Thus, proceeding as we have done for the others systems previously considered,
we obtain the following:

######  Corollary 2.26 (CLT for INB lattices).

Consider the Lax matrices $L^{(+,r)},L^{(\times,r)}$ (2.113)-(2.117) of the
additive and multiplicative INB lattices respectively distributed according to
their Generalized Gibbs Ensemble (2.120)-(2.121). Then, defining the Free
energies ${\mathcal{F}}_{+,r}(\alpha,P),{\mathcal{F}}_{\times,r}(\alpha,P)$ as

$\displaystyle{\mathcal{F}}_{+,r}(\alpha,P)=$
$\displaystyle-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{(+,r)}_{N}(\alpha,P))\,,$
(2.123) $\displaystyle{\mathcal{F}}_{\times,r}(\alpha,P)=$
$\displaystyle-\lim_{N\to\infty}\frac{1}{N}\ln(Z^{(\times,r)}_{N}(\alpha,P))\,,$
(2.124)

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

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left((L^{(+,r)})^{(r+1)j}\right)-\mathbb{E}\left[\mathrm{Tr}\left((L^{(+,r)})^{(r+1)j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2}_{+,r})\,,$
(2.126)

$\lim_{N\to\infty}\frac{\mathrm{Tr}\left((L^{(\times,r)})^{(r+1)j}\right)-\mathbb{E}\left[\mathrm{Tr}\left((L^{(\times,r)})^{(r+1)j}\right)\right]}{\sqrt{N}}\rightharpoonup{\mathcal{N}}(0,\sigma^{2}_{\times,r})\,,$
(2.127)

where

$\displaystyle\mathbb{E}\left[\mathrm{Tr}\left((L^{(+,r)})^{(r+1)j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{+,r}(\alpha,P+itx^{(r+1)j})_{|_{t=0}}\,,\quad\sigma^{2}_{+,r}=|\partial^{2}_{t}{\mathcal{F}}_{+,r}(\alpha,P+itx^{(r+1)j})_{|_{t=0}}|\,,$
(2.128)
$\displaystyle\mathbb{E}\left[\mathrm{Tr}\left((L^{(\times,r)})^{(r+1)j}\right)\right]=iN\partial_{t}{\mathcal{F}}_{\times,r}(\alpha,P+itx^{(r+1)j})_{|_{t=0}}\,,\quad\sigma^{2}_{\times,r}=|\partial^{2}_{t}{\mathcal{F}}_{\times,r}(\alpha,P+itx^{(r+1)j})_{|_{t=0}}|\,.$
(2.129)

######  Remark 2.27.

We recall that in [33], it was shown that the density of states for this model
has support on the complex plane, but despite that all the moments of the
Generalized Gibbs ensemble are reals. Furthermore, in this case, we lack a
$\beta$-ensemble to compare with.

##  3 Transfer Operator Method

In this section, we develop a transfer operator approach to obtain the
asymptotic behaviour of the following integrals

$\displaystyle
I_{1}^{(N)}(H)=\int_{X^{N}}\left(\prod_{j=1}^{N}F(x_{j},\alpha)\right)e^{-H({\bf
x})}{\rm d}{\bf x}\,,$ (3.1) $\displaystyle
I_{2}^{(N)}(H)=\int_{X^{N}}\left(\prod_{j=1}^{N-1}F\left(x_{j},\alpha\left(1-\frac{j}{N}\right)\right)\right)R(x_{N})e^{-H({\bf
x})}{\rm d}{\bf x}\,,$ (3.2)

where $F,H,X$ satisfies the following set of hypotheses, which are closely
related to hypotheses 1.2

######  Hypotheses 3.1.

In this section, we assume the following hypotheses for $F,H,X$:

  1. HP. 1

$X$ is a convex subset of ${\mathbb{R}}^{d}$, $d=1,2$ (we identify
${\mathbb{C}}$ 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.

$H\,:\,X^{N}\to{\mathbb{R}}$ is $k$-circular with seed $h$ and weed
${\widetilde{h}}$ which are both 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

     * •

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

     * •

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

     * •

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

     * •

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

$h({\bf x},{\bf y})=w({\bf x})+w({\bf y})+o(\varepsilon)\,.$ (3.3)

     * •

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 $X$.

We want to study the asymptotic behaviour of the integrals
$I_{1}^{(N)},I_{2}^{(N)}$ (3.1)-(3.2) which are a generalization of the
partition functions of the type 1-2 systems at hand. In particular, we use the
knowledge of the asymptotic behaviour of these integrals to compute the two
ratios

$\mathbb{E}_{1}\left[e^{-it\Re\mathrm{Tr}\left(L^{s}\right)}\right]=\frac{Z^{(1)}_{N}(\alpha,\Re
P+it\Re z^{s})}{Z^{(1)}_{N}(\alpha,\Re
P)},\quad\mathbb{E}_{2}\left[e^{-it\Re\mathrm{Tr}\left(T^{s}\right)}\right]=\frac{Z^{(2)}_{N}(\alpha,\Re
P+it\Re z^{s})}{Z^{(2)}_{N}(\alpha,\Re P)}\,,$ (3.4)

which is one of the key steps of our proof. Under these assumptions, we prove
the following Theorem:

######  Theorem 3.1.

Under hypotheses 3.1. Consider a $k$-circular function
$U\,:\,X^{N}\to{\mathbb{R}}$ with seed $u$ and weed ${\widetilde{u}}$. Then,

$J_{1}^{(N)}=\frac{I_{1}^{(N)}(H+itU)}{I_{1}^{(N)}(H)}\,,\quad
J_{2}^{(N)}=\frac{I_{2}^{(N)}(H+itU)}{I_{2}^{(N)}(H)}$ (3.5)

are finite for all fixed $N$. Furthermore, assume that for some $d\geq 3$,
$|u|^{a}\exp\left(-h\right)$ is bounded for any $1\leq a\leq d$. Form the
Euclidean division $N=kM+\ell$. Then, there exists an $\varepsilon>0$, and
three complex-valued functions $\lambda\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$, $c_{k,\ell}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ and
$c_{k,\ell,M}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that for
all $q\in{\mathbb{N}}$ :

$\displaystyle
J_{1}=c_{k,\ell}(\alpha,t)\lambda(\alpha,t)^{M-2}\left(1+o(M^{-q})\right)\,,\quad\text{as
}M\to\infty\,,$ (3.6) $\displaystyle
J_{2}=c_{k,\ell,M}(\alpha,t)\prod_{j=1}^{M-2}\lambda\left(\alpha\frac{j}{M},t\right)\left(1+o_{M}(1)\right)\,,\quad\text{as
}M\to\infty\,,$ (3.7)

for $|t|<\varepsilon$. Furthermore, we have for all $\alpha>0$

$\displaystyle\bullet\lambda(\alpha,0)=1\,,$ (3.8) $\displaystyle\bullet
c_{k,\ell}(\alpha,0)=1\,,$ (3.9) $\displaystyle\bullet\lim_{t\to
0}c_{k,\ell,M}(\alpha,t)=1\,\quad\text{uniformly in }M\,,$ (3.10)
$\displaystyle\bullet\text{the remainder }o_{M}(1)\text{ is uniformly bounded
in }t\in(-\varepsilon,\varepsilon).$ (3.11)

Moreover, there exist three functions ${\widetilde{c}}_{k,\ell}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$,
${\widetilde{\lambda}}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ and
${\widetilde{c}}_{k,\ell,M}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that there
exist four constants $C_{1}^{(1)},C_{2}^{(1)},C_{1}^{(2)},C_{2}^{(2)}>0$ and
$p\in{\mathbb{N}}$ such that for all $q\in{\mathbb{N}}$:

$\displaystyle\bullet
C^{(1)}_{1}<{\widetilde{c}}_{k,\ell}(\alpha,t)<C^{(1)}_{2}\,,$ (3.12)
$\displaystyle\bullet
C_{1}^{(1)}N^{p}<{\widetilde{c}}_{k,\ell,M}(\alpha,t)<C^{(2)}_{2}N^{p}\,,$
(3.13)
$\displaystyle\bullet\lambda(\alpha,t)=\frac{{\widetilde{\lambda}}(\alpha,t)}{{\widetilde{\lambda}}(\alpha,0)}\,,$
(3.14) $\displaystyle\bullet
I_{1}^{(N)}(H+itU)={\widetilde{c}}_{k,\ell}(\alpha,t){\widetilde{\lambda}}(\alpha,t)^{M-2}\left(1+o(M^{-q})\right)\,.$
(3.15) $\displaystyle\bullet
I_{2}^{(N)}(H+itU)={\widetilde{c}}_{k,\ell,M}(\alpha,t)\prod_{j=1}^{M-2}{\widetilde{\lambda}}\left(\alpha\frac{j}{M},t\right)\left(1+o_{M}(1)\right)\,.$
(3.16)

The previous theorem is the central tool that we use to prove Theorem 1.5 in
Section 5. Since the proof is rather technical, we defer it to Section 6.

