}(\alpha,\Re P),{\mathcal{F}}^{(2)}(\alpha,\Re P)$ as

$\displaystyle{\mathcal{F}}^{(1)}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{1}{N}\ln\left(Z_{N}^{(1)}(\alpha,\Re P)\right)\,,$
(1.41) $\displaystyle{\mathcal{F}}^{(2)}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{1}{N}\ln\left(Z_{N}^{(2)}(\alpha,\Re P)\right)\,,$
(1.42)

then

  1. i.

${\mathcal{F}}^{(1)}(\alpha,\Re
P)=\partial_{\alpha}\left(\alpha{\mathcal{F}}^{(2)}(\alpha,\Re P)\right)$

  2. ii.

$A(\alpha)=i\partial_{t}{\mathcal{F}}^{(1)}(\alpha,\Re P+it\Re
z^{s})_{|_{t=0}}$

  3. iii.

$\int_{0}^{1}A(\alpha x){\rm d}x=i\partial_{t}{\mathcal{F}}^{(2)}(\alpha,\Re
P+it\Re z^{s})_{|_{t=0}}$

  4. iv.

$\sigma^{2}(\alpha)=\partial^{2}_{t}{\mathcal{F}}^{(1)}(\alpha,\Re P+it\Re
z^{s})_{|_{t=0}}$

  5. v.

$\int_{0}^{1}\sigma^{2}(\alpha x){\rm
d}x=\partial^{2}_{t}{\mathcal{F}}^{(2)}(\alpha,\Re P+it\Re z^{s})_{|_{t=0}}$

######  Remark 1.6.

Let us notice that

  * •

The previous theorem also holds for $\Im z^{s}$ in place of $\Re z^{s}$.

  * •

As a consequence of Theorem 1.5, we deduce the convergence in law of both

$\frac{1}{N}\Re\mathrm{Tr}\left(L^{s}\right)\text{ and
}\frac{1}{N}\Im\mathrm{Tr}\left(M^{s}\right)$

as $N\to+\infty$, for any $L$ among the considered models. Furthermore, the
limits are deterministic, therefore we deduce the convergence of the moments,
as $N\to+\infty$

$\lim_{N\to+\infty}\frac{1}{N}\mathrm{Tr}\left(L^{s}\right)=\int_{\mathbb{C}}z^{s}{\rm
d}\nu_{N}(z)\,.$

In the case of the Ablowitz-Ladik lattice (matrix of type 1-iii) or of the
circular $\beta$-ensemble in the high-temperature regime (matrix of type
2-iii), this is a consequence of the following facts

    * 1)

The empirical measures are supported on the circle (as empirical measures of
unitary matrices)

    * 2)

They converge (see for [53] for Ablowitz-Ladik, [38] for circular
$\beta$-ensemble for example)

    * 3)

Polynomial functions are bounded on the circle.

In general, even in cases where we know the empirical measure to converge, the
fact that the previous limit exists is not trivial, since polynomial functions
are not bounded on non-compact sets.

######  Remark 1.7.

Let us check here that Theorem 1.5 applies to the case of the Toda lattice,
and its random matrix counterpart, i.e. the real $\beta$-ensemble in the high-
temperature regime. The points that need to be discussed are HP 3., HP 4. and
that $|h|^{s}e^{-W}$ is bounded for $1\leq s\leq 3$.

Let $L$ be a type $1$—i matrix with coordinates ${\bf x}=({\bf a},{\bf b})\in
X_{0}^{N}=({\mathbb{R}}\times{\mathbb{R}}_{+})^{N}$ and consider
$\mu^{(1)}_{N}$ with $P$ satisfying Hypotheses 1.1 and

$F(x,\alpha)=F(a,b,\alpha)=b^{2\alpha-1}\,.$

The latter function is not integrable on $X_{0}$, so it doesn’t satisfy
Hypotheses 1.2 HP3. Instead, we take advantage of the following fact: there
are some $c>0$, $C\in{\mathbb{R}}$ such that for all $x\in{\mathbb{R}}$,

$P(x)\geq C+cx^{2}.$

Thus, $\mathrm{Tr}\left(P(L)\right)\geq
CN+c\mathrm{Tr}\left(L^{2}\right)=CN+c\sum_{j=1}^{N}a_{j}^{2}+2b_{j}^{2}$,
therefore setting

${\widetilde{F}}(a,b,\alpha)=b^{2\alpha-1}e^{-\frac{c}{2}(a^{2}+2b^{2})}\,,$

the measure $\mu^{(1)}_{N}$ reads

$\frac{1}{Z^{(1)}_{N}(\alpha,P)}\left(\prod_{j=1}^{N}{\widetilde{F}}(x_{j},\alpha)\right)e^{-(\mathrm{Tr}\left(P(L)\right)-\frac{c}{2}\mathrm{Tr}\left(L^{2}\right))}{\rm
d}{\bf x}\,.$

The modified potential $P(x)-\frac{c}{2}x^{2}$ still fulfils Hypotheses 1.1,
and the function ${\widetilde{F}}$ then satisfies the prescribed assumptions.
Indeed,
$\partial_{\alpha}\sqrt{F(a,b,\alpha)}=\ln(b)b^{\alpha-\frac{1}{2}}e^{-\frac{c}{4}(a^{2}+2b^{2})}$
thus we can easily find, for any compact $K\subset{\mathbb{R}}_{+}^{*}$, a
function $g_{K}\in L^{2}({\mathbb{R}}_{+}\times{\mathbb{R}})$ such that for
any $\alpha\in K$, and for any $(a,b)\in{\mathbb{R}}_{+}\times{\mathbb{R}}$

$\sqrt{F(a,b,\alpha)},\ \left|\partial_{\alpha}\sqrt{F(a,b,\alpha)}\right|\leq
g_{K}(a,b)\,.$

The remaining points of HP3. hold for ${\mathtt{c}}=1$. Indeed,
$\|F(\cdot,\alpha)\|_{L^{1}(X)}=c^{-\alpha}2^{1-\alpha}\Gamma(\alpha)$, that
as $\alpha\to 0$ is of order $\alpha^{-1}$, and with
$\mathcal{O}_{\varepsilon}=[0,\varepsilon]^{2}$, we have

$\int_{\mathcal{O}_{\varepsilon}}b^{2\alpha-1}e^{-\frac{c}{2}(a^{2}+2b^{2})}{\rm
d}a{\rm d}b\geq\int_{0}^{\varepsilon}e^{-\frac{ca^{2}}{2}}{\rm
d}a\int_{0}^{\varepsilon}b^{2\alpha-1}e^{-c}{\rm
d}b=:\frac{d_{\varepsilon}}{\alpha}\left(1+o(1)\right)\,,$

as $\alpha\to 0$; moreover, $d_{\varepsilon}\to 0$ as $\varepsilon\to 0$ and
it is independent of $\alpha$.

HP4. and the boundedness of $|h|^{s}e^{-W}$, $1\leq s\leq 3$ is a consequence
of Theorem 2.1, see Corollary 2.2.

######  Remark 1.8.

In the central part of the proof, we introduce a family of operators that act
on $L^{2}(X^{k})$ as follows:

${\mathcal{L}}_{\alpha,t}f({\bf y})=\int_{X^{k}}f({\bf
x})\prod_{j=1}^{k}\sqrt{F(x_{j},\alpha)F(y_{j},\alpha)}e^{-(W+ith)({\bf
y},{\bf x})}{\rm d}{\bf x}{\rm d}{\bf y}\,.$

Based on our assumptions, each ${\mathcal{L}}_{\alpha,t}$ is a Hilbert-Schmidt
operator, meaning that the kernel

$({\bf x},{\bf
y})\mapsto\prod_{j=1}^{k}\sqrt{F(x_{j},\alpha)F(y_{j},\alpha)}e^{-(W+ith)({\bf
y},{\bf x})}$

belongs to $L^{2}(X^{k}\times X^{k})$. Consequently,
${\mathcal{L}}_{\alpha,t}$ is a compact operator. Then, we have the following:

  * •

Under Hypotheses 1.2 and the assumption that $|h|^{a}\exp(-W)$ is bounded for
$1\leq a\leq 3$, we show in Lemma 6.6 that
$(\alpha,t)\mapsto{\mathcal{L}}_{\alpha,t}$ is a regular, operator valued
function: specifically, $(\alpha,t)\mapsto{\mathcal{L}}_{\alpha,t}$ is
differentiable with respect to $\alpha$ and three times differentiable with
respect to $t$;

  * •

As a consequence of Jentzsh’s theorem [72, Theorem 137.4] and an approximation
argument, ${\mathcal{L}}_{\alpha,0}$ possesses a simple dominant eigenvalue
with a spectral gap uniform in $\alpha$.

These two properties ensure that for sufficiently small $t$, the operator
${\mathcal{L}}_{\alpha,t}$ possesses a simple dominant eigenvalue
${\widetilde{\lambda}}(\alpha,t)$, and the function
$t\mapsto{\widetilde{\lambda}}(\alpha,t)$ is three times differentiable. The
function ${\widetilde{\lambda}}(\alpha,t)$ is the key to our analysis, because
of the equality, established in Lemma 5.2

${\mathcal{F}}^{(1)}(\alpha,\Re P+it\Re
z^{s})=-\frac{1}{k}\ln\left({\widetilde{\lambda}}(\alpha,t)\right)\,,$

where ${\mathcal{F}}^{(1)}$ is the free energy defined in (1.41).

Note that we only require the existence of a second derivative with respect to
$t$ in the proof of the main theorem, while the existence of a third
derivative is used in the proof of the Berry-Esseen bound, as stated in
Theorem 6.10. The differentiability with respect to $\alpha$ is utilized in
the proof of Theorem 6.8.

As a by-product, we can also compute the so-called susceptibility matrix
$\mathcal{C}$ for integrable models. This is the matrix of the space-
correlation functions of the conserved fields, i.e.

$\mathcal{C}_{m,n}=\lim_{N\to\infty}\frac{1}{N}\left({\mathbb{E}}_{1}\left[\mathrm{Tr}\left(\Re{L^{m}}\right)\mathrm{Tr}\left(\Re
L^{n}\right)\right]-{\mathbb{E}}_{1}\left[\mathrm{Tr}\left(\Re
L^{m}\right)\right]{\mathbb{E}}_{1}\left[\mathrm{Tr}\left(\Re
L^{n}\right)\right]\right)\,,$ (1.43)

where $L$ is the Lax matrix of the integrable system at hand and the mean
values are taken according to the corresponding Generalized Gibbs ensemble.
The computation of such quantities is relevant to obtain the decay of the
correlation functions for these integrable systems, as shown by Spohn in [63].
In particular, we can prove the following:

######  Theorem 1.9.

Under the same hypotheses as Theorem 1.5. Consider $\mu^{(1)}_{N}$ (1.31) and
define the free energy ${\mathcal{F}}^{(1)}(\alpha,\Re P)$ as in (1.41), then

$\mathcal{C}_{m,n}=\partial_{t_{1}}\partial_{t_{2}}\left({\mathcal{F}}^{(1)}(\alpha,\Re
P+it_{1}\Re z^{m}+it_{2}\Re z^{n})\right)_{|_{t_{1}=t_{2}=0}}\,.$ (1.44)

######  Remark 1.10.

In view of Theorem 1.5, we can rewrite (1.44) as

$\mathcal{C}_{m,n}=\partial_{\alpha}\partial_{t_{1}}\partial_{t_{2}}\left(\alpha\left({\mathcal{F}}^{(2)}(\alpha,\Re
P+it_{1}\Re z^{m}+it_{2}\Re z^{n})\right)\right)_{|_{t_{1}=t_{2}=0}}\,.$
(1.45)

In our context, the previous equality implies that we can compute the
susceptibility matrix of the integrable systems that we are considering in
terms of just the free energy of the corresponding classical $\beta$-ensemble
in the high-temperature regime.