##  4 Nagaev–Guivarc’h theory

In this section, we adapt some standard results from the fluctuation theory of
Nagaev–Guivarc’h [32] to our situation. Specifically, we prove the following:

######  Theorem 4.1.

Let $(X_{n})_{n\geq 1}$ be a sequence of real random variables with partial
sums $(S_{n})_{n\geq 1}$. Assume that there exists $\varepsilon>0$, two
functions $\lambda\in C^{1}([0,\varepsilon))$, $c\in C^{0}([0,\varepsilon))$
and $h_{n}\in C^{0}([0,\varepsilon))$, such that for all
$t\in[-\varepsilon,\varepsilon]$, and all $n\geq 1$ we have

$\mathbb{E}\left[e^{-itS_{n}}\right]=c(t)\lambda(t)^{m}\left(1+h_{n}(t)\right)\,,$
(4.1)

Where $\lim_{n\to\infty}n/m=k\in{\mathbb{N}}_{\geq 1}$.

Furthermore, assume that:

  * a.

there exists $A,\,\sigma^{2}\in{\mathbb{C}}$ such that
$\lambda(t)=\exp\left(-iAt-\sigma^{2}t^{2}/2+o(t^{3})\right)$ as $t\to 0$;

  * b.

$h_{n}\xrightarrow{n\to\infty}0$ uniformly in $[-\varepsilon,\varepsilon]$,
and $h_{n}(0)=0$;

  * c.

$c(0)=1$.

Then $A\in{\mathbb{R}},\,\sigma^{2}\geq 0$, and
$\left(S_{n}-nA/k\right)/\sqrt{n}$ converges to a Gaussian distribution
${\mathcal{N}}(0,\sigma^{2}/k)$ as $n$ tends to infinity.

###### Proof.

First, evaluating (4.1) at $t=0$, we deduce that $\lambda(0)=1$. Then, we use
the asymptotic expansion of $\lambda(t)$, and properties b.-c. to prove that

$\mathbb{E}\left[\exp\left(-it\frac{S_{n}-mA}{n}\right)\right]\xrightarrow[n\to\infty]{}1\,.$

Thus, by Lévy theorem [71], we deduce that $S_{n}/n-A/k$ converges in
distribution to $0$. So, since $S_{n}$ is real, then $A\in{\mathbb{R}}$.
Exploiting again the asymptotic expansion of $\lambda(t)$ and properties
b.-c., we show that
$\mathbb{E}\left[\exp\left(-it\frac{S_{n}-mA}{\sqrt{n}}\right)\right]$
converges to the function $\exp\left(-\frac{\sigma^{2}t^{2}}{2k}\right)$. By
Lévy theorem [71], this must be the characteristic function of a real random
variable, proving that $\sigma^{2}\geq 0$, and that $(S_{n}-nA/k)/\sqrt{n}$
converges to a Gaussian distribution $\mathcal{N}(0,\sigma^{2}/k)$. ∎

Further, we prove the following:

######  Theorem 4.2.

Let $(X_{n})_{n\geq 1}$ be a sequence of random variables with partial sums
$(S_{n})_{n\geq 1}\in{\mathbb{R}}$. Assume that there exists $\varepsilon>0$
and functions $(x,t)\mapsto\lambda(x,t)\in C^{1,0}([0,1)\times{\mathbb{R}})$,
$c_{n}\in C^{0}({\mathbb{R}})$, and $h_{n}$ continuous at $0$, such that for
all $t\in[-\varepsilon,\varepsilon]$, and all $n\geq 1$ we have

$\mathbb{E}\left[e^{-itS_{n}}\right]=c_{n}(t)\left(\prod_{j=1}^{m}\lambda\left(j/m,t\right)\right)\left(1+h_{n}(t)\right)\,,$
(4.2)

where $\lim_{n\to\infty}n/m=k\in{\mathbb{N}}_{\geq 1}$.

Furthermore, assume that:

  * a.

there exists two continuous functions $A,\,\sigma^{2}\,:[0,1]\to{\mathbb{C}}$
such that

$\lambda(x,t)=\exp\left(-iA(x)t-\sigma^{2}(x)t^{2}/2+o(t^{2})\right)$ as $t\to
0$;

  * b.

$||h_{n}||_{\infty}\xrightarrow{n\to\infty}0$ uniformly in
$[-\varepsilon,\varepsilon]$, and $h_{n}(0)=0$;

  * c.

$c_{n}(0)=1$ and
$\lim_{n\to\infty}c_{n}(t/\sqrt{n})=\lim_{n\to\infty}c_{n}(t/n)=1$.

Then $\int_{0}^{1}A(x){\rm d}x\in{\mathbb{R}},\,\int_{0}^{1}\sigma^{2}(x){\rm
d}x\in{\mathbb{R}}_{+}$, and
$\sqrt{n}\left(\frac{S_{n}}{n}-\frac{\int_{0}^{1}A(x){\rm d}x}{k}\right)$
converges to a Gaussian distribution
${\mathcal{N}}\left(0,\frac{\int_{0}^{1}\sigma^{2}(x){\rm d}x}{k}\right)$ as
$n$ tends to infinity.

###### Proof.

First, let $t={\widetilde{t}}/n$, then by hypothesis c.

$\lim_{n\to\infty}\mathbb{E}\left[e^{-i{\widetilde{t}}\left(S_{n}/n-\frac{1}{n}\sum_{\ell=1}^{m}A\left(\ell/m\right)\right)}\right]=\lim_{n\to\infty}\mathbb{E}\left[e^{-i{\widetilde{t}}\left(S_{n}/n-\frac{\int_{0}^{1}A(x){\rm
d}x}{k}\right)}\right]=1\,.$ (4.3)

Thus, by Levy theorem, $S_{n}/n\to\frac{\int_{0}^{1}A(x){\rm d}x}{k}$ almost
surely, thus, since $S_{n}\in{\mathbb{R}}$, this implies that
$\int_{0}^{1}A(x){\rm d}x\in{\mathbb{R}}$. Consider now
$t={\widetilde{t}}/\sqrt{n}$, following the same reasoning one conclude that

$\begin{split}\lim_{n\to\infty}\mathbb{E}\left[e^{-i{\widetilde{t}}\sqrt{n}\left(S_{n}/n-\frac{1}{n}\sum_{\ell=1}^{n}A\left(\ell/m\right)\right)}\right]&=\lim_{n\to\infty}\mathbb{E}\left[e^{-i{\widetilde{t}}\sqrt{n}\left(S_{n}/n-\frac{\int_{0}^{1}A(x){\rm
d}x}{k}\right)}\right]\\\
&=\lim_{n\to\infty}e^{-\frac{{\widetilde{t}}^{2}}{2n}\sum_{\ell=1}^{m}\sigma^{2}(\ell/m)}=e^{-\frac{{\widetilde{t}}^{2}}{2k}\int_{0}^{1}\sigma^{2}(x){\rm
d}x}\,,\end{split}$ (4.4)

thus, by Lévy theorem [71],
$e^{-\frac{{\widetilde{t}}^{2}}{2k}\int_{0}^{1}\sigma^{2}(x){\rm d}x}$ must be
the characteristic function of a real random variable, proving that
$\int_{0}^{1}\sigma^{2}(x)\in{\mathbb{R}}_{+}$. ∎

##  5 Proof Of Theorem 1.5-1.9

We are ready to prove Theorem 1.5-1.9. For convenience, we split the proof
into two Lemmas, which combined give the full proof of our results.

######  Lemma 5.1.

Under hypotheses 1.2. Consider $\mu^{(1)}_{N},\,\mu^{(2)}_{N}$ (1.31)-(1.32),
and let $s\in{\mathbb{N}}$. Assume that
$\Re\mathrm{Tr}\left(P(L)\right)+it\Re\mathrm{Tr}\left(L^{s}\right)$ is a
$k$-circular function, that $W,u$ are the compatible seeds of
$\Re\mathrm{Tr}\left(P(L)\right),\mathrm{Tr}\left(L^{s}\right)$ and that
$|u|^{a}e^{-h}$ is bounded for $1\leq a\leq 2$. Then, there exist four
continuous functions

$\displaystyle
A\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}\,,\qquad{\widetilde{A}}\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}\,,$
(5.1)
$\displaystyle\sigma\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}_{+}\,,\qquad{\widetilde{\sigma}}\,:\,{\mathbb{R}}_{+}\longrightarrow{\mathbb{R}}_{+}\,,$
(5.2)

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

$\left(S_{N}-NA(\alpha)/k\right)/\sqrt{N}$

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

$\left(S_{N}-N{\widetilde{A}}(\alpha)/k\right)/\sqrt{N}$

converges to a Gaussian distribution
${\mathcal{N}}(0,{\widetilde{\sigma}}^{2}(\alpha)/k)$ as $N$ tends to
infinity.

The proof of the previous result is a trivial application of Theorem
3.1-4.1-4.2 and of Lemma 1.3 where $H$ is the polynomial $\Re P$. Furthermore,
we can interpret the previous relations through the free energies of
$\mu^{(1)}_{N},\,\mu^{(2)}_{N}$ (1.31) -(1.32):

######  Lemma 5.2.

Under the same hypotheses and notation of Lemma 5.1. Consider the two measures
$\mu^{(1)}_{N},\,\mu^{(2)}_{N}$ (1.31)-(1.32), and define the free energies as

${\mathcal{F}}^{(1)}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{\ln(Z_{N}^{(1)}(\alpha,\Re
P))}{N}\,,\quad{\mathcal{F}}^{(2)}(\alpha,\Re
P)=-\lim_{N\to\infty}\frac{\ln(Z_{N}^{(2)}(\alpha,\Re P))}{N}\,,$ (5.3)

then, using the same notation as in Lemma 5.1, the following holds:

  1. i.

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

  2. ii.

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

  3. iii.

${\widetilde{A}}(\alpha)=ik\partial_{t}{\mathcal{F}}^{(2)}(\alpha,\Re P+it\Re
z^{s})_{|_{t=0}}$

  4. iv.

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

  5. v.

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

######  Remark 5.3.

The previous theorem implies that

$A(\alpha)=\partial_{\alpha}(\alpha{\widetilde{A}}(\alpha))\,,\qquad\sigma^{2}(\alpha)=\partial_{\alpha}(\alpha{\widetilde{\sigma}}^{2}(\alpha))\,.$
(5.4)

###### Proof.

To prove $i.$, we can just compute the free energy of
$\mu^{(1)}_{N},\,\mu^{(2)}_{N}$ using Theorem 3.1. For
${\mathcal{F}}^{(1)}(\alpha,\Re P)$ we deduce immediately that

$\begin{split}{\mathcal{F}}^{(1)}(\alpha,\Re
P)&=-\lim_{M\to\infty}\frac{1}{N}\ln(Z^{(1)}_{N}(\alpha,\Re
P))\stackrel{{\scriptstyle\eqref{eq:Z_periodic}}}{{=}}-\lim_{M\to\infty}\frac{M}{N}\ln({\widetilde{\lambda}}(\alpha,0))\\\
&=-\frac{1}{k}\ln({\widetilde{\lambda}}(\alpha,0))\,.\end{split}$ (5.5)

Where we recall that $N=kM+\ell$. The computation for
${\mathcal{F}}^{(2)}(\alpha,\Re P)$ follows in the same way, establishing i..

We now prove $ii.-iv$. First, we notice that following the notation of Theorem
3.1 \- 4.1:

$c(t)=c_{k,\ell}(\alpha,t)\,,\quad
h_{n}=1+o(M^{-q})\,,\quad\lambda(t)=\lambda(\alpha,t)\,,$ (5.6)

thus to compute $A(\alpha),\,\sigma(\alpha)$ we expand $\lambda(\alpha,t)$
around $t=0$

$\lambda(\alpha,t)=1+t\partial_{t}\lambda(\alpha,0)+\frac{t^{2}}{2}\partial_{t}^{2}\lambda(\alpha,0)+o(t^{3})\,,$
(5.7)

which implies that

$\lambda(\alpha,t)=\exp\left(t\partial_{t}\ln(\lambda(\alpha,t))_{|_{t=0}}+\frac{t^{2}}{2}\partial^{2}_{t}\ln(\lambda(\alpha,t))_{|_{t=0}}+o(t^{3})\right)\,.$
(5.8)

This implies that
$A(\alpha)=-i\partial_{t}\ln(\lambda(\alpha,t))_{|_{t=0}}\,,\sigma^{2}(\alpha)=-\partial^{2}_{t}\ln(\lambda(\alpha,t))_{|_{t=0}}$.
From (6.9), we deduce that
$\partial_{t}\ln(\lambda(\alpha,t))_{|_{t=0}}=\partial_{t}\ln({\widetilde{\lambda}}(\alpha,t))_{|_{t=0}}$,
thus from the previous expressions and the explicit form of the free energy
(5.5) we conclude.

To prove $iii.-v.$ we proceed in the same way, thus following the notation of
Theorem 3.1-4.2:

$c_{n}(t)=c_{k,\ell,M}(\alpha,t)\,,\quad
h_{n}=1+o(1)\,,\quad\lambda\left(j/M,t\right)=\lambda\left(\alpha\frac{j}{M},t\right)\,.$
(5.9)

Thus, as in (5.8) except that $\alpha\to\alpha\frac{j}{M}$, we expand
$\lambda\left(\alpha\frac{j}{M},t\right)$ around $t=0$, leading to

$\begin{split}{\widetilde{A}}(\alpha)&=\partial_{t}\left(\int_{0}^{1}\ln(\lambda(\alpha
x,t)){\rm d}x\right)_{|_{t=0}}\,,\\\
{\widetilde{\sigma}}^{2}(\alpha)&=\partial^{2}_{t}\left(\int_{0}^{1}\ln(\lambda(\alpha
x,t)){\rm d}x\right)_{|_{t=0}}\end{split}$ (5.10)

which concludes the proof. ∎

######  Remark 5.4.

We notice that the Lemma 5.2, and Lemma 5.1 imply that we can compute the
expected values, and the variances of $\mathrm{Tr}\left(L^{s}\right)$
according to $\mu^{(1)}_{N},\,\mu^{(2)}_{N}$ just computing derivatives of the
corresponding free energy. This property is broadly used in the physics
literature, but we lacked a precise proof of the general result. Furthermore,
we can compute the expected value, and the variance of
$\mathrm{Tr}\left(L^{s}\right)$ according to $\mu^{(1)}_{N}$ starting from the
corresponding values for $\mu^{(2)}_{N}$. Thus, we have reduced all this
problem to the computation of the free energy of $\mu^{(2)}_{N}$.

The proof of both Theorem 1.5 and Theorem 1.9 follows from the four previous
lemmas. Thus, we have completed the proof of our main theorems.

##  6 Technical Results

In this section, we prove the technical results that we used to prove our main
Theorems 1.5-1.9, the proof follows the same line as the proof of [35,
Proposition 4.2], and we prove Theorem 1.11, whose proof uses the same
machinery as the latter proofs. In the last part, we prove a Berry-Esseen type
bound for the type 1 measure $\mu_{N}^{(1)}$. We start by proving Theorem 3.1.