Furthermore, considering the type 1 measures (1.31), we investigate the space-
correlations for local functions, defined as follows. Given a differentiable
function $F\colon X^{N}\to{\mathbb{C}}$, we define its support as the set

${\rm supp}\,F:=\left\\{\ell\in\\{1,\ldots,N\\}\colon\ \ \ \frac{\partial
F}{\partial x_{\ell}}\nequiv 0\right\\}$ (1.46)

and its diameter as

${\rm diam}\left({\rm supp}\,F\right):=\sup_{i,j\in{\rm
supp}\,F}{\mathtt{d}}_{N}(i,j),$ (1.47)

where ${\mathtt{d}}$ is the periodic distance

${\mathtt{d}}(i,j):=\min\left(|i-j|,\ N-|i-j|\right).$ (1.48)

Note that $0\leq{\mathtt{d}}(i,j)\leq N/2$.

We say that a function $F$ is local if ${\rm diam}\left({\rm supp}\,F\right)$
is uniformly bounded in $N$, i.e. there exists a constant
${\mathtt{c}}\in{\mathbb{N}}$ such that ${\rm diam}\left({\rm
supp}\,F\right)\leq{\mathtt{c}}$, and ${\mathtt{c}}$ is independent of $N$. We
remark that these definitions were already introduced in [34].

Given this definition, we can prove the following lemma

######  Theorem 1.11 (Decay of correlations).

Consider a matrix $L$ of type 1 with polynomial potential $\Re P$ such that
$\mu_{N}^{(1)}$ (1.31) is well-defined, let $I,J:X^{k}\to{\mathbb{R}}$ two
local functions such that $\int_{X^{k}\times X^{k}}\left|I({\bf
x})\prod_{i=1}^{k}F(x_{i},\alpha)e^{-W({\bf x},{\bf y})}\right|^{2}{\rm d}{\bf
x}{\rm d}{\bf y}<\infty$, where $W$ is one seed of $\mathrm{Tr}\left(\Re
P(L)\right)$, and analogously for $J({\bf x})$. Write $N=kM+\ell$, and let
$j\in\\{1,\ldots,M\\}$. Then there exists some $0<\mu<1$ such that

${\mathbb{E}}_{1}\left[I({\bf x}_{1})J({\bf
x}_{j})\right]-{\mathbb{E}}_{1}\left[I({\bf
x}_{1})\right]{\mathbb{E}}_{1}\left[J({\bf
x}_{j})\right]=O(\mu^{M-j}+\mu^{j})\,.$

This result establishes the exponential decay of space correlations between
two local functions acting on distinct segments of the chain, with the decay
rate determined by the distance between the corresponding particle sets.
Thanks to this Theorem, we rigorously justify the assumption made by H. Spohn
regarding the decay of space correlations between the local conserved fields
and their currents in Section 2.1, where we apply the aforementioned result.
We remark that the same reasoning can be applied to other integrable systems
considered in our study.

Finally, we also prove a Berry-Essen type bound for the aforementioned
integrable models. Specifically, we prove the following

######  Theorem 1.12.

Fix $s\in{\mathbb{N}}$. Under Hypotheses 1.2. Consider a matrix $L$ of type 1
with polynomial potential $\Re P$ such that $\mu_{N}^{(1)}$ (1.31) is well-
defined, and let $W,h$ be the compatible seed of $\mathrm{Tr}\left(\Re
P(L)\right)$ and $\mathrm{Tr}\left(\Re L^{s}\right)$. Assume that
$|h|^{a}e^{-W}$ is bounded for $1\leq a\leq 3$, so that

$\mathbb{E}_{1}\left[e^{-it\mathrm{Tr}\left(\Re
L^{s}\right)}\right]=\frac{Z_{kM+\ell}^{(1)}(\alpha,\Re P+it\Re
x^{s})}{Z_{kM+\ell}^{(1)}(\alpha,\Re P)}\,.$ (1.49)

Then, there exists $A\in{\mathbb{R}}$, $\sigma,C>0$ such that if
$Y\sim{\mathcal{N}}(0,\sigma^{2})$ we have for any interval $J$ of the real
line

$\left|\mathbb{P}\left(\left[\mathrm{Tr}\left(L^{s}\right)-NA\right]/\sqrt{N}\in
J\right)-\mathbb{P}\left(Y\in J\right)\right|\leq\frac{C}{\sqrt{N}}\,.$ (1.50)

The paper is organized as follows. In Section 2 we deduce a central limit
theorem for the systems in table 1 by applying Theorem 1.5. In Section 3, we
use the transfer operator method to explicitly compute the partition functions
for the models in table 1. In Section 4, we prove some generalization of
Nagaev–Guivarc’h Theorem, which is the main probability tool that allows us to
prove our main result. In Section 5, we prove the main results of our paper,
i.e Theorems 1.5-1.9. In Section 6, we prove the technical result that we
stated in Section 3 and we prove Theorems 1.11-1.12. Finally, in Section 7, we
provide concluding remarks and outline potential avenues for future
advancements in this field. We defer some of the technical results of the
paper in Appendix A-B.

##  2 Application

In this section, we show how to apply Theorem 1.5 to obtain a CLT for the
integrable systems and for the classical $\beta$-ensembles in the high-
temperature regime in Table 1.

Specifically, we are able to prove that all the integrable systems in Table 1
in the periodic case have a Generalized Gibbs ensemble of the form
$\mu^{(1)}_{N}$ (1.31). Meanwhile, the $\beta$-ensembles at high-temperature
are characterized by a probability distribution of the form $\mu^{(2)}_{N}$
(1.32). In this way, we proved a further connection between the theory of
integrable systems and Random Matrix Theory. Indeed, in view of Theorem 1.5
and Theorem 1.9, for any integrable system in the previous table, we can
relate its free energy, moments, variances and covariances with the
corresponding quantities of the random matrix model on the same line.
Moreover, in the final part of this section, we consider the family of INB
lattices that do not have a known $\beta$-ensemble counterpart. Despite that,
we are still able to derive the existence of a polynomial central limit
theorem. Finally, applying the decay of correlations of Theorem 1.11, we are
able to compute the currents of the Toda chain in terms of the limiting
variance in the CLT, justifying a computation by Spohn in [66].

###  2.1 The Toda lattice, and the real $\beta$-ensemble at high-temperature

##### The Toda lattice.

In this subsection, we apply the previous results to deduce the central limit
theorem for the empirical measure in the setup of the Toda chain. We then
apply the decay of correlations established in Theorem 1.11 to deduce a
formulation of the currents of the Toda chain in terms of the limiting
variance in the CLT, justifying a computation by Spohn, [66]. Finally, we
relate the limiting variance and free energy of the Toda chain with the ones
of the $\beta$-ensemble at high-temperature.

To begin with, our aim is to obtain a central limit theorem for the conserved
fields when the initial data is sampled according to the Generalized Gibbs
ensemble of the Toda chain, which we rewrite here for convenience. To ease the
notations, we denote it by $\mu_{T}$ instead of
$\mu_{N,\text{Toda}}^{\alpha,P}$ in the sequel, and we denote the associated
partition function by $Z_{N}^{T}(\alpha,P)$.

${\rm d}\mu_{T}:=\frac{1}{Z_{N}^{T}(\alpha,P)}\
\prod_{j=1}^{N}b_{j}^{2\alpha-1}\mathbbm{1}_{b_{j}>0}e^{-\mathrm{Tr}\left(P(L)\right)}{\rm
d}{\bf a}\,{\rm d}{\bf b}\,,$ (2.1)

where $P$ is a polynomial of even degree with positive leading coefficient,
and $\alpha>0$. Also recall that $L$ is a periodic Jacobi matrix, i.e. a
matrix of type 1-i.

We want to apply Theorem 1.5 to this model. To do that, we need some
preparation. First, we notice that the Lax matrix of the periodic Toda lattice
is a circular-like matrix, thus we can apply Lemma 1.3 to ensure that the
traces of powers of $L$ are circular functions. Furthermore, from the
following Theorem, whose proof can be found in [34], one can deduce that the
seeds of even powers of the trace are bounded from below, see Remark 2.3.

######  Theorem 2.1 (cf. Theorem 3.1 [34]).

For any $1\leq m\leq N-1$, consider the matrix $L$ given by (1.4). One has

$\mathrm{Tr}\left(L^{m}\right)=\sum_{j=1}^{N}h_{j}^{(m)}\,,$ (2.2)

where $h_{j}^{(m)}:=[L^{m}]_{jj}$ is given explicitly by

$h_{j}^{(m)}({\bf p},{\bf r})=\sum_{({\bf n},{\bf
q})\in{\mathcal{A}}^{(m)}}\rho^{(m)}({\bf n},{\bf
q})\prod_{i=-{\widetilde{m}}}^{{\widetilde{m}}-1}a_{j+1}^{q_{i}}\prod_{i=-{\widetilde{m}}+1}^{{\widetilde{m}}-1}b_{j+i}^{2n_{i}}\,,$
(2.3)

where it is understood $a_{j}\equiv a_{j{\,{\rm mod}\,}N+1},\,b_{j}\equiv
b_{j{\,{\rm mod}\,}N+1}$ and ${\mathcal{A}}^{(m)}$ is the set

$\begin{split}{\mathcal{A}}^{(m)}:=\Big{\\{}({\bf n},{\bf
q})\in{\mathbb{N}}^{\mathbb{Z}}_{0}\times{\mathbb{N}}^{\mathbb{Z}}_{0}\
\colon\ \ \
&\sum_{i=-{\widetilde{m}}}^{{\widetilde{m}}-1}\left(2n_{i}+q_{i}\right)=m,\\\
&\forall i\geq 0,\ \ \ n_{i}=0\Rightarrow n_{i+1}=q_{i+1}=0,\,\\\ &\forall
i<0,\ \ \ n_{i+1}=0\Rightarrow n_{i}=q_{i}=0\Big{\\}}.\end{split}$ (2.4)

The quantity ${\widetilde{m}}:=\lfloor m/2\rfloor$,
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$ and $\rho^{(m)}({\bf n},{\bf
q})\in{\mathbb{N}}$ is given by

$\displaystyle\rho^{(m)}({\bf n},{\bf q}):=$
$\displaystyle\binom{n_{-1}+n_{0}+q_{0}}{q_{0}}\binom{n_{-1}+n_{0}}{n_{0}}\prod_{i=-{\widetilde{m}}\atop
i\neq-1}^{{\widetilde{m}}-1}\binom{n_{i}+n_{i+1}+q_{i+1}-1}{q_{i+1}}\binom{n_{i}+n_{i+1}-1}{n_{i+1}}\,.$
(2.5)

The combination of these results leads to the following corollary

######  Corollary 2.2.

Fix $m\in{\mathbb{N}}$, and consider the matrix $L$ (1.4). Then for $N$ big
enough, there exists some $k=k(m)\in{\mathbb{N}}$, such that
$\mathrm{Tr}\left(L^{m}\right)$ is a $k$-circular function, with seed $V$ and
weed $\widetilde{V}$. Furthermore, both $V,\widetilde{V}$ are bounded from
below, away from $-\infty$. Furthermore, for $m$ even, for any local
polynomial $u({\bf a},{\bf b})$ with diameter ${\mathtt{d}}<k$ the function
$|u|^{s}e^{-V},\,s\in\mathbb{N}$ is bounded from above.

######  Remark 2.3.

We notice that one can prove that the function $V,\widetilde{V}$ are lower
bounded away from $-\infty$ using the explicit formula in Theorem 2.1 or
applying the properties of super-Motzkin paths used for the proof of the
Theorem in [34].