To prove these results, we follow the same ideas as in [32, Theorem 2.4],
which we apply, for $k\geq 1$, to a family $(\mathcal{L}_{t})_{t\in I}$ of
compact operators ${\mathcal{L}}_{t}:L^{2}(X^{k})\to L^{2}(X^{k})$, where $I$
is an interval containing a neighbourhood of $0$. Denoting for $u\in
L^{p}(X^{k})$ $\|u\|_{p}=\left(\int_{X^{k}}|u({\bf x})|^{p}{\rm d}{\bf
x}\right)^{\frac{1}{p}}$, we endow the space of linear bounded operators of
$L^{2}(X^{k})$, denoted by $\mathbf{L}(L^{2}(X^{k}))$ with the norm

$\|{\mathcal{L}}\|:=\sup_{u\in L^{2}(X^{k}),\ u\neq
0}\frac{\|{\mathcal{L}}u\|_{2}}{\|u\|_{2}}\,.$

######  Definition 6.1.

Let $I\subset{\mathbb{R}}$ be open. We say that the
$\mathbf{L}(L^{2}(X^{k}))$-valued function $t\in I\mapsto{\mathcal{L}}_{t}$ is
$a$-times continuously differentiable, and denote it by ${\mathcal{L}}\in
C^{a}(\mathbf{L}(L^{2}(X^{k})))$ if ${\mathcal{L}}$ has a Taylor expansion of
order $a$ around any $t\in I$, i.e.

${\mathcal{L}}_{t+h}={\mathcal{L}}_{t}+hD^{(1)}_{t}+\frac{h^{2}}{2}D^{(2)}_{t}+\ldots+\frac{h^{a}}{a!}D^{(a)}_{t}+o(h^{a})\qquad\text{as
}h\to 0\,,$

with $t\in I\mapsto D_{t}^{(a)}\in\mathbf{L}(L^{2}(X^{k}))$ continuous, where
the $o(h^{a})$ is with respect to the norm $\|\cdot\|$. The operator
$D^{(j)}_{t}$ is then by definition the $j$-th derivative of
${\mathcal{L}}_{t}$: $\partial_{t}^{(j)}{\mathcal{L}}_{t}:=D_{t}^{(j)}$.

######  Remark 6.2.

We define in the same way $C^{a}$ functions with values in $L^{2}(X^{k})$.

We enforce the following proposition, which can be easily deduced from [32,
Proposition 2.3]:

######  Proposition 6.3.

Let $\lambda(0)$ be an isolated eigenvalue of the operator ${\mathcal{L}}_{0}$
with multiplicity one, and assume that the family of operators $t\in
I\mapsto{\mathcal{L}}_{t}\in\left(\mathbf{L}(L^{2}(X^{k})),\|\cdot\|\right)$
depends on $t$ in a $C^{d}$ way. Then, for $t$ small enough,
${\mathcal{L}}_{t}$ has an isolated eigenvalue $\lambda(t)$ with multiplicity
one, and $\lambda(t)$, the corresponding eigenprojection $\pi_{t}$ and its
eigenfunction $\varphi_{t}$ are $C^{d}$ with respect to $t$.

Moreover, assume that the rest of the spectrum of ${\mathcal{L}}_{0}$ is
contained in a disk of radius $|\lambda(0)|-\delta$. Writing
${\mathcal{Q}}_{t}=(I-\pi_{t}){\mathcal{L}}_{t}$, so that for $n\geq 1$
${\mathcal{L}}_{t}^{n}=\lambda(t)^{n}\pi_{t}+{\mathcal{Q}}_{t}^{n}$. For any
$r>|\lambda(0)|-\delta$, there exists a constant $C>0$ independent of $t,n$
such that $\|{\mathcal{Q}}^{n}_{t}\|\leq Cr^{n}$ for all $n\in\mathbb{N}$.

######  Remark 6.4.

We notice that in the previous proposition the size of the set of $t$ such
that the statements holds is not specified: it is an existence result. One of
the crucial step of the following analysis is to ensure that such set is
uniform with respect to some parameters.

In the sequel, for $I,J\subset{\mathbb{R}}$ open, and for $E$ a normed space,
we will denote by $C^{a,b}(I\times J,E)$ the set of functions with values in
$E$ which are $C^{a}$ with respect to the first parameter, and $C^{b}$ with
respect to the second parameter. We will denote $C^{a,b}(I\times J)$ for
functions with values in ${\mathbb{C}}$.

The following families of kernel operators play a crucial role in the proof of
the main theorems of this section.

######  Definition 6.5.

Under Hypotheses 3.1. Consider a $k$-circular function
$U\,:\,X^{N}\to{\mathbb{R}}$ with seed $u$, such that $u$ is compatible with
$h$, the seed of $H$.

For $t\in{\mathbb{R}}$, $\alpha>0$ define

  * •

${\mathcal{L}}_{t,\alpha}\in\mathbf{L}(L^{2}(X^{k})$ as

${\mathcal{L}}_{\alpha,t}f({\bf y})=\int_{X^{k}}f({\bf
x})\sqrt{\prod_{q=1}^{k}F(x_{q},\alpha)F(y_{q},\alpha)}e^{-h({\bf y},{\bf
x})-itu({\bf y},{\bf x})}{\rm d}{\bf x}=:\int_{X^{k}}k_{\alpha,t}({\bf x},{\bf
y})f({\bf x}){\rm d}{\bf x}\,.$ (6.1)

  * •

With the writing $N=kM+\ell$, $0\leq\ell<k$, for $1\leq j\leq M-1$ define
${\mathcal{L}}_{t,\alpha}^{(j)}\in\mathbf{L}(L^{2}(X^{k})$ as

${\mathcal{L}}_{\alpha,t}^{(j)}f({\bf y})={\widetilde{F}}({\bf
y},j+1)\int_{X^{k}}f({\bf x}){\widetilde{F}}({\bf x},j)e^{-h({\bf y},{\bf
x})-itu({\bf y},{\bf x})}{\rm d}{\bf x}\,,$

where

${\widetilde{F}}({\bf
x},j)=\sqrt{\prod_{q=1}^{k}F\left(x_{q},\alpha\left(1-\frac{(j-1)k+q}{N}\right)\right)}\,.$

For a kernel operator $T\in\mathbf{L}(L^{2}(X^{k}))$

$(Tu)({\bf y})=\int_{X^{k}}k({\bf x},{\bf y})u({\bf x}){\rm d}{\bf x}\,$

where $k:X^{k}\times X^{k}\to{\mathbb{R}}$, define its Hilbert-Shmidt norm as

$\|T\|_{\mathrm{HS}}^{2}=\iint_{X^{2k}}k({\bf x},{\bf y})^{2}{\rm d}{\bf
x}{\rm d}{\bf y}\,.$

Then, $\|T\|\leq\|T\|_{\mathrm{HS}}$. Furthermore, if $T$ is a kernel operator
with finite Hilbert-Schmidt norm, then it is compact, see [44].

######  Lemma 6.6.

Under assumptions 3.1. For $\alpha>0$, $t\in{\mathbb{R}}$, consider the
operators ${\mathcal{L}}_{\alpha,t},{\mathcal{L}}_{t,\alpha}^{(j)}$, $1\leq
j\leq M-1$ given by Definition 6.5. Assume that for some $d\geq 2$,
$|u|^{a}\exp\left(-h\right)$ is bounded for any $1\leq a\leq d$.

Then, those operators have finite Hilbert-Schmidt norm, and
${\mathcal{L}}_{\alpha,t},{\mathcal{L}}_{t,\alpha}^{(j)}\in
C^{1,d}((0,+\infty)\times{\mathbb{R}},\mathbf{L}(L^{2}(X^{k})))$, i.e.

  * 1)

For any fixed $t\in{\mathbb{R}}$, the $\mathbf{L}(L^{2}(X^{K}))$-valued
functions $\alpha\mapsto{\mathcal{L}}_{\alpha,t}$ and
$\alpha\mapsto{\mathcal{L}}_{\alpha,t}^{(j)}$ are continuously differentiable.