We apply the previous Corollary to the Gibbs measure of the Toda lattice
(2.1), so it can be written as

$\begin{split}{\rm
d}\mu_{T}&=\frac{1}{Z_{N}^{T}(\beta,P)}\prod_{j=1}^{N}b_{j}^{2\alpha-1}\mathbbm{1}_{b_{j}>0}\exp\Big{(}-\sum_{j=1}^{M}V({\bf
a}_{j},{\bf b}_{j},{\bf a}_{j+1},{\bf b}_{j+1})\\\ &-{\widetilde{V}}({\bf
a}_{M},a_{kM+1},\ldots,a_{kM+\ell},{\bf a}_{1},{\bf
b}_{M},b_{kM+1},\ldots,b_{kM+\ell},{\bf b}_{1})\Big{)}{\rm d}{\bf a}\,{\rm
d}{\bf b}\,.\end{split}$ (2.6)

Now we can apply Theorem 1.5, see Remark 1.7, and we deduce the following

######  Corollary 2.4 (CLT for the Toda lattice).

Consider the Lax matrix L (1.4) of the Toda lattice distributed according to
the Generalized Gibbs Ensemble (2.1), and assume that $P(x)$ is a polynomial
of even degree with positive leading order coefficient. Then, defining the
Free energy ${\mathcal{F}}_{T}(\alpha,P)$ as

${\mathcal{F}}_{T}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{T}_{N}(\alpha,P))\,,$
(2.7)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(L^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,.$
(2.8)

Where

$\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{T}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{T}(\alpha,P+itx^{j})_{|_{t=0}}|\,.$
(2.9)

Moreover, we can also apply Theorem 1.9 to compute the correlation between the
conserved fields at time zero, indeed the theorem immediately implies that

$\lim_{N\to\infty}\frac{\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\mathrm{Tr}\left(L^{n}\right)\right]-\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]\mathbb{E}\left[\mathrm{Tr}\left(L^{n}\right)\right]}{N}=\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{T}(\alpha,P+it_{1}x^{j}+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}\,,$
(2.10)

where the mean value is taken with respect to the Gibbs measure of the Toda
lattice (2.1). We notice that this implies that we can compute the
susceptibility matrix of the Toda lattice (1.44) in terms of the derivative of
the Free energy.

####  2.1.1 The Toda chain’s currents

Since the conserved fields are local quantities, they must satisfy a local
conservation law. Following the notation of [67], we define

$Q^{[n],N}_{j}=L^{n}_{j,j}\,,$ (2.11)

where $L\in\textrm{Mat}(N,{\mathbb{R}})$ is (1.4). We can easily compute the
evolution equation for such quantities as

$\frac{d}{dt}Q^{[n],N}_{j}=(BL^{n}-L^{n}B)=b_{j-1}L^{n}_{j,j-1}-b_{j}L^{n}_{j+1,j}\,.$
(2.12)

Defining $J^{[n],N}_{j}=b_{j-1}L^{n}_{j,j-1}$, we have

$\frac{d}{dt}Q^{[n],N}_{j}=J^{[n],N}_{j}-J^{[n],N}_{j+1}$

and we say that $J^{[n],N}_{j}$ is the current of the local conserved field
$Q^{[n],N}_{j}$. In particular, defining the matrix $L^{\downarrow}$ as

$L^{\downarrow}_{i,j}=\begin{cases}L_{i,j}\quad\text{if}\;j<i\;\text{or}\;i=1,j=N\\\
0\quad\text{otherwise}\end{cases}\,$ (2.13)

we can recast the previous definition as

$J^{[n],N}_{j}=(L^{n}L^{\downarrow})_{j,j}\,.$ (2.14)

We notice that both $Q^{[n],N}_{j}$ and $J^{[n],N}_{j}$ depend on time, and we
adopt the convention that if not explicitly written the evaluation is at time
$0$. Furthermore, we define

$Q^{[n],N}=\sum_{j=1}^{N}Q^{[n],N}_{j}\,,\qquad
J^{[n],N}=\sum_{j=1}^{N}J^{[n],N}_{j}\,,$ (2.15)

and we refer to $Q^{[n],N}$ as the $n^{th}$-conserved field, and to
$J^{[n],N}$ as the $n^{th}$-total current.

The evaluation of the expected values of both the currents $J^{[n],N}_{j}$ and
the total current $J^{[n],N}$ according to the Generalized Gibbs ensemble
(2.1) is one of the crucial steps to apply the theory of Generalized
Hydrodynamics to the Toda lattice, as it is explained in [67]. In this
manuscript, the author used some heuristic arguments to explicitly derive the
expression for these quantities, here we rigorously justify his argument
applying Theorem 1.11.

First, we extend the definition of $Q_{j}^{[n],N}$ and $J_{j}^{[n],N}$ for
$n=0$, setting $Q_{j}^{[0],N}=r_{j}$, and
$J_{j}^{[0],N}=-p_{j}=-Q_{j}^{[1],N}$. We notice that
$\sum_{j=1}^{N}J_{j}^{[0],N}=-\sum_{j=1}^{N}Q_{j}^{[1],N}$ is still a
conserved field. We are now in position to show how to compute the limiting
Toda average current

$\lim_{N\to\infty}\frac{1}{N}\mathbb{E}\left[J^{[n],N}\right]\,,$ (2.16)

in terms of the susceptibility matrix (1.44) of the Toda chain, so in
particular of the derivative of the Free energy (2.10). Indeed, we prove the
following:

######  Lemma 2.5.

Consider the Lax matrix L (1.4) of the Toda lattice distributed according to
the Generalized Gibbs Ensemble (2.1), and assume that $P(x)$ is a polynomial
of even degree with positive leading order coefficient. Then, for any fixed
$n\in{\mathbb{N}}$, and $\alpha\in{\mathbb{R}}_{+}$ defining the total
currents $J^{[n],N}$ as in (2.15) we have the following equality

$\lim_{N\to\infty}\frac{1}{N}{\mathbb{E}}\left[J^{[n],N}\right]=\int_{0}^{\alpha}\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{T}(s,P+it_{1}x+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}{\rm
d}s.$ (2.17)

###### Proof.

In view of the cyclic structure of the measure $\mu_{T}$ and of the total
current, we deduce that

$\frac{1}{N}\mathbb{E}\left[J^{[n],N}\right]=\mathbb{E}\left[J^{[n],N}_{1}\right]\,.$
(2.18)

Furthermore, for any fixed $N$, we deduce, by differentiating with respect to
the parameter $\alpha$, the following equality

$\partial_{\alpha}\mathbb{E}\left[J^{[n],N}_{1}\right]=-\mathrm{Cov}\left(J^{[n],N}_{1}\
;\ \sum_{j=1}^{N}r_{j}\right)=-\sum_{j=1}^{N}\mathrm{Cov}\left(J^{[n],N}_{1}\
;\ Q^{[0],N}_{j}\right)\,,$ (2.19)

where we defined for any functions $f,g\in L^{2}(X^{N},\mu_{T})$

$\textrm{Cov}(f\,;\,g)=\mathbb{E}\left[fg\right]-\mathbb{E}\left[f\right]\mathbb{E}\left[g\right]\,.$
(2.20)

We show now that the following limits coincide

$\lim_{N\to\infty}\sum_{j=1}^{N}\mathrm{Cov}\left(J^{[n],N}_{1}\,;\,Q^{[0],N}_{j}\right)=\lim_{N\to\infty}\sum_{j=1}^{N}\mathrm{Cov}\left(J^{[0],N}_{1}\,;\,Q^{[n],N}_{N-j+2}\right)\,.$
(2.21)

Indeed, for any $n,m\geq 0$ and $t\in{\mathbb{R}}$

$\begin{split}\mathrm{Cov}\left(J_{j+1}^{[n],N}(t)-J_{j}^{[n],N}(t)\ ;\
Q^{[m],N}_{1}(0)\right)&=-\frac{d}{dt}\mathrm{Cov}\left(Q^{[n],N}_{j}(t)\ ;\
Q^{[m],N}_{1}(0)\right)\\\ &=-\frac{d}{dt}\mathrm{Cov}\left(Q^{[n],N}_{1}(0)\
;\ Q^{[m],N}_{N-j+2}(-t)\right)\\\ &=\mathrm{Cov}\left(Q^{[n],N}_{1}(0)\ ;\
J^{[m],N}_{N-j+3}(-t)-J^{[m],N}_{N-j+2}(-t)\right),\end{split}$ (2.22)

where we used that $s\mapsto Q^{[n],N}_{j}(t+s)Q^{[m],N}_{1}(s)$ is constant
in law under the Toda dynamic, and the periodicity of the matrix $L$ (1.4).
Denoting the difference operator $\partial_{j}f(j)=f(j+1)-f(j)$, equation
(2.22) shows that

$\partial_{j}\left(\mathrm{Cov}\left(J_{j}^{[n],N}(t)\ ;\
Q^{[m],N}_{1}(0)\right)-\mathrm{Cov}\left(Q^{[n],N}_{1}(0);\
J^{[m],N}_{N-j+2}(-t)\right)\right)=0$ (2.23)

Evaluating the previous expression at $t=0$, we deduce that there is some
constant $c_{N}$, independent of $j$, such that

$\mathrm{Cov}\left(J_{j}^{[n],N}\ ;\
Q^{[m],N}_{1}\right)-\mathrm{Cov}\left(Q^{[n],N}_{1};\
J^{[m],N}_{N-j+2}\right)=c_{N}\,.$ (2.24)

Furthermore, since both $Q^{[n],N}_{j}$, and $J^{[m],N}_{j}$ are local
quantities, in view of Theorem 1.11, we deduce that
$\lim_{N\to\infty}Nc_{N}=0$. So, evaluating the previous expression for $m=0$,
we deduce (2.21). Thus, in the large $N$ limit, we can recast (2.19) as

$\lim_{N\to\infty}\partial_{\alpha}{\mathbb{E}}\left[J^{[n],N}_{1}\right]=-\lim_{N\to\infty}\sum_{j=1}^{N}\mathrm{Cov}\left(J^{[0],N}_{1}\
;\
Q_{j}^{[n],N}\right)=\lim_{N\to\infty}\sum_{j=1}^{N}\mathrm{Cov}\left(Q^{[1],N}_{1}\
;\ Q_{j}^{[n],N}\right)\,.$ (2.25)

Moreover, in view of the periodicity properties ot the conserved fields and
(2.10)

$\lim_{N\to\infty}\partial_{\alpha}{\mathbb{E}}\left[J^{[n],N}_{1}\right]=\lim_{N\to\infty}\frac{1}{N}\mathrm{Cov}\left(Q^{[1],N}\
;\
Q^{[n],N}\right)=\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{T}(\alpha,P+it_{1}x+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}\,.$
(2.26)

Noticing that $\lim_{\alpha\to 0}{\mathbb{E}}\left[J^{[n],N}_{1}\right]=0$,
and that we can always uniformly bound
${\mathbb{E}}\left[J^{[n],N}_{1}\right]$ by a constant independent of $N$, the
previous equation implies that

$\lim_{N\to\infty}{\mathbb{E}}\left[J^{[n],N}_{1}\right]=\int_{0}^{\alpha}\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{T}(s,P+it_{1}x+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}{\rm
d}s.$ (2.27)

So, we conclude. ∎

##### The real $\beta$-ensemble in the high-temperature regime.

We consider the real $\beta$-ensemble introduced in Section 1.1 as the
probability measure on the real line

${\rm
d}\mathbf{P}^{\beta,V}_{N}(\lambda_{1},\ldots,\lambda_{N})=\frac{1}{\mathbf{Z}^{\beta,V}_{N}}\prod_{1\leq
i<j\leq
N}|\lambda_{i}-\lambda_{j}|^{\beta}e^{-\sum_{j=1}^{N}V(\lambda_{j})}{\rm
d}\boldsymbol{\lambda}\,,$ (2.28)

where $\beta>0$ and $V$ is a continuous function such that the partition
function

$\mathbf{Z}^{\beta,V}_{N}=\int_{{\mathbb{R}}^{N}}\prod_{i<j}|\lambda_{j}-\lambda_{i}|^{\beta}e^{-\sum_{j=1}^{N}V(\lambda_{j})}{\rm
d}\boldsymbol{\lambda}$

is finite. This is the case if $V$ grows to infinity fast enough, namely if
for some $\beta^{\prime}>\max(1,\beta)$,

$\liminf_{|x|\to\infty}\frac{V(x)}{N\beta^{\prime}\ln|x|}>1\,,$ (2.29)

see [6, equation (2.6.2)].