  * 2)

For any fixed $\alpha>0$, $t\mapsto{\mathcal{L}}_{\alpha,t}$ and
$\alpha\mapsto{\mathcal{L}}_{\alpha,t}^{(j)}$ are $d$-times continuously
differentiable.

Furthermore, for all $\alpha>0$, define

${\mathcal{B}}_{\alpha}=\frac{1}{\|{\mathcal{L}}_{\alpha,0}\|_{\mathrm{HS}}}{\mathcal{L}}_{\alpha,0}$

and

${\mathcal{B}}_{\alpha}^{(j)}=\frac{1}{\|{\mathcal{L}}_{\alpha,0}^{(j)}\|_{\mathrm{HS}}}{\mathcal{L}}_{\alpha,0}^{(j)},\
1\leq j\leq M-1\,.$

Then, ${\mathcal{B}}_{\alpha}$ (resp. ${\mathcal{B}}_{\alpha}^{(j)}$) has a
dominant, simple eigenvalue $\mu(\alpha)>0$ (resp. $\mu^{(j)}(\alpha)$).
Moreover, for all $A>0$ there exists $\delta>0$ such that and all
$\alpha\in(0,A]$, any other eigenvalue $\mu$ satisfies
$|\mu-\mu(\alpha)|\geq\delta$ (resp. $|\mu-\mu^{(j)}(\alpha)|\geq\delta$). We
say that those operators have a uniform spectral gap in $\alpha\in(0,A]$.

We defer the proof of this lemma to Appendix B.

Applying the previous proposition, we can prove both asymptotic estimates of
Theorem 3.1. For convenience, we split the theorem into two; the first one is
related to the proof of the result involving $I_{N}^{(1)}$, and the second
$I_{N}^{(2)}$:

######  Theorem 6.7.

Under hypotheses 3.1. Consider a $k$-circular function
$U\,:\,X^{N}\to{\mathbb{R}}$ with seed $u$ and weed ${\widetilde{u}}$, such
that $u$ is compatible with $h$, the seed of $H$. Define

$J_{1}^{(N)}=\frac{I_{1}^{(N)}(H+itU)}{I_{1}^{(N)}(H)}$ (6.2)

where $I_{1}^{(N)}$ is defined as

$I_{1}^{(N)}(H)=\int_{X^{N}}\left(\prod_{j=1}^{N}F(x_{j},\alpha)\right)e^{-H({\bf
x})}{\rm d}{\bf x}\,.$ (6.3)

Assume that for some $d\geq 2$, $|u|^{a}\exp\left(-h\right)$ is bounded for
any $1\leq a\leq d$. Form the Euclidean division $N=kM+\ell$. Then, there
exists an $\varepsilon>0$, and two complex-valued functions $\lambda\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$, $c_{k,\ell}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that for
all $q\in{\mathbb{N}}$ :

$J_{1}=c_{k,\ell}(\alpha,t)\lambda(\alpha,t)^{M-2}\left(1+o(M^{-q})\right)\,,\quad\text{as
}M\to\infty\,,\\\ $ (6.4)

for $|t|<\varepsilon$. Furthermore, for any $\alpha>0$

$\displaystyle\bullet\lambda(\alpha,0)=1\,,$ (6.5) $\displaystyle\bullet
c_{k,\ell}(\alpha,0)=1\,.$ (6.6)

Moreover, there exist two functions ${\widetilde{c}}_{k,\ell}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$, and
${\widetilde{\lambda}}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that there
exist two constants $C_{1}^{(1)},C_{2}^{(1)}>0$ such that for all
$q\in{\mathbb{N}}$:

$\displaystyle\bullet
C^{(1)}_{1}<|{\widetilde{c}}_{k,\ell}(\alpha,t)|<C^{(1)}_{2}\,,$ (6.7)
$\displaystyle\bullet\lambda(\alpha,t)=\frac{{\widetilde{\lambda}}(\alpha,t)}{{\widetilde{\lambda}}(\alpha,0)}\,,$
(6.8) $\displaystyle\bullet
I_{1}^{(N)}(H+itU)={\widetilde{c}}_{k,\ell}(\alpha,t){\widetilde{\lambda}}(\alpha,t)^{M-2}\left(1+o(M^{-q})\right)\,.$
(6.9)

###### Proof.

For $\alpha>0$ and $t\in{\mathbb{R}}$, consider
${\mathcal{L}}_{t,\alpha}\,:\,L^{2}(X^{k})\to L^{2}(X^{k})$ given by
Definition 6.5. By Lemma 6.6, each ${\mathcal{L}}_{t,\alpha}$ is Hilbert-
Schmidt, and in particular, compact. Moreover, since the kernel $k_{\alpha,0}$
(6.1) is positive, we apply a generalization of Jentzsch’s theorem (showing
that ${\mathcal{L}}_{\alpha,0}$ has a simple, positive, dominant eigenvalue)
[72, Theorem 137.4] in combinations with Lemma 6.6 and Proposition 6.3 to
deduce that:

  * •

There exists $\varepsilon>0$ such that for $|t|<\varepsilon$,
${\mathcal{L}}_{\alpha,t}$ has a simple, dominant eigenvalue
${\widetilde{\lambda}}(\alpha,t)$ with associated eigenfunction
$\varphi(\cdot,\alpha,t)\in L^{2}(X^{k})$,
$\|\varphi(\cdot,\alpha,t)\|_{2}=1$,

  * •

Its dominant eigenvalue ${\widetilde{\lambda}}$ varies smoothly with respect
to $(\alpha,t)$: ${\widetilde{\lambda}}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$,

  * •

The associated eigenfunction $\varphi(\cdot,\alpha,t)$ varies smoothly:
$(\alpha,t)\mapsto\varphi(\cdot,\alpha,t)\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon),L^{2}(X^{k}))$,

and there exists an operator ${\mathcal{Q}}_{t}\,:\,L^{2}(X^{k})\to
L^{2}(X^{k})$ such that

${\mathcal{L}}_{\alpha,t}\phi({\bf
y})={\widetilde{\lambda}}\left(\alpha,t\right)\left\langle\phi,\varphi\left(\cdot,\alpha,t\right)\right\rangle\varphi\left({\bf
y},\alpha,t\right)+{\mathcal{Q}}_{t}\phi({\bf y})\,,\quad\forall\,\phi\in
L^{2}(X^{k})\,,$ (6.11)

denoting by $\left\langle\cdot,\cdot\right\rangle$ the standard scalar product
in $L^{2}(X^{k})$.

Furthermore, at $t=0$ we have ${\widetilde{\lambda}}\left(\alpha,0\right)>0$,
$\varphi(\cdot,\alpha,t)>0$ almost everywhere, and there exists a $\delta>0$
such that for all $t\in(-\varepsilon,\varepsilon)$, one has
$||{\mathcal{Q}}_{t}||\leq|{\widetilde{\lambda}}\left(\alpha,t\right)|-\delta$.

For ${\bf z}\in X^{k}$ define $G_{{\bf z}}({\bf y})$ as

$G_{\bf z}({\bf y})=\begin{cases}\begin{aligned}
&\sqrt{\prod_{q=1}^{k}F(y_{q},\alpha)}\int_{X^{\ell}}\prod_{j=kM+1}^{kM+\ell}F(x_{j},\alpha)\\\
&\times\exp\left(-({\widetilde{h}}+it{\widetilde{u}})({\bf
y},x_{kM+1},\ldots,x_{kM+\ell},{\bf z})\right)\prod_{j=kM+1}^{kM+\ell}{\rm
d}x_{j}\,,\end{aligned}&\quad\ell>0\,,\\\
\sqrt{\prod_{q=1}^{k}F(y_{q},\alpha)}\exp\left(-h({\bf y},{\bf z})-itu({\bf
y},{\bf z})\right)\,,&\quad\ell=0\,,\end{cases}$ (6.12)

and the linear operator ${\mathcal{S}}\;:L^{2}(X^{k})\to{\mathbb{C}}$ as

${\mathcal{S}}\varphi=\int_{X^{k}}\prod_{j=1}^{k}\sqrt{F(x_{j},\alpha)}\varphi({\bf
x}){\rm d}{\bf x}\,.$ (6.13)

We notice that

$\|{\mathcal{S}}\|:=\sup_{\|\varphi\|_{2}=1}\left|{\mathcal{S}}\varphi\right|\leq\|F\|_{1}^{k}<+\infty$

therefore ${\mathcal{S}}$ is continuous.