Dumitriu and Edelman showed in [19] that the $\beta$-ensemble admits a
tridiagonal representation

$H=\begin{pmatrix}a_{1}&b_{1}&&&0\\\ b_{1}&a_{2}&b_{2}&&\\\
&\ddots&\ddots&\ddots&\\\ &&\ddots&\ddots&b_{N-1}\\\
0&&&b_{N-1}&a_{N}\end{pmatrix}\,,$ (2.30)

where the entries of the matrix are distributed according to the following
probability measure

${\rm
d}\mu^{\beta,V}_{N,\text{beta}}=\frac{1}{Z^{\text{beta}}_{N}(\beta,V)}\prod_{j=1}^{N-1}b_{j}^{\beta(N-j)-1}\mathbbm{1}_{b_{j}\geq
0}\exp\left(-\mathrm{Tr}\left(V(H)\right)\right){\rm d}{\bf a}{\rm d}{\bf
b}\,.$ (2.31)

Then, the eigenvalues of $H$ are distributed according to ${\rm
d}\mathbf{P}^{\beta,V}_{N}$ (2.28). An important example is the case
$V(x)=x^{2}/2$ for which we recover the classical Gaussian $\beta$-ensemble
(also called Hermite $\beta$-ensemble), see [6, Section 2.5], and the
distribution $\mu^{\beta,V}_{N,\text{beta}}$ factorizes in the following way:
the entries of $H$ can be seen to be independent (modulo the symmetry of the
matrix), Gaussian $\mathcal{N}(0,1)$ on the diagonal, and the law of the off-
diagonal elements is given by renormalized chi variables

$b_{j}\sim\frac{1}{\sqrt{2}}\chi_{(N-j)\beta}\,,$

where the variable $X$ is $\chi_{\kappa}$-distributed if its law is given by
the density function

$f(x)=\frac{x^{\kappa-1}e^{-x^{2}/2}}{2^{\kappa/2-1}\Gamma(\kappa/2)}\,.$

We are interested in the so-called high-temperature regime for this model,
specifically, we are interested in the infinite size $N$ limit, in such a way
that $\beta=\frac{2\alpha}{N}$ for some $\alpha>0$. In this regime the
probability distribution (2.31) becomes $\mu_{N,\text{HT}}^{\alpha,V}$ (1.11),
which we recall:

${\rm
d}\mu_{N,\text{HT}}^{\alpha,V}=\frac{1}{Z^{\text{HT}}_{N}(\beta,V)}\prod_{j=1}^{N-1}b_{j}^{2\alpha\left(1-\frac{j}{N}\right)-1}\mathbbm{1}_{b_{j}\geq
0}\exp\left(-\mathrm{Tr}\left(V(H)\right)\right){\rm d}{\bf a}{\rm d}{\bf
b}\,.$ (2.32)

This regime has drawn a lot of attention from the random matrix and
statistical physics communities lately. Introducing the empirical measure by

${\rm d}\nu_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\lambda_{i}}\,,$

this model was first considered in [4], where the authors were able to compute
the limiting empirical measure for this model when $V(x)=x^{2}/2$. Recently,
Garcia-Zelada showed in [29] that under a general choice of $V$, the sequence
of empirical measures satisfies a large deviation principle with strictly
convex rate function, ensuring the convergence of $\nu_{N}$. Although the
limiting measure is not explicit, its density $\rho^{V}_{\alpha}$ satisfies
for almost every $x$ the nonlinear equation

$V(x)-2\alpha\int_{\mathbb{R}}\log|x-y|\rho^{V}_{\alpha}(y){\rm
d}y+\log\rho^{V}_{\alpha}(x)=\lambda^{V}_{\alpha}$

for some constant $\lambda^{V}_{\alpha}$, see [36, Lemma 3.2] for example.

The fluctuations of the eigenvalues in the bulk and at the edge of a
configuration were studied for example in [9, 56, 57, 60, 47]. These
fluctuations were shown to be described by Poisson statistics in this regime.
With the choice $V(x)=x^{2}/2$, Nakano and Trinh proved in [56] a Central
Limit theorem for this ensemble, namely y they proved that for smooth enough
$f:{\mathbb{R}}\to{\mathbb{R}}$, the random variables

$\sqrt{N}\left(\int_{\mathbb{R}}f{\rm
d}\nu_{N}-\int_{\mathbb{R}}f\rho^{V}_{\alpha}{\rm d}x\right)$

converge towards a centred Gaussian random variable with variance depending
both on $\alpha$ and $f$. In [23], the authors showed this central limit
theorem for general confining potentials and smooth enough, decaying at
infinity test functions. In this paper, we consider the case where $V$ is a
polynomial of even degree greater than $2$ as in Hypotheses 1.1. We deduce a
central limit theorem for polynomial test functions from section 4.

Indeed, we notice that the matrix $H$ (2.30) is a Toeplitz-like matrix (see
Appendix A), thus, in view of Lemma 1.3 and Corollary 2.2, we can apply
Theorem 1.5 to the real $\beta$-ensemble in the high-temperature regime, thus
we deduce that

######  Corollary 2.6 (CLT for real, high temperature $\beta$-ensemble).

Consider the matrix representation (2.30) of the real $\beta$-ensemble in the
high-temperature regime with potential $P$, polynomial of even degree with
positive leading order coefficient. Then, defining the Free energy
${\mathcal{F}}_{\mathrm{HT}}(\alpha,P)$ as

${\mathcal{F}}_{\mathrm{HT}}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{\mathrm{HT}}_{N}(\beta,P))\,,$
(2.33)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(H^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(H^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.34)

where

$\mathbb{E}\left[\mathrm{Tr}\left(H^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\mathrm{HT}}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{\mathrm{HT}}(\alpha,P+itx^{j})_{|_{t=0}}|\,.$
(2.35)

Thus, we obtained a central limit theorem for the real $\beta$-ensemble in the
high-temperature regime with polynomial potential.

Furthermore, all the hypotheses to apply the second part of our results are
satisfied, so we deduce the following identities

$\begin{split}&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{\mathrm{HT}}(\alpha,P+itx^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{T}(\alpha,P+itx^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{\mathrm{HT}}(\alpha,P+itx^{j})_{|_{t=0}})=\partial^{2}_{t}{\mathcal{F}}_{T}(\alpha,P+itx^{j})_{|_{t=0}}\,,\end{split}$
(2.36)

where ${\mathcal{F}}_{T}$ is the free energy of the Toda lattice (2.7). So we
can compute both the moments and their variances of the Toda lattice starting
from the one of the real $\beta$-ensemble at high temperature.

######  Remark 2.7.

Applying the second part of Theorem 1.5, we deduce the following equality
valid for the currents of the Toda lattice:

$\lim_{N\to\infty}{\mathbb{E}}\left[J^{[n],N}_{1}\right]=\int_{0}^{\alpha}\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{T}(s,P+it_{1}x+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}{\rm
d}s=\alpha\partial_{t_{1}}\partial_{t_{2}}{\mathcal{F}}_{\mathrm{HT}}(\alpha,P+it_{1}x+it_{2}x^{n})_{|_{t_{1},t_{2}=0}}.$
(2.37)

###  2.2 The exponential Toda lattice, and the Laguerre $\beta$-ensemble at
high-temperature

In this subsection, we focus on the Exponential Toda lattice and its relation
with the Laguerre $\beta$-ensemble in the high-temperature regime [28]. These
two systems were considered in [33]. In this paper, the authors considered the
classical Gibbs ensemble for the Exponential Toda lattice and were able to
compute the density of states for this model connecting it to the Laguerre
$\alpha$ ensemble [51], which is related to the classical $\beta$ one in the
high-temperature regime. Here we consider both the Generalized Gibbs ensemble
for the integrable lattice and the Laguerre $\beta$-ensemble at high-
temperature with polynomial potential, and we obtain a CLT for both systems,
furthermore, we connect the two in the same way as we did for the Toda lattice
and the real $\beta$-ensemble.

##### The exponential Toda lattice.

The exponential Toda lattice is the Hamiltonian system on $\mathbb{R}^{2N}$
described by the Hamiltonian

$H_{E}({\mathbf{p}},{\mathbf{q}})=\sum_{j=1}^{N}e^{-p_{j}}+\sum_{j=1}^{N}e^{q_{j}-q_{j+1}}\,,\quad
p_{j},q_{j}\in{\mathbb{R}}\,,$ (2.38)

with canonical Poisson bracket. Here, we consider periodic boundary conditions

$q_{j+N}=q_{j}+\Omega,\quad p_{j+N}=p_{j},\qquad\forall\,j\in{\mathbb{Z}},$
(2.39)

and $\Omega\geq 0$ is an arbitrary constant. The equations of motion are given
in Hamiltonian form as

$\begin{split}&\dot{q}_{j}=\frac{\partial H_{E}}{\partial
p_{j}}=-e^{-p_{j}}\,,\\\ &\dot{p}_{j}=-\frac{\partial H_{E}}{\partial
q_{j}}=e^{q_{j-1}-q_{j}}-e^{q_{j}-q_{j+1}}\,.\end{split}$ (2.40)

Following [33], we perform the non-canonical change of coordinates

$x_{j}=e^{-\frac{p_{j}}{2}},\quad
y_{j}=e^{\frac{q_{j}-q_{j+1}}{2}}=e^{-\frac{r_{j}}{2}},\quad
r_{j}=q_{j+1}-q_{j},\quad j=1,\dots,N,$ (2.41)

to obtain a Lax Pair for this system. Indeed, in these variables, the
Hamiltonian (2.38) transform into

$H_{E}({\mathbf{x}},{\mathbf{y}})=\sum_{j=1}^{N}(x_{j}^{2}+y_{j}^{2})\,,$
(2.42)

and the Hamilton’s equations (2.40) become

$\dot{x_{j}}=\frac{x_{j}}{2}\left(y_{j}^{2}-y_{j-1}^{2}\right),\quad\dot{y_{j}}=\frac{y_{j}}{2}\left(x_{j+1}^{2}-x_{j}^{2}\right),\quad
j=1,\dots,N,$ (2.43)

where $x_{N+1}=x_{1},\,y_{0}=y_{N}$.

Let us introduce the matrices $L,A\in\textrm{Mat}(N)$ as

$\displaystyle L=\begin{pmatrix}x_{1}^{2}+y_{N}^{2}&x_{1}y_{1}&&&x_{N}y_{N}\\\
x_{1}y_{1}&x_{2}^{2}+y_{1}^{2}&x_{2}y_{2}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&\ddots&x_{N-1}y_{N-1}\\\
x_{N}y_{N}&&&x_{N-1}y_{N-1}&x_{N}^{2}+y_{N-1}^{2}\end{pmatrix}\,,$ (2.44)
$\displaystyle A=\frac{1}{2}\begin{pmatrix}0&x_{1}y_{1}&&&-x_{N}y_{N}\\\
-x_{1}y_{1}&0&x_{2}y_{2}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&\ddots&x_{N-1}y_{N-1}\\\
x_{N}y_{N}&&&-x_{N-1}y_{N-1}&0\end{pmatrix}\,,$ (2.45)

The system of equations (2.43) admits the Lax representation

$\dot{L}=[A,L].$ (2.46)

Hence, the quantities $H_{m}=\mathrm{Tr}\left(L^{m}\right)$, $m=1,\ldots,N$
are constants of motion as well as the eigenvalues of $L$. For this integrable
model, we define the generalized Gibbs ensemble as

${\rm
d}\mu_{ET}=\frac{1}{Z^{H_{E}}_{N}(\alpha,P)}\prod_{j=1}^{N}x_{j}^{2\alpha-1}y_{j}^{2\alpha-1}\mathbbm{1}_{x_{j}\geq
0}\mathbbm{1}_{y_{j}\geq 0}e^{-\mathrm{Tr}\left(P(L)\right)}{\rm d}{\bf x}{\rm
d}{\bf y}\,,$ (2.47)

where $\alpha>0$, and $P$ is a real-valued polynomial with positive leading
coefficient. $Z_{N}^{H_{E}}(\alpha,P)$ is the normalization constant. We
notice that the matrix $L$ is of type 1-ii.

######  Remark 2.8.

The definition of our Gibbs ensemble is more general than the one given in
[33], indeed there the authors were considering just the classical Gibbs
ensemble for this model, so the case $P(x)=x/2$.

We notice that the structure of (2.47) resembles the one of $\mu_{N}^{(1)}$
(1.31), thus we want to apply Theorem 1.5. To do this, we have to identify the
functions $F,W$. First, as an application of Lemma 1.3 and Theorem 2.1, we
obtain the following corollary

######  Corollary 2.9.

Fix $m\in{\mathbb{N}}$, and consider the matrix $L$ (2.44). Then for $N$ big
enough, there exists some $k=k(m)\in{\mathbb{N}}$, such that
$\mathrm{Tr}\left(L^{m}\right)$ is a $k$-circular function, with seed $V$ and
weed $\widetilde{V}$. Furthermore, both $V,\widetilde{V}$ are bounded from
below away from $-\infty$.

As in the Toda lattice case, if we naively set
$F(x,y)=x^{2\alpha-1}y^{2\alpha-1}$, this would not fit in the hypotheses of
our theorem. As in the previous case, we have just to consider a slight
modification of the measure:

${\rm
d}\mu_{ET}=\frac{1}{Z^{H_{E}}_{N}(\alpha,P)}\prod_{j=1}^{N}x_{j}^{2\alpha-1}y_{j}^{2\alpha-1}\exp\left(-\varepsilon\frac{x_{j}^{2}+y_{j}^{2}}{2}\right)\mathbbm{1}_{x_{j}\geq
0}\mathbbm{1}_{y_{j}\geq
0}e^{-\mathrm{Tr}\left(P(L)\right)+\varepsilon\frac{x_{j}^{2}+y_{j}^{2}}{2}}{\rm
d}{\bf x}{\rm d}{\bf y}\,,$ (2.48)

for fixed $\varepsilon>0$, but small. In this way, defining
$F(x,y,\alpha)=x^{2\alpha-1}y^{2\alpha-1}\exp(-\varepsilon\frac{x^{2}+y^{2}}{2})$,
and $W({\bf x}_{1},{\bf y}_{1},{\bf x}_{2},{\bf y}_{2})=V({\bf x}_{1},{\bf
y}_{1},{\bf x}_{2},{\bf
y}_{2})-\frac{\varepsilon}{2}\sum_{j=1}^{2k}x_{j}^{2}+y_{j}^{2}$ we are in the
same hypotheses as Theorem 1.5, thus we deduce the following corollary

######  Corollary 2.10 (CLT for the Exponential Toda lattice).

Consider the Lax matrix L (2.44) of the Exponential Toda lattice distributed
according to the Generalized Gibbs Ensemble (2.47). Then, defining the Free
energy ${\mathcal{F}}_{HE}(\alpha,P)$ as

${\mathcal{F}}_{ET}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{H_{E}}_{N}(\alpha,P))\,,$
(2.49)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(L^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.50)

where

$\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{ET}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{ET}(\alpha,P+itx^{j})_{|_{t=0}}|$
(2.51)

##### The Laguerre $\beta$-ensemble in the high-temperature regime.

The Laguerre $\beta$-ensemble is a random matrix ensemble introduced by
Dumitriu and Edelman in [19]. It has the following matrix representation

$Q=\begin{pmatrix}x_{1}^{2}&x_{1}y_{1}&&&\\\
x_{1}y_{1}&x_{2}^{2}+y_{1}^{2}&x_{2}y_{2}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&\ddots&x_{N-1}y_{N-1}\\\
&&&x_{N-1}y_{N-1}&x_{N}^{2}+y_{N-1}^{2}\end{pmatrix}\,,$ (2.52)

where the entries of $Q$ are distributed according to

${\rm
d}\mu_{L}=\frac{1}{Z^{L}_{N}(\beta)}\prod_{j=1}^{N}x_{j}^{\beta(M-j+1)-1}\mathbbm{1}_{x_{j}\geq
0}\prod_{j=1}^{N-1}y^{\beta(N-j)-1}\mathbbm{1}_{y_{j}\geq
0}\exp\left(-\mathrm{Tr}\left(P(Q)\right)\right){\rm d}{\bf x}{\rm d}{\bf
y}\,,$ (2.53)

here we consider the case $M=N$; $P$ can be any continuous function such that
the partition function is well-defined. For our purpose we consider $P$ to be
a polynomial of degree greater than one with positive leading coefficient, as
in Hypotheses 1.1.

The remarkable property of this ensemble is that it is possible to explicitly
compute the joint eigenvalue density as

${\rm
d}\mathbb{P}_{L}=\frac{1}{\mathfrak{Z}^{L}_{N}(\beta,P)}\prod_{j=1}^{N}\lambda_{j}^{\frac{\beta}{2}\left(M-N+1\right)-1}\mathbbm{1}_{\lambda_{j}\geq
0}\prod_{j<i}|\lambda_{j}-\lambda_{i}|^{\beta}e^{-\sum_{j=1}^{N}P(\lambda_{j})}{\rm
d}\boldsymbol{\lambda}\,.$ (2.54)

We are interested in the so-called high-temperature limit, i.e. when
$\beta=\frac{2\alpha}{N}$, $\alpha\in{\mathbb{R}}_{+}$, which was considered
in [5], where the authors were able to compute the density of states for the
particular case $P(x)=x/2$.

In this regime, the density (2.53) takes the form

${\rm
d}\mu_{L}=\frac{1}{Z^{L}_{N}(\alpha,P)}\prod_{j=1}^{N}x_{j}^{2\alpha\left(1-\frac{j+1}{N}\right)-1}\mathbbm{1}_{x_{j}\geq
0}\prod_{j=1}^{N-1}y_{j}^{2\alpha\left(1-\frac{j}{N}\right)-1}\mathbbm{1}_{y_{j}\geq
0}\exp\left(-\mathrm{Tr}\left(P(Q)\right)\right){\rm d}{\bf x}{\rm d}{\bf
y}\,.$ (2.55)

We notice that the structure of this density resembles the one of
$\mu^{(2)}_{N}$ (1.32). Furthermore, the matrix $Q$ is Toeplitz-like and
tridiagonal, thus applying Lemma 1.3 and Theorem 2.1, we deduce the following
corollary

######  Corollary 2.11 (CLT for Laguerre $\beta$-ensemble).

Consider the matrix representation (2.52) of the Laguerre $\beta$-ensemble in
the high-temperature regime, and let $P$ be a real polynomial of degree at
least $1$ with positive leading coefficient. Then, defining the Free energy
${\mathcal{F}}_{L}(\alpha,P)$ as

${\mathcal{F}}_{L}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{L}_{N}(\alpha,P))\,,$
(2.56)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(Q^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(Q^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.57)

where

$\mathbb{E}\left[\mathrm{Tr}\left(Q^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{L}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{L}(\alpha,P+itx^{j})_{|_{t=0}}|$
(2.58)

Which is the perfect analogue of the result for the Exponential Toda lattice.
Furthermore, we are ready to apply the second part of our result, indeed we
can deduce the following identities

$\begin{split}&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{L}(\alpha,P+itx^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{ET}(\alpha,P+itx^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{L}(\alpha,P+itx^{j})_{|_{t=0}})=\partial^{2}_{t}{\mathcal{F}}_{ET}(\alpha,P+itx^{j})_{|_{t=0}}\,,\end{split}$
(2.59)

thus, we can compute all the quantities involved in the previous theorems from
the Free Energy of the Laguerre ensemble.

######  Remark 2.12.

We notice that in the Laguerre $\beta$-ensemble one can consider a more
general regime $M=\gamma N$ with $\gamma\in(0,1]$. It is possible to obtain an
analogous result for this situation, but one has to consider a slightly
different $F$ and add an extra parameter depending on $\alpha,\gamma$

###  2.3 The Volterra lattice, and the antisymmetric $\beta$-ensemble at high-
temperature

In this subsection, we focus on the Volterra lattice and its relation with the
Antisymmetric $\beta$-ensemble [21] in the high-temperature regime [28]. These
two systems were considered in [33]. In this paper, the authors considered the
classical Gibbs ensemble for the Volterra lattice and were able to compute the
density of states for this model connecting it to the Antisymmetric $\alpha$
ensemble [28], which is related to the classical $\beta$ one introduced by
Dumitriu and Forrester [21].

##### The Volterra Lattice.

The Volterra lattice (or _discrete KdV equation_) is the following systems of
$N$ coupled ODEs

$\dot{a_{j}}=a_{j}\left(a_{j+1}-a_{j-1}\right),\qquad j=1,\dots,N,$ (2.60)

here $a_{j}\in{\mathbb{R}}_{+}$ for $j=1,\ldots,N$, and we consider periodic
boundary conditions $a_{j}=a_{j+N}$ for all $j\in{\mathbb{Z}}$. Volterra
introduced it to study evolution of populations in a hierarchical system of
competing species. This system was considered by Kac and van Moerbeke in [43],
who solved it explicitly using a discrete version of the inverse scattering
transform introduced by Flaschka [25].

Introducing on the phase space ${\mathbb{R}}^{N}_{+}$ the following Poisson
bracket

$\\{a_{j},a_{i}\\}_{\textrm{Volt}}=a_{j}a_{i}(\delta_{i,j+1}-\delta_{i,j-1})\,,$
(2.61)

and defining the Hamiltonian $H_{1}=\sum_{j=1}^{N}a_{j}\,$, we can rewrite the
equations of motion (2.60) in Hamiltonian form as

$\dot{a}_{j}=\\{a_{j},H_{1}\\}_{\textrm{Volt}}\,.$ (2.62)

An elementary constant of motion for the system is
$H_{0}=\prod_{j=1}^{N}a_{j}$ which is independent of $H_{1}$.

The Volterra lattice is a completely integrable system, and it admits several
equivalents _Lax representations_ , see e.g. [43, 54, 33]. We use the one
presented in [33]. Specifically, we introduce the matrices
$L,A\in\textrm{Mat}({\mathbb{R}},N)$ as

$\displaystyle L=\begin{pmatrix}0&\sqrt{a_{1}}&&&-\sqrt{a_{N}}\\\
-\sqrt{a_{1}}&0&\sqrt{a_{2}}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&\ddots&\sqrt{a_{N-1}}\\\
\sqrt{a_{N}}&&&-\sqrt{a_{N-1}}&0\end{pmatrix}\,,$ (2.63) $\displaystyle
A=\frac{1}{2}\sum_{j=1}^{N}\sqrt{a_{j}a_{j+1}}(E_{j,j+2}-E_{j+2,j})\,,$ (2.64)

where $E_{r,s}$ is defined as
$\left(E_{r,s}\right)_{ij}=\delta^{i}_{r}\delta^{j}_{s}$ and
$E_{j+N,i}=E_{j,i+N}=E_{j,i}$. Then, it follows that the equations of motion
(2.62) are equivalent to

$\dot{L}=[L;A]\,.$ (2.65)

In view of this Lax pair, we know that $\mathrm{Tr}\left(L^{k}\right)$ are
constant of motion for the model.

Following [33], we introduce the Generalized Gibbs Ensemble of the Volterra
lattice (2.60) as

${\rm d}\mu_{\textrm{Volt}}({\bf
a})=\frac{e^{-\mathrm{Tr}\left(P(L)\right)}\prod_{j=1}^{N}a_{j}^{\alpha-1}\mathbbm{1}_{a_{j}>0}{\rm
d}{\bf a}}{Z_{N}^{\textrm{Volt}}(\alpha,P)}\,,$ (2.66)

where $\alpha>0$, $P$ is a polynomial as in Hypotheses 1.1 of the form
$P(x)=(-1)^{j}x^{2j}+\textrm{l.o.t}$, where l.o.t stands for lower order
terms. Moreover, we notice that, given the antisymmetric nature of $L$,
$\mathrm{Tr}\left(L^{2j+1}\right)=0$. For this reason, we perform the change
of coordinates $\sqrt{a_{j}}=x_{j}$, where we take just the positive root, so
the previous measure read

${\rm d}\mu_{\textrm{Volt}}({\bf
x})=\frac{e^{-\mathrm{Tr}\left(P(L)\right)}\prod_{j=1}^{N}x_{j}^{2\alpha-1}\mathbbm{1}_{x_{j}>0}{\rm
d}{\bf x}}{Z_{N}^{\textrm{Volt}}(\alpha,P)}\,.$ (2.67)

This Generalized Gibbs ensemble resembles the structure of $\mu_{N}^{(1)}$
(1.31), we have just to identify $F,W$. We notice that it is possible to
generalize Theorem 2.1 also for the antisymmetric situation, moreover, the
matrix $L$ is circular-like. Thus, by applying Lemma 1.3 and Theorem 2.1, we
deduce the following Corollary:

######  Corollary 2.13.

Fix $m\in{\mathbb{N}}$, and consider the matrix $L$ (2.63). Then for $N$ big
enough, there exists some $k=k(m)\in{\mathbb{N}}$, such that
$\mathrm{Tr}\left(L^{m}\right)$ is a $k$-circular function, with seed $V$ and
weed $\widetilde{V}$. Furthermore, both $V,\widetilde{V}$ are bounded from
below away from $-\infty$.