In this notation, we can recast (6.17), applying ${\mathcal{S}}$ to $({\bf
x},{\bf y})\mapsto\left({\mathcal{L}}^{M-1}_{\alpha,t}G_{\bf x}\right)({\bf
y})$, as

$\begin{split}I_{1}^{(N)}(\alpha,H+itU)&={\mathcal{S}}\left(\left({\mathcal{L}}_{\alpha,t}^{M-1}G_{{\bf
x}_{M}}\right)({\bf x}_{M})\right)\\\
&={\widetilde{\lambda}}^{M-1}(\alpha,t){\mathcal{S}}\big{(}\left\langle\varphi\left(\cdot,\alpha,t\right);G_{{\bf
x}_{M}}(\cdot)\right\rangle\varphi({\bf
x}_{M},\alpha,t)\big{)}+{\mathcal{S}}\big{(}{\mathcal{Q}}_{t}^{M-2}G_{{\bf
x}_{M}}({\bf x}_{M})\big{)}\,,\end{split}$ (6.14)

where here and in the sequel, if $f\in L^{2}(X^{k}\times X^{k})$, using a
slight notational liberty, we denoted ${\mathcal{S}}(f({\bf x})$ for
${\mathcal{S}}(f)$. Defining

$\begin{split}c_{k,\ell}(\alpha,t)&=\frac{{\mathcal{S}}\big{(}\left\langle\varphi\left(\cdot,\alpha,t\right);G_{{\bf
x}_{M}}(\cdot)\right\rangle\varphi({\bf
x}_{M},\alpha,t)\big{)}}{{\mathcal{S}}\big{(}\left\langle\varphi\left(\cdot,\alpha,0\right);G_{{\bf
x}_{M}}(\cdot)\right\rangle\varphi({\bf x}_{M},\alpha,0)\big{)}}\,,\\\
\lambda(\alpha,t)&=\frac{{\widetilde{\lambda}}^{M-1}(\alpha,t)}{{\widetilde{\lambda}}^{M-1}(\alpha,0)}\,.\end{split}$
(6.15)

Because $\varphi(\cdot,\alpha,0)>0$ almost everywhere, the denominator in
$c_{k,\ell}$ is positive, and by direct computation, using that
$h,{\widetilde{h}}$ are bounded from below, one sees that

${\mathcal{S}}\big{(}\left\langle\varphi\left(\cdot,\alpha,t\right);G_{{\bf
x}_{M}}(\cdot)\right\rangle\varphi({\bf
x}_{M},\alpha,t)\big{)}\leq\|F(\cdot,\alpha)\|_{1}^{2k+\ell}\,.$

In view of Proposition 6.3,
$\|{\mathcal{Q}}_{t}^{n}\|\leq(|{\widetilde{\lambda}}(t)|-\delta)^{n}$, and we
conclude. ∎

We prove now another theorem related to the expansion of $I_{2}^{(N)}$, which,
combined with the previous one, conclude the proof of Theorem 3.1

######  Theorem 6.8.

Under hypotheses 3.1. Consider a $k$-circular function
$U\,:\,X^{N}\to{\mathbb{R}}$ with seed $u$ and weed ${\widetilde{u}}$, such
that $u$ is compatible with $h$, the seed of $H$. Define

$J_{2}^{(N)}=\frac{I_{2}^{(N)}(H+itU)}{I_{2}^{(N)}(H)}$ (6.16)

where $I_{2}^{(N)}$ is defined as

$I_{2}^{(N)}(H)=\int_{X^{N}}\left(\prod_{j=1}^{N-1}F\left(x_{j},\alpha\left(1-\frac{j}{N}\right)\right)\right)R(x_{N})e^{-H({\bf
x})}{\rm d}{\bf x}\,.$ (6.17)

Assume that for some $d\geq 2$, $|u|^{a}\exp\left(-h\right)$ is bounded for
any $1\leq a\leq d$. Form the Euclidean division $N=kM+\ell$. Then, there
exists an $\varepsilon>0$, and a complex-valued $c_{k,\ell,M}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that with
the same $\boldsymbol{\lambda(\alpha,t)}$ as in Theorem 6.7,

$J_{2}^{(N)}=c_{k,\ell,M}(\alpha,t)\prod_{j=1}^{M-2}\lambda\left(\alpha\frac{j}{M},t\right)\left(1+o_{M}(1)\right)\quad\text{as
}M\to\infty\,,$ (6.18)

for $|t|<\varepsilon$. Furthermore, for any $\alpha>0$,

$\displaystyle\bullet\lambda(\alpha,0)=1\,,$ (6.19)
$\displaystyle\bullet\lim_{t\to
0}c_{k,\ell,M}(\alpha,t)=1\,\quad\text{uniformly in }M\,,$ (6.20)
$\displaystyle\bullet\text{the remainder }o_{M}(1)\text{ is uniformly bounded
in }t\in(-\varepsilon,\varepsilon).$ (6.21)

Moreover, there exist two functions ${\widetilde{\lambda}}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ and
${\widetilde{c}}_{k,\ell,M}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$ such that there
exist two constants $C_{1}^{(2)},C_{2}^{(2)}>0$ and $p\in{\mathbb{N}}$ such
that:

$\displaystyle\bullet
C_{1}^{(1)}N^{p}<|{\widetilde{c}}_{k,\ell,M}(\alpha,t)|<C^{(2)}_{2}N^{p}\,,$
(6.22)
$\displaystyle\bullet\lambda(\alpha,t)=\frac{{\widetilde{\lambda}}(\alpha,t)}{{\widetilde{\lambda}}(\alpha,0)}\,,$
(6.23) $\displaystyle\bullet
I_{2}^{(N)}(H+itU)={\widetilde{c}}_{k,\ell,M}(\alpha,t)\prod_{j=1}^{M-2}{\widetilde{\lambda}}\left(\alpha\frac{j}{M},t\right)\left(1+o_{M}(1)\right)\,.$
(6.24)

###### Proof.

Let $W({\bf x})=H({\bf x})+itU({\bf x})$, and let
$w\,:\,X^{2k}\to{\mathbb{C}}$ be its seed, and
${\widetilde{w}}\,:\,X^{2k+\ell}\to{\mathbb{C}}$ be its weed. For $1\leq j\leq
M-1$, define the function ${\widetilde{F}}({\bf
x},j)\,:\,X^{k}\times{\mathbb{N}}\to{\mathbb{R}}_{+}$ as

${\widetilde{F}}({\bf
x},j)=\sqrt{\prod_{q=1}^{k}F\left(x_{q},\alpha\left(1-\frac{(j-1)k+q}{N}\right)\right)}\,,$
(6.25)

we notice that ${\widetilde{F}}({\bf x},j)\in L^{2}(X^{k})$. Define the family
of kernel operators, ${\mathcal{L}}_{t,\alpha}^{(j)}\,:\,L^{2}(X^{k})\to
L^{2}(X^{k})$ as

${\mathcal{L}}_{t,\alpha}^{(j)}f({\bf y})={\widetilde{F}}({\bf
y},j+1)\int_{X^{k}}f({\bf x}){\widetilde{F}}({\bf x},j)e^{-w({\bf y},{\bf
x})}{\rm d}{\bf x}\,.$ (6.26)

By Lemma 6.6, $(\alpha,t)\mapsto{\mathcal{L}}_{\alpha,t}^{(j)}$ is $C^{1,d}$.
As in the proof of Theorem 6.7, since for $t=0$ the kernel is positive, apply
a generalization of Jentzsch’s theorem [72, Theorem 137.4] in combinations
with Proposition 6.3 and Lemma 6.6 deducing that:

  * •

By the existence of a uniform spectral gap (Lemma 6.6), there exists some
$\varepsilon>0$ independent of $\boldsymbol{j}$ (but depending on $\alpha$)
such that for $|t|<\varepsilon$, ${\mathcal{L}}_{\alpha,t}^{(j)}$ has a
simple, dominant eigenvalue
$\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),t\right)$ with
associated eigenfunction
$\varphi\left(\cdot,\alpha\left(1-\frac{j}{M}\right),t\right)\in L^{2}(X^{k})$
with unit $L^{2}$ norm.

  * •

$\hat{\lambda}\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon))$

  * •

The associated eigenfunction varies smoothly:
$(\alpha,t)\mapsto\varphi\left(\cdot,\alpha\left(1-\frac{j}{M}\right),t\right)\in
C^{1,d}({\mathbb{R}}_{+}^{*}\times(-\varepsilon,\varepsilon),L^{2}(X^{k}))$ ,

and there exists an operator ${\mathcal{Q}}_{t}^{(j)}\,:\,L^{2}(X^{k})\to
L^{2}(X^{k})$ such that $\forall\,\phi\in L^{2}(X^{k})$,
$\forall|t|<\varepsilon$,

${\mathcal{L}}_{t,\alpha}^{(j)}\phi({\bf
y})=\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),t\right)\pi_{t}^{(j)}\phi({\bf
y})+{\mathcal{Q}}_{t}^{(j)}\phi({\bf y})$ (6.27)

with

$\pi_{t}^{(j)}\phi({\bf
y})=\left\langle\phi;\varphi\left(\cdot,\alpha\left(1-\frac{j}{M}\right),t\right)\right\rangle\varphi\left({\bf
y},\alpha\left(1-\frac{j}{M}\right),t\right)\,,$

recalling that we denote by $\left\langle\cdot,\cdot\right\rangle$ the
standard scalar product in $L^{2}(X^{k})$.

Furthermore, we have
$\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),0\right)>0$,
$\varphi\left(\cdot,\alpha\left(1-\frac{j}{M}\right),0\right)>0$ almost
everywhere, and there exists $\delta>0$ such that
$\|{\mathcal{Q}}_{t}^{(j)}\|\leq\left|\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),t\right)\right|-\delta$.

Now, with ${\widetilde{\lambda}}$ the function of Theorem 6.7, we have

$\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),t\right)={\widetilde{\lambda}}\left(\alpha\left(1-\frac{j}{M}\right),t\right)+O\left(\frac{1}{M}\right)\,,$
(6.28)

Where the $O\left(\frac{1}{M}\right)$ term is uniform in
$t\in(-\varepsilon,\varepsilon)$. Indeed, recalling that
${\mathcal{L}}_{t,\alpha}$ is defined in (6.1), by the integrability
assumptions on $F(\cdot,\alpha)$, $\partial_{\alpha}\sqrt{F}$ (see Assumptions
3.1), we have

$\|{\mathcal{L}}^{(j)}_{t,\alpha}-{\mathcal{L}}_{t,\alpha(1-j/M)}\|\leq
C\frac{\alpha}{M}$ (6.29)

where $C\geq 0$ is independent of $t$. We then deduce (6.28) by applying
Proposition 6.3.

Analogously as the previous proof, define the function $G_{{\bf z}}({\bf
y},t)$ on $X^{k}$ by

$G_{\bf z}({\bf y},t)=\begin{cases}\begin{aligned} &{\widetilde{F}}({\bf
y},M)\int_{X^{\ell}}\prod_{j=1}^{\ell-1}F\left(x_{kM+j},\alpha\left(1-\frac{j+kM}{N}\right)\right)\\\
&\times\exp\left(-{\widetilde{w}}({\bf y},x_{kM+1},\ldots,x_{N},{\bf
z})\right)R(x_{N})\prod_{j=kM+1}^{kM+\ell}{\rm
d}x_{j}\,,\end{aligned}&\quad\ell>0\,,\\\ {\widetilde{F}}({\bf
y},M)\int_{X}\exp\left(-w({\bf y},{\bf z})\right)R(z_{k}){\rm
d}z_{k}\,,&\quad\ell=0\,,\end{cases}$ (6.30)

We recall that $W=H+itU$ so $G_{\bf z}({\bf y},t)$ depends on $t$ , and the
linear operator ${\mathcal{S}}\,:\,L^{2}(X^{k})\to{\mathbb{C}}$ as

${\mathcal{S}}(\psi)=\int_{X^{k}}{\widetilde{F}}({\bf x},1)\psi({\bf
x})\prod_{j=1}^{k}{\rm d}x_{j}\,.$ (6.31)

Given the assumptions 3.1, the operator ${\mathcal{S}}$ is uniformly bounded
in $M$. In this notation, we can rewrite $I_{2}^{(N)}(H+itU)$ as

$I_{2}^{(N)}(H+itU)={\mathcal{S}}\left({\mathcal{L}}_{t,\alpha}^{(2)}\ldots{\mathcal{L}}_{t,\alpha}^{(M-1)}G_{{\bf
x}_{1}}({\bf
x}_{M},t)\right)={\mathcal{S}}\left(\prod_{j=2}^{M-1}{\mathcal{L}}_{t,\alpha}^{(j)}G_{{\bf
x}_{1}}({\bf x}_{M},t)\right)$ (6.32)

Applying the decomposition (6.27), it follows that we can decompose the
previous expression as

$\begin{split}I_{2}^{(N)}(H+itU)&=\prod_{j=2}^{M-1}\widehat{\lambda}_{t}\left(\alpha\left(1-\frac{j}{M}\right)\right){\mathcal{S}}\left(\pi_{t}^{(2)}\ldots\pi_{t}^{(M-1)}G_{{\bf
x}_{1}}({\bf x}_{M},t)\right)\\\
&+{\mathcal{S}}\left({\mathcal{L}}_{t,\alpha}^{(2)}\ldots{\mathcal{L}}_{t,\alpha}^{(M-2)}{\mathcal{Q}}_{t}^{(M-1)}G_{{\bf
x}_{1}}({\bf x}_{M},t)\right)\\\
&+\sum_{n=2}^{M-2}\prod_{j=n+1}^{M-1}\widehat{\lambda}_{t}\left(\alpha\left(1-\frac{j}{M}\right)\right){\mathcal{S}}\left({\mathcal{L}}_{t,\alpha}^{(2)}\ldots{\mathcal{L}}_{t,\alpha}^{(n-1)}{\mathcal{Q}}_{t}^{(n)}\pi_{t}^{(n+1)}\ldots\pi_{t}^{(M-1)}G_{{\bf
x}_{1}}({\bf x}_{M},t)\right)\,,\end{split}$ (6.33)

where we arranged the terms of the product of the
${\mathcal{L}}^{(j)}_{t,\alpha}$’s by order of the first appearance from the
right of a factor ${\mathcal{Q}}^{(j)}_{t}$ (the first term being the product
where no ${\mathcal{Q}}^{(j)}_{t}$ appears). We notice that

${\mathcal{S}}\left(\pi_{t}^{(2)}\ldots\pi_{t}^{(M-1)}G_{{\bf x}_{1}}({\bf
x}_{M},t)\right)=\braket{\varphi_{t}^{1/M};G_{{\bf x}_{1}}({\bf
x}_{M},t)}\prod_{i=2}^{M-2}\braket{\varphi_{t}^{\alpha(1-i/M)};\varphi_{t}^{\alpha(1-(i+1)/M)}}{\mathcal{S}}\left(\varphi_{t}^{\alpha\left(1-\frac{2}{M}\right)}\right)\,,$

where we set
$\varphi_{t}^{\alpha(1-i/M)}\equiv\varphi\left(\cdot,\alpha(1-i/M),t\right)$
to shorten the notation. Furthermore, the ratio

$\prod_{i=2}^{M-1}\frac{\langle\varphi_{t}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{t}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle}{\langle\varphi_{0}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{0}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle}$

stays bounded as $M\to\infty$ uniformly in $t\in(-\varepsilon,\varepsilon)$.
This is due to the fact that

$\langle\varphi_{t}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{t}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle=1+O\left(\frac{\alpha}{M}\right),$

because of (6.29) and Proposition 6.3, thus the product

$\prod_{i=2}^{M-1}\langle\varphi_{t}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{t}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle$

stays from bounded below and above uniformly on $M\geq 1$,
$t\in(-\varepsilon,\varepsilon)$.