Thus, following the same kind of reasoning as in the Toda lattice, section
2.1, and the Exponential Toda lattice, section 2.2, we deduce the following:

######  Corollary 2.14 (CLT for Volterra lattice).

Consider the Lax matrix L (2.63) of the Volterra lattice distributed according
to the Generalized Gibbs Ensemble (2.66). Then, defining the Free energy
${\mathcal{F}}_{\textrm{Volt}}(\alpha,P)$ as

${\mathcal{F}}_{\textrm{Volt}}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{\textrm{Volt}}_{N}(\alpha,P))\,,$
(2.68)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(L^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.69)

where

$\mathbb{E}\left[\mathrm{Tr}\left(L^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\textrm{Volt}}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{\textrm{Volt}}(\alpha,P+tx^{j})_{|_{t=0}}|\,.$
(2.70)

##### The Antisymmetric $\beta$-ensemble in the high-temperature regime

The Antisymmetric $\beta$-ensemble is a random matrix ensemble introduced by
Dumitriu and Forrester in [21]; it has the following matrix representation

$Q=\begin{pmatrix}0&x_{1}&&&\\\ -x_{1}&0&x_{2}&&\\\ &\ddots&\ddots&\ddots&\\\
&&\ddots&\ddots&x_{N-1}\\\ &&&-x_{N-1}&0\end{pmatrix}\,,$ (2.71)

and the entries of the matrix $Q$ are distributed according to

${\rm
d}\mu_{AG}=\frac{1}{Z^{AG}_{N}(\beta,P)}\prod_{j=1}^{N-1}x_{j}^{\beta(N-j)-1}\mathbbm{1}_{x_{j}\geq
0}\exp(-\mathrm{Tr}\left(P(Q)\right)){\rm d}{\bf x}\,,$ (2.72)

here $P$ can be any function that makes (2.72) normalizable. For our purpose,
we consider as in Hypotheses 1.1 $P$ polynomial of the form
$P(x)=(-1)^{j}x^{2j}+\textrm{l.o.t}$.

As in the previous cases, we are interested in the high-temperature regime for
this model, so we set $\beta=\frac{2\alpha}{N}$, and we rewrite the previous
density as

${\rm
d}\mu_{AG}=\frac{1}{Z^{AG}_{N}(\alpha,P)}\prod_{j=1}^{N-1}x_{j}^{2\alpha\left(1-\frac{j}{N}\right)-1}\mathbbm{1}_{x_{j}\geq
0}\exp(-\mathrm{Tr}\left(P(Q)\right)){\rm d}{\bf x}\,.$ (2.73)

This regime was introduced in [33], where the author computed the density of
states for this model in the case $P(x)=-x^{2}/2$. The structure of this last
density (2.73) resembles the one of $\mu^{(2)}_{N}$(1.32), indeed proceeding
as in the case of the Volterra lattice, we deduce the following corollary

######  Corollary 2.15 (CLT for Antisymmetric $\beta$-ensemble).

Consider the matrix representation (2.71) of the Antisymmetric
$\beta$-ensemble in the high-temperature regime, endowed with the probability
distribution ${\rm d}\mu_{AG}$ (2.73), and let $P(x)$ be a polynomial of the
form $P(x)=(-1)^{\ell}x^{2\ell}+\textrm{l.o.t.}$. Then, defining the Free
energy ${\mathcal{F}}_{AG}(\alpha,P)$ as

${\mathcal{F}}_{AG}(\alpha,P)=-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{AG}_{N}(\alpha,P))\,,$
(2.74)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left(Q^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left(Q^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,.$
(2.75)

where

$\mathbb{E}\left[\mathrm{Tr}\left(Q^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}}|\,.$
(2.76)

Which is the perfect analogue of the result for the Volterra lattice.

######  Remark 2.16.

In view of Theorem 1.5, we deduce the following identities

$\begin{split}&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}})=\partial^{2}_{t}{\mathcal{F}}_{AG}(\alpha,P+itx^{j})_{|_{t=0}}\end{split}$
(2.77)

###  2.4 The defocusing Ablowitz-Ladik lattice, and the Circular
$\beta$-ensemble at high-temperature

In this subsection, we focus on the defocusing Ablowitz-Ladik lattice, and its
relation to the Circular $\beta$-ensemble at high-temperature [38, 35, 67].
This relation was highlighted by one of the present authors and T. Grava [35],
and independently by H. Spohn [67]. In these papers, the authors were able to
characterize the density of states of the Ablowitz-Ladik lattice in terms of
the one of the Circular $\beta$-ensemble in the high-temperature regime.
Moreover, in [35] the authors were able to compute explicitly the density of
states in the case of linear potential in terms of the solution of the Double
Confluent Heun Equation [17] highlighting a connection with the Painlevé
equations [49, 27]. In [53], the two present authors obtained a large
deviations principles for the empirical spectral measure for any continuous
and bounded potential.

##### The defocusing Ablowitz-Ladik lattice.

The defocusing Ablowitz-Ladik (dAL) lattice is defined by the following system
of nonlinear equations

$i\dot{a}_{j}=-(a_{j+1}+a_{j-1}-2a_{j})+|a_{j}|^{2}(a_{j-1}+a_{j+1})\,,$
(2.78)

where $a_{j}(t)\in{\mathbb{C}}$. We assume $N$-periodic boundary conditions
$a_{j+N}=a_{j}$, for all $j\in{\mathbb{Z}}$. The dAL lattice was introduced by
Ablowitz and Ladik [2, 3] as the spatial integrable discretization of the
defocusing cubic nonlinear Schrödinger Equation for the complex function
$\psi(x,t)$, $x\in S^{1}$ and $t\in{\mathbb{R}}$:

$i\partial_{t}\psi(x,t)=-\partial^{2}_{x}\psi(x,t)+2\lvert\psi(x,t)\rvert^{2}\psi(x,t).$
(2.79)

As for the others dynamical systems that we considered, the dAL is an
integrable system. Its integrability was proved by Ablowitz and Ladik by
discretizing the $2\times 2$ Zakharov-Shabat Lax pair [1, 2] of the cubic
nonlinear Schrödinger equation. Furthermore, Nenciu and Simon [59, 62]
constructed a new Lax pair for this lattice. Following their construction we
double the size of the chain according to the periodic boundary condition,
thus we consider a chain of $2N$ particles $a_{1},\ldots,a_{2N}$ such that
$a_{j}=a_{j+N}$ for $j=1,\ldots,N$. Define the $2\times 2$ unitary matrix
$\Xi_{j}$

$\Xi_{j}=\begin{pmatrix}{\overline{a}}_{j}&\rho_{j}\\\
\rho_{j}&-a_{j}\end{pmatrix}\,,\quad
j=1,\dots,2N\,,\quad\rho_{j}=\sqrt{1-|a_{j}|^{2}}$ (2.80)

and the $2N\times 2N$ matrices

${\mathcal{M}}=\begin{pmatrix}-a_{2N}&&&&&\rho_{2N}\\\ &\Xi_{2}\\\
&&\Xi_{4}\\\ &&&\ddots\\\ &&&&\Xi_{2N-2}\\\
\rho_{2N}&&&&&{\overline{a}}_{2N}\end{pmatrix}\,,\qquad{\mathcal{L}}=\begin{pmatrix}\Xi_{1}\\\
&\Xi_{3}\\\ &&\ddots\\\ &&&\Xi_{2N-1}\end{pmatrix}\,.$ (2.81)

Now let us define the unitary Lax matrix

${\mathcal{E}}={\mathcal{L}}{\mathcal{M}}\,,$ (2.82)

that has the structure of a $5$-band periodic diagonal matrix. The matrix
${\mathcal{E}}$ is a periodic CMV matrix [14]. The equations of motion (2.78)
are equivalent to the following Lax equation for the matrix ${\mathcal{E}}$:

$\dot{\mathcal{E}}=i\left[{\mathcal{E}},{\mathcal{E}}^{+}+({\mathcal{E}}^{+})^{\dagger}\right]\,,$
(2.83)

where † stands for hermitian conjugate and

${\mathcal{E}}^{+}_{j,k}=\begin{cases}\frac{1}{2}{\mathcal{E}}_{j,j}\quad
j=k\\\ {\mathcal{E}}_{j,k}\quad k=j+1\,{\,{\rm
mod}\,}\,2N\,\mbox{or}\,k=j+2\,{\,{\rm mod}\,}\,2N\\\
0\quad\mbox{otherwise}.\end{cases}$ (2.84)

The matrix ${\mathcal{E}}$ is a circular-like matrix, this can be deduced from
the fact that both ${\mathcal{L}},{\mathcal{M}}$ are circular-like (2.81), and
the entries are continuous functions on a compact set, so by applying Lemma
1.3, we deduce the following:

######  Corollary 2.17.

Fix $m\in{\mathbb{N}}$, and consider the matrix ${\mathcal{E}}$ (2.82). Then
for $N$ big enough, there exists some $k=k(m)\in{\mathbb{N}}$, such that
$\mathrm{Tr}\left({\mathcal{E}}^{m}\right)$ is a $k$-circular function, with
seed $V$ and weed $\widetilde{V}$.

Following [67, 35, 53], we notice that the quantity
$K_{0}=\prod_{j=1}^{N}(1-|a_{j}|^{2})$ is conserved, so this means that if
$|a_{j}(0)|<1$ for all $j=1,\dots,N$ then $|a_{j}(t)|<1$ for all $j=1,\dots,N$
for all $t\in{\mathbb{R}}$, so we can consider ${\mathbb{D}}^{N}$ as our phase
space, here ${\mathbb{D}}=\\{z\in{\mathbb{C}}\,|\,|z|<1$}. On this phase
space, we introduce the Generalized Gibbs ensemble for the defocusing AL
lattice as

${\rm
d}\mu_{\textrm{dAL}}=\frac{\prod_{j=1}^{N}\left(1-|a_{j}|^{2}\right)^{\alpha-1}\mathbbm{1}_{a_{j}\in{\mathbb{D}}}\exp\left(-\mbox{Tr}\left(\Re
P\left({\mathcal{E}}\right)\right)\right){\rm d}^{2}{\bf
a}}{Z_{N}^{\textrm{dAL}}(\alpha,\Re P)}\,,$ (2.85)

where $P\in{\mathbb{C}}[X]$ is a polynomial. In view of Corollary 2.17, we are
in the hypotheses of Theorem 1.5, thus we deduce the following:

######  Corollary 2.18 (CLT for defocusing Ablowitz–Ladik lattice).

Consider the Lax matrix ${\mathcal{E}}$ (2.82) of the defocusing
Ablowitz–Ladik lattice distributed according to the Generalized Gibbs Ensemble
(2.85). Then, defining the Free energy
${\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re P)$ as

${\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{\ln(Z^{\textrm{dAL}}_{N}(\alpha,\Re P))}{2N}\,,$
(2.86)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limits

$\begin{split}&\lim_{N\to\infty}\frac{\mathrm{Tr}\left(\Re{\mathcal{E}}^{j}\right)-\mathbb{E}\left[\Re\mathrm{Tr}\left({\mathcal{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma_{\text{real}}^{2})\,,\\\
&\lim_{N\to\infty}\frac{\mathrm{Tr}\left(\Im{\mathcal{E}}^{j}\right)-\mathbb{E}\left[\Im\mathrm{Tr}\left({\mathcal{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma_{\text{imag}}^{2})\,,\end{split}$
(2.87)

where

$\begin{split}&\mathbb{E}\left[\mathrm{Tr}\left(\Re{\mathcal{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Re
x^{j})_{|_{t=0}}\,,\quad\sigma_{\text{real}}^{2}=|\partial^{2}_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Re x^{j})_{|_{t=0}}|\,.\\\
&\mathbb{E}\left[\mathrm{Tr}\left(\Im{\mathcal{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Im
x^{j})_{|_{t=0}}\,,\quad\sigma_{\text{imag}}^{2}=|\partial^{2}_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Im x^{j})_{|_{t=0}}|\,.\end{split}$ (2.88)

##### The circular $\beta$-ensemble at high-temperature.

The circular $\beta$-ensemble was introduced by Killip and Nenciu in [45]; as
the other $\beta$-ensembles that we considered, it possesses a matrix
representation. Consider the two block diagonal matrices

${\mathfrak{M}}=\mbox{diag}\left(\Xi_{1},\Xi_{3},\Xi_{5}\ldots,\right)\quad\mbox{
and
}\quad{\mathfrak{L}}=\mbox{diag}\left(\Xi_{0},\Xi_{2},\Xi_{4},\ldots\right)\,,$
(2.89)

where the blocks $\Xi_{j}$, $j=1,\dots,N-1$ are defined in (2.80), while
$\Xi_{0}=(1)$ and $\Xi_{N}=({\overline{\alpha}}_{N})$ are $1\times 1$
matrices. Then, we define ${\mathfrak{E}}$ as follows

${\mathfrak{E}}={\mathfrak{L}}{\mathfrak{M}}.$ (2.90)

The entries of this matrix are distributed according to

${\rm
d}\mu_{C}=\frac{\prod_{j=1}^{N-1}\left(1-|a_{j}|^{2}\right)^{\frac{\beta}{2}(N-j)-1}\mathbbm{1}_{a_{j}\in{\mathbb{D}}}\exp\left(-\mbox{Tr}\left(\Re
P\left({\mathfrak{E}}\right)\right)\right)\prod_{j=1}^{N-1}{\rm
d}^{2}a_{j}\frac{{\rm d}a_{N}}{ia_{N}}}{Z_{N}^{C}(\beta,\Re P)}\,.$ (2.91)

As for the other $\beta$-ensembles, one can explicitly compute the joint
eigenvalue density for this ensemble as

${\rm d}\mathbb{P}_{C}=\frac{1}{\mathfrak{Z}^{C}_{N}(\beta,\Re
P)}\prod_{j<\ell}|e^{i\theta_{j}}-e^{i\theta_{\ell}}|^{\beta}\mathbbm{1}_{\theta_{j}\in\mathbb{T}}e^{-\sum_{j=1}^{N}\Re
P(e^{i\theta_{j}})}{\rm d}\boldsymbol{\theta}\,,$ (2.92)

here $\mathbb{T}=[-\pi,\pi)$, $e^{i\theta_{j}}$ are the eigenvalues of
${\mathfrak{E}}$, and $P$ can be any continuous function. We restrict our
attention to the class of polynomials $P\in{\mathbb{C}}[X]$.

We are interested in the high-temperature limit for this ensemble [35, 67], so
we set $\beta=\frac{2\alpha}{N}$, obtaining

${\rm
d}\mu_{C}=\frac{\prod_{j=1}^{N-1}\left(1-|a_{j}|^{2}\right)^{\alpha\left(1-\frac{j}{N}\right)-1}\mathbbm{1}_{a_{j}\in{\mathbb{D}}}\exp\left(-\mbox{Tr}\left(\Re
P\left({\mathfrak{E}}\right)\right)\right)\prod_{j=1}^{N-1}{\rm
d}^{2}a_{j}\frac{{\rm d}a_{N}}{ia_{N}}}{Z_{N}^{C}(\alpha,P)}\,.$ (2.93)

So, in view of Corollary 2.17, we are in the hypotheses of Theorem 1.5, so we
deduce the following

######  Corollary 2.19 (CLT for Circular $\beta$-ensemble).

Consider the matrix representation (2.90) of the Circular $\beta$-ensemble in
the high-temperature regime, endowed with the probability distribution ${\rm
d}\mu_{C}$ (2.93), and let $P\in{\mathbb{C}}[X]$ be a polynomial. Then,
defining the Free energy ${\mathcal{F}}_{C}(\alpha,\Re P)$ as

${\mathcal{F}}_{C}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{\ln(Z^{C}_{N}(\alpha,\Re P))}{2N}\,,$ (2.94)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\begin{split}&\lim_{N\to\infty}\frac{\mathrm{Tr}\left(\Re{\mathfrak{E}}^{j}\right)-\mathbb{E}\left[\Re\mathrm{Tr}\left({\mathfrak{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma_{\text{real}}^{2})\,,\\\
&\lim_{N\to\infty}\frac{\mathrm{Tr}\left(\Im{\mathfrak{E}}^{j}\right)-\mathbb{E}\left[\Im\mathrm{Tr}\left({\mathfrak{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma_{\text{imag}}^{2})\,,\end{split}$
(2.95)

where

$\begin{split}&\mathbb{E}\left[\mathrm{Tr}\left(\Re{\mathfrak{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{C}(\alpha,\Re
P+it\Re
x^{j})_{|_{t=0}}\,,\quad\sigma_{\text{real}}^{2}=|\partial^{2}_{t}{\mathcal{F}}_{C}(\alpha,\Re
P+it\Re x^{j})_{|_{t=0}}|\,.\\\
&\mathbb{E}\left[\mathrm{Tr}\left(\Im{\mathcal{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{C}(\alpha,\Re
P+it\Im
x^{j})_{|_{t=0}}\,,\quad\sigma_{\text{imag}}^{2}=|\partial^{2}_{t}{\mathcal{F}}_{C}(\alpha,\Re
P+it\Im x^{j})_{|_{t=0}}|\,.\end{split}$ (2.96)

######  Remark 2.20.

  * We notice that

  * •

Hardy and Lambert in [38] already proved a CLT theorem for the Circular
$\beta$-ensemble in the high-temperature regime for a wider class of functions
and potentials than we can consider with our result. Nevertheless, we
highlight the fact that in our case we can explicitly compute the means, and
the variances in terms of the Free energy.

  * •

The following identities hold in view of the last part of Theorem 1.5

$\begin{split}&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{C}(\alpha,\Re
P+it\Re x^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Re x^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{C}(\alpha,\Re P+it\Re
x^{j})_{|_{t=0}})=\partial^{2}_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Re x^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{C}(\alpha,\Re P+it\Im
x^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re P+it\Im
x^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{C}(\alpha,\Re P+it\Im
x^{j})_{|_{t=0}})=\partial^{2}_{t}{\mathcal{F}}_{\textrm{dAL}}(\alpha,\Re
P+it\Im x^{j})_{|_{t=0}}\,.\end{split}$ (2.97)

This relation was already proved in [35] with the same kind of argument that
we followed.

###  2.5 The defocusing Schur flow, and the Jacobi $\beta$-ensemble at high-
temperature

In this subsection, we focus on the defocusing Schur flow [31], and its
relation to the Jacobi $\beta$-ensemble at high-temperature [28]. This
relation was first noticed in [67], and then the two present authors obtained
a large deviations principles for the empirical spectral measure for the
defocusing Schur flow, and they were able to link it to the one of the Jacobi
$\beta$-ensemble in the high-temperature regime [53].

##### The defocusing Schur flow.

The defocusing Schur flow is the system of ODEs [31]

$\dot{a_{j}}=\rho_{j}^{2}(a_{j+1}-a_{j-1})\,,\quad\rho_{j}=\sqrt{1-|a_{j}|^{2}}\,,$
(2.98)

and, as before, we consider periodic boundary conditions, namely
$a_{j}=a_{j+N}$ for all $j\in{\mathbb{Z}}$.

We notice that, if one chooses an initial data such that
$a_{j}(0)\in{\mathbb{R}}$ for all $j=1,\ldots,N$, then
$a_{j}(t)\in{\mathbb{R}}$ for all times. Moreover, it is straightforward to
verify that $K_{0}=\prod_{j=1}^{N}\left(1-|a_{j}|^{2}\right)$ is conserved
along the Schur flow. This implies that we can choose as phase space for the
Schur flow the $N$-cube $\mathbb{I}^{N}$, where ${\mathbb{I}}:=(-1,1)$.
Furthermore, it was shown in [31], that the Schur flow has the same Lax matrix
as the focusing Ablowitz–Laddik lattice.

Following [67, 33], on ${\mathbb{I}}^{N}$ we define the finite volume limit
GGE as

${\rm d}\mu_{\textrm{dS}}({\bf
a})=\frac{\prod_{j=1}^{N}\left(1-a_{j}^{2}\right)^{\alpha-1}\mathbbm{1}_{a_{j}\in{\mathbb{I}}}\exp\left(-\mbox{Tr}\left(P\left({\mathcal{E}}\right)\right)\right){\rm
d}{\bf a}}{Z_{N}^{\textrm{dS}}(\alpha,P)}\,,$ (2.99)

where $P\,:\,{\mathbb{R}}\to{\mathbb{R}}$ is a polynomial. Thanks to Corollary
2.17, we can apply Theorem 1.5 obtaining a CLT theorem for the defocusing
Schur flow

######  Corollary 2.21 (CLT for defocusing Schur flow).

Consider the Lax matrix ${\mathcal{E}}$ (2.82) of the defocusing Schur flow
distributed according to the Generalized Gibbs Ensemble (2.99). Then, defining
the Free energy ${\mathcal{F}}_{\textrm{dS}}(\alpha,P)$ as

${\mathcal{F}}_{\textrm{dS}}(\alpha,P)=-\lim_{N\to\infty}\frac{\ln(Z^{\textrm{dS}}_{N}(\alpha,P))}{2N}\,,$
(2.100)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left({\mathcal{E}}^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left({\mathcal{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.101)

where

$\mathbb{E}\left[\mathrm{Tr}\left({\mathcal{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\textrm{dS}}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{\textrm{dS}}(\alpha,P+itx^{j})_{|_{t=0}}|\,.$
(2.102)

##### The Jacobi $\beta$-ensemble in the high-temperature regime.

The Jacobi $\beta$-ensemble is a random matrix ensemble introduced by Killip
and Nenciu in [45]. It has two slightly different matrix representations. The
first one is the same as the Circular $\beta$-ensemble (2.90), but the
distribution of the entries of the matrix is

${\rm d}\mu_{J}({\bf
a})=\frac{\prod_{j=1}^{2N-1}\left(1-a_{j}^{2}\right)^{\beta(2N-j)/4-1}\prod_{j=1}^{2N-1}(1-a_{j})^{{\mathtt{a}}+1-\beta/4}(1+(-1)^{j}a_{j})^{{\mathtt{b}}+1-\beta/4}\mathbbm{1}_{a_{j}\in{\mathbb{I}}}\exp\left(-\mbox{Tr}\left(P\left({\mathfrak{E}}\right)\right)\right){\rm
d}{\bf a}}{Z_{N}^{J}(\beta,P)}\,,$ (2.103)

where ${\mathtt{a}},{\mathtt{b}}>-1$, $P\in{\mathbb{R}}[X]$ is a polynomial.
We notice that we are considering an even number of random variables, and
$a_{j}\in{\mathbb{R}}$; for these reasons, all the eigenvalues of
${\mathfrak{E}}$ come in pairs, meaning that if $e^{i\theta}$ is an
eigenvalue, then $e^{-i\theta}$ is another one. Exploiting this symmetry,
Killip and Nenciu found another matrix representation for this ensemble

$J=\begin{pmatrix}c_{1}&b_{1}&&&\\\ b_{1}&c_{2}&b_{2}&&\\\
&\ddots&\ddots&\ddots&\\\ &&\ddots&\ddots&b_{N-1}\\\
&&&b_{N-1}&c_{N}\end{pmatrix}\,,\quad\begin{cases}b_{j}=\left((1-a_{2j-2})(1-a_{2j-1}^{2})(1+a_{2j})\right)^{1/2}\\\
c_{j}=(1-a_{2j-2})a_{2j-1}-(1+a_{2j-2})a_{2j-3}\end{cases}\,,$ (2.104)

where $a_{0}=a_{2N}=-1$, and the eigenvalues $\\{\lambda_{j}\\}_{j=1}^{N}$ of
$J$ are related to the one of ${\mathfrak{E}}$ as
$\lambda_{j}=\cos(\theta_{j})$.

Also, in this case, it is possible to compute explicitly the joint eigenvalue
density for this model as

${\rm
d}\mathbb{P}_{J}=\frac{1}{\mathfrak{Z}^{J}_{N}(\beta,P)}\prod_{j<\ell}|\cos(\theta_{j})-\cos(\theta_{\ell})|^{\beta}\mathbbm{1}_{\theta_{j}\in\mathbb{T}}e^{-2\sum_{j=1}^{N}P(\cos(\theta_{j}))}{\rm
d}\boldsymbol{\theta}\,.$ (2.105)

As in the previous cases, we are interested in the high-temperature regime for
this ensemble, so we wet $\beta=\frac{2\alpha}{N}$, thus the measure (2.103)
read

${\rm d}\mu_{J}({\bf
a})=\frac{\prod_{j=1}^{2N-1}\left(1-a_{j}^{2}\right)^{\alpha\left(1-\frac{j}{2N}\right)}\prod_{j=1}^{2N-1}(1-a_{j})^{{\mathtt{a}}+1-\frac{\alpha}{2N}}(1+(-1)^{j}a_{j})^{{\mathtt{b}}+1-\frac{\alpha}{2N}}\mathbbm{1}_{a_{j}\in{\mathbb{I}}}\exp\left(-\mbox{Tr}\left(P\left({\mathfrak{E}}\right)\right)\right){\rm
d}{\bf a}}{Z_{N}^{J}(\beta,P)}\,.$ (2.106)

This regime was considered in [69] and in the recent paper [55], where the
authors established a CLT for polynomial test functions in the absence of
external potential ($P=0$ in (2.106) ) by considering orthogonal polynomials,
obtaining an explicit recurrence relation for the limiting variance.

Again, thanks to Corollary 2.17, we can apply Theorem 1.5 deducing the
following

######  Corollary 2.22 (CLT for Jacobi $\beta$-ensemble in the high-
temperature).

Consider the matrix representation ${\mathfrak{E}}$ (2.90) of the Jacobi
$\beta$-ensemble in the high-temperature regime (2.106) . Then, defining the
Free energy ${\mathcal{F}}_{J}(\alpha,P)$ as

${\mathcal{F}}_{J}(\alpha,P)=-\lim_{N\to\infty}\frac{\ln(Z^{J}_{N}(\alpha,P))}{N}\,,$
(2.107)

for all $j\in{\mathbb{N}}$ fixed, we have the following weak limit

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left({\mathfrak{E}}^{j}\right)-\mathbb{E}\left[\mathrm{Tr}\left({\mathfrak{E}}^{j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2})\,,$
(2.108)

where

$\mathbb{E}\left[\mathrm{Tr}\left({\mathfrak{E}}^{j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{J}(\alpha,P+itx^{j})_{|_{t=0}}\,,\quad\sigma^{2}=|\partial^{2}_{t}{\mathcal{F}}_{J}(\alpha,P+itx^{j})_{|_{t=0}}|\,.$
(2.109)

######  Remark 2.23.

We notice that for $N$ even, for
${\mathtt{a}}+{\mathtt{b}}=-1+\frac{\beta}{4}$ we can apply the final part of
Theorem 1.5, thus we deduce that

$\begin{split}&\partial_{\alpha}(\alpha\partial_{t}{\mathcal{F}}_{J}(\alpha,P+itx^{j})_{|_{t=0}})=\partial_{t}{\mathcal{F}}_{\textrm{dS}}(\alpha,P+itx^{j})_{|_{t=0}}\,,\\\
&\partial_{\alpha}(\alpha\partial^{2}_{t}{\mathcal{F}}_{J}(\alpha,P+itx^{j})_{|_{t=0}})=\partial_{t}^{2}{\mathcal{F}}_{\textrm{dS}}(\alpha,P+itx^{j})_{|_{t=0}}\end{split}$
(2.110)

###  2.6 The Itoh–Narita–Bogoyavleskii lattices

In this section, we apply our results to two families of integrable lattices
with short-range interaction that generalize the Volterra one (2.60). These
families are described in [11] (see also [10, 40, 58]).

One is called _additive Itoh–Narita–Bogoyavleskii (INB) $r$-lattice_ and is
defined by the following equations

$\displaystyle\dot{a}_{i}$
$\displaystyle=a_{i}\left(\sum_{j=1}^{r}a_{i+j}-\sum_{j=1}^{r}a_{i-j}\right),\quad
i=1,\dots,N,\;N\geq r\in{\mathbb{N}}.$ (2.111)

The second family is called the _multiplicative Itoh–Narita–Bogoyavleskii
(INB) $r$-lattice_ and is defined by the equations

$\displaystyle\dot{a}_{i}$
$\displaystyle=a_{i}\left(\prod_{j=1}^{r}a_{i+j}-\prod_{j=1}^{r}a_{i-j}\right),\quad
i=1,\dots,N,\;N\geq r\in{\mathbb{N}}.$ (2.112)

In both cases we consider the periodicity condition $a_{j+N}=a_{j}$. We notice
that setting $r=1$, we recover in both cases the Volterra lattice. Moreover,
both families admit the KdV equation as continuum limits, see [11].

In both cases the interaction is short-range, but in the additive case (2.111)
the nonlinearity is quadratic as in the Volterra lattice, instead in the
multiplicative one (2.112) it is of polynomial order.

As we already mentioned, both families are integrable for all
$r\in{\mathbb{N}}$, indeed both families admits a Lax pair formulation. For
the additive INB lattice (2.111), it reads

$\displaystyle L^{(+,r)}$
$\displaystyle=\sum_{i=1}^{N}\left(a_{i+r}E_{i+r,i}+E_{i,i+1}\right)$ (2.113)
$\displaystyle=\begin{pmatrix}0&1&0&\cdots&\leavevmode\hbox to14.94pt{\vbox
to6.22pt{\pgfpicture\makeatletter\hbox{\hskip 7.47229pt\lower-3.10944pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.47229pt}{-1.1961pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a_{N-r}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&0&0\\\
0&0&1&\cdots&0&a_{N-r+1}&0&0\\\ 0&0&0&1&\cdots&0&a_{N-r+2}&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&&\\\
a_{r+1}&0&\cdots&\cdots&0&1&0&\leavevmode\hbox to5pt{\vbox
to6.44pt{\pgfpicture\makeatletter\hbox{\hskip 2.5pt\lower-3.22221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
0&a_{r+2}&0\cdots&&\ddots&\ddots&\ddots&\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0&0&1\\\
1&0&\cdots&a_{N-r-1}&\cdots&0&0&0\\\ \end{pmatrix}\leavevmode\hbox
to71.18pt{\vbox to20.78pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}{}{}{}}{}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{2.7pt}{0.0pt}\pgfsys@moveto{33.15273pt}{0.0pt}\pgfsys@lineto{59.60551pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{33.15273pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.10948pt}{4.36633pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$r+1$ row }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{3.30943pt}\pgfsys@moveto{0.0pt}{8.15485pt}\pgfsys@lineto{0.0pt}{20.38126pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{8.15485pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{9.79585pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$N-r$ column }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.114)
$\displaystyle A^{(+,r)}$
$\displaystyle=\sum_{i=1}^{N}\left(\sum_{j=0}^{r}a_{i+j}\right)E_{i,i}+E_{i,i+r+1}\,,$
(2.115)

we recall that we are always considering periodic boundary conditions, so for
all $j\in{\mathbb{Z}}$, $a_{j+N}=a_{j}$ and $E_{i,j+N}=E_{i+N,j}=E_{i,j}$. In
this notation, the equations of motion (2.111) are equivalent to

$\dot{L}^{(+,r)}=[L^{(+,r)};A^{(+,r)}]\,.$ (2.116)

Analogously, the multiplicative INB $r$-lattices have a Lax Pair formulation,
which reads

$\displaystyle L^{(\times,r)}$
$\displaystyle=\sum_{i=1}^{N}\left(a_{i}E_{i,i+1}+E_{i+r,i}\right),$ (2.117)
$\displaystyle=\begin{pmatrix}0&a_{1}&0&\cdots&\leavevmode\hbox to5pt{\vbox
to6.44pt{\pgfpicture\makeatletter\hbox{\hskip 2.5pt\lower-3.22221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}&0&0&0\\\
0&0&a_{2}&\cdots&0&1&0&0\\\ 0&0&0&a_{3}&\cdots&0&1&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&&\\\
1&0&\cdots&\cdots&0&a_{r}&0&\leavevmode\hbox to5pt{\vbox
to6.44pt{\pgfpicture\makeatletter\hbox{\hskip 2.5pt\lower-3.22221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-3.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{0}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
0&1&0\cdots&&\ddots&\ddots&\ddots&\\\
\vdots&\ddots&\ddots&\ddots&\ddots&0&0&a_{N-1}\\\
a_{N}&0&\cdots&1&\cdots&0&0&0\\\ \end{pmatrix}\leavevmode\hbox to71.18pt{\vbox
to20.89pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}{}{}{}{}}{}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{2.7pt}{0.0pt}\pgfsys@moveto{33.15273pt}{0.0pt}\pgfsys@lineto{59.60551pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{33.15273pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{23.10948pt}{4.36633pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$r+1$ row }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}{}{}}{}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{3.42221pt}\pgfsys@moveto{0.0pt}{8.26762pt}\pgfsys@lineto{0.0pt}{20.49403pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{0.0pt}{8.26762pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.533pt}{9.90863pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$N-r$ column }}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox
to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$ (2.118)
$\displaystyle A^{(\times,r)}$
$\displaystyle=\sum_{i=1}^{N}\left(\prod_{j=0}^{r}a_{i+j}\right)E_{i,i+r+1}\,.$
(2.119)

Following the construction made in [33], where the authors numerically
computed the density of states for these two families of lattices, we
introduce the generalized Gibbs ensemble for these models as

$\displaystyle{\rm d}\mu_{+,r}$
$\displaystyle=\frac{\exp(-\mathrm{Tr}\left(P(L^{(+,r))}\right))\prod_{j=1}^{N}a_{j}^{\alpha-1}\mathbbm{1}_{a_{j}\geq
0}{\rm d}{\bf a}}{Z_{N}^{(+,r)}(a,P)}\,,$ (2.120) $\displaystyle{\rm
d}\mu_{\times,r}$
$\displaystyle=\frac{\exp(-\mathrm{Tr}\left(P(L^{(\times,r))}\right))\prod_{j=1}^{N}a_{j}^{\alpha-1}\mathbbm{1}_{a_{j}\geq
0}{\rm d}{\bf a}}{Z_{N}^{(\times,r)}(\alpha,P)}\,,$ (2.121)

where $P$ is a polynomial of degree $j(r+1)$ for some $j\in{\mathbb{N}}$ as in
Hypotheses 1.1. We restrict to this potential in view of the following Lemma
proved in [33]

######  Lemma 2.24.

Fix $\ell\in{\mathbb{N}}$. Then for $N$ large enough

$\mathrm{Tr}\left((L^{(+,r)})^{\ell}\right)=\mathrm{Tr}\left((L^{(\times,r)})^{\ell}\right)=0\,,$
(2.122)

if $\ell$ is not an integer multiple of $r+1$.

We notice that both $L^{(+,r)}$ and $L^{(\times,r)}$ are circular-like, and
that the entries are all positive, so from Lemma 1.3, we deduce the following:

######  Lemma 2.25.

Fix $m\in{\mathbb{N}}$, and consider the matrices $L^{(+,r)},L^{(\times,r)}$
(2.113)-(2.117). Then for $N$ big enough, there exist
$k^{(+,r)}=k^{(+,r)}(m),k^{(\times,r)}=k^{(\times,r)}(m)\in{\mathbb{N}}$, such
that

  * •

$\mathrm{Tr}\left((L^{(+,r)})^{m}\right)$ is $k^{(+,r)}$-circular with seed
$V^{(+,r)}$ and weed ${\widetilde{V}}^{(+,r)}$

  * •

$\mathrm{Tr}\left((L^{(\times,r)})^{m}\right)$ is $k^{(\times,r)}$-circular
with seed $V^{(\times,r)}$ and weed ${\widetilde{V}}^{(\times,r)}$