Denoting the first term of (6.33) by $f(\alpha,t)$, and the second and third
terms by $g_{1}(\alpha,t)$ and $g_{2}(\alpha,t)$, we can rewrite $J_{2}^{(N)}$
(6.16) as

$J_{2}^{(N)}(H+itU)=\frac{f(\alpha,t)}{f(\alpha,0)}\left(\frac{1+\frac{g_{1}(\alpha,t)}{f(\alpha,t)}+\frac{g_{2}(\alpha,t)}{f(\alpha,t)}}{1+\frac{g_{1}(\alpha,0)}{f(\alpha,0)}+\frac{g_{2}(\alpha,0)}{f(\alpha,0)}}\right)\,.$
(6.34)

Thus, to prove our result we need to show that there exist $3$ constants
$c_{1},c_{2},c_{3}$ independent of $M$ such that for all
$t\in(-\varepsilon,\varepsilon)$,

$\displaystyle\left|\frac{g_{1}(\alpha,t)}{f(\alpha,t)}\right|\leq c_{1}\,,$
(6.35) $\displaystyle\left|\frac{g_{2}(\alpha,t)}{f(\alpha,t)}\right|\leq
c_{2}\,,$ (6.36)
$\displaystyle\left|\frac{\left\langle\varphi^{(1/M)}_{t};G_{{\bf x}_{1}}({\bf
x}_{M},t)\right\rangle{\mathcal{S}}\varphi_{t}^{\alpha\left(1-\frac{2}{M}\right)}}{\left\langle\varphi_{0}^{(1/M)};h_{0}\right\rangle{\mathcal{S}}\varphi_{0}^{\alpha\left(1-\frac{2}{M}\right)}}\right|\leq
c_{3}\,.$ (6.37)

If we are able to show this, then defining

$\begin{split}c_{k,\ell,M}(\alpha,t)&=\frac{\left\langle\varphi^{(1/M)}_{t};G_{{\bf
x}_{1}}({\bf
x}_{M},t)\right\rangle{\mathcal{S}}\varphi_{t}^{\alpha\left(1-\frac{2}{M}\right)}}{\left\langle\varphi_{0}^{(1/M)};h_{0}\right\rangle{\mathcal{S}}\varphi_{0}^{\alpha\left(1-\frac{2}{M}\right)}}\prod_{i=2}^{M-1}\frac{\langle\varphi_{t}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{t}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle}{\langle\varphi_{0}^{\alpha\left(1-\frac{i}{M}\right)};\varphi_{0}^{\alpha\left(1-\frac{i+1}{M}\right)}\rangle}\,,\\\
\lambda(y,t)&=\frac{{\widetilde{\lambda}}\left(y,t\right)}{{\widetilde{\lambda}}\left(y,0\right)}\,,\end{split}$
(6.38)

we obtain (6.18) with the wanted properties. Notice that in the definition of
$\lambda$ we took
$\dfrac{{\widetilde{\lambda}}(y,t)}{{\widetilde{\lambda}}(y,0)}$ instead of
$\dfrac{\widehat{\lambda}(y,t)}{\widehat{\lambda}(y,0)}$. This is indeed
possible because of equation (6.28).

First, we focus on (6.36). The term $g_{2}(\alpha,t)$ is given by

$\braket{\varphi^{(1/M)};G_{{\bf x}_{1}}({\bf
x}_{M},t)}\sum_{n=2}^{M-2}\prod_{j=n+1}^{M-1}\widehat{\lambda}\left(\alpha\left(1-\frac{j}{M}\right),t\right)\prod_{i=1}^{M-n-2}\braket{\varphi_{t}^{\frac{\alpha
i}{M}};\varphi_{t}^{\frac{\alpha(i+1)}{M}}}{\mathcal{S}}\left({\mathcal{L}}_{t,\alpha}^{(2)}\ldots{\mathcal{L}}_{t,\alpha}^{(n-1)}{\mathcal{Q}}_{t}^{(n)}\varphi_{t}^{\alpha\left(1-\frac{n+1}{M}\right)}\right)\,.$

Because $\varphi\left({\bf x},y,t\right)$ is regular with respect to $y$, we
deduce that there exists a function $(y,t)\mapsto\psi(\cdot,y,t)\in
C^{\infty}({\mathbb{R}}^{+}\times[-\varepsilon,\varepsilon],L^{2}(X^{k}))$
with $\|\psi_{t}^{\alpha(1-n/M)}\|_{2}$ uniformly bounded in $n,M$ and $t$
such that

${\mathcal{Q}}_{t}^{n}\left(\varphi_{t}^{\alpha\left(1-\frac{n+1}{M}\right)}\right)=Q_{t}^{(n)}\left(\varphi_{t}^{\alpha\left(1-\frac{n}{M}\right)}\right)+\frac{1}{M}{\mathcal{Q}}_{t}^{(n)}\psi_{t}^{\alpha(1-n/M)}=\frac{1}{M}{\mathcal{Q}}_{t}^{(n)}\psi_{t}^{\alpha(1-n/M)}\,.$
(6.39)

given this equality, it is trivial to prove (6.36), recalling that for any
$t,j$, $\dfrac{1}{\widehat{\lambda}_{t}^{(j)}}{\mathcal{L}}_{t,\alpha}^{(j)}$
has operator norm smaller than one.

For (6.35), it suffices to show that there exists a constant $c_{2}$
independent of $M$ such that

$\left\|\frac{{\mathcal{Q}}_{t}^{(M-1)}G_{{\bf x}_{1}}({\bf
x}_{M},t)}{\left\langle\varphi^{(1/M)}_{t},G_{{\bf x}_{1}}({\bf
x}_{M},t)\right\rangle}\right\|\leq c_{2}\,.$ (6.40)

From the assumptions, (6.27) and the definition of $G_{{\bf x}_{1}}({\bf
x}_{M},t)$ (6.30), we deduce that there exists a constant $d_{1}$ such that

$\left\|{\mathcal{Q}}_{j}^{(M-1)}G_{{\bf x}_{1}}({\bf x}_{M},t)\right\|\leq
d_{1}\left(\lambda\left(\frac{\alpha}{M}\right)-\delta\right)M^{{\mathtt{c}}(k+\ell)}\,.$
(6.41)

On the other hand, given the previous proof and the assumptions, we conclude
that, for $t$ small enough, there exists a constant $d_{2}$ such that

$\left|\left\langle\varphi^{(1/M)}_{t},G_{{\bf x}_{1}}({\bf
x}_{M},t)\right\rangle\right|\geq d_{2}M^{{\mathtt{c}}(k+\ell)}\,.$ (6.42)

Indeed, for $t=0$ the previous inequality follows from the definition of
$G_{{\bf x}_{1}}({\bf x}_{M},t)$, and we deduce that for $t$ small enough the
same holds true

Combining the two previous estimates, and setting $p={\mathtt{c}}(k+\ell)$ we
deduce (6.40), which leads to (6.36). The proof of (6.37) is analogous, thus
we conclude. ∎

We now turn on the proof of Theorem 1.11, which we rewrite here for
convenience.

######  Theorem 6.9 (Decay of correlations).

Under hypotheses 1.2. Let $W$ be the seed of $\mathrm{Tr}\left(\Re
P(L)\right)$ and $I,J:X^{k}\to{\mathbb{R}}$ two local functions such that
$\int_{X^{k}\times X^{k}}\left|I({\bf
x})\prod_{i=1}^{k}\sqrt{F(x_{i},\alpha)F(y_{i},\alpha)}e^{-W({\bf x},{\bf
y})}\right|^{2}{\rm d}{\bf x}{\rm d}{\bf y}<\infty$, 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

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

###### Proof.

Let ${\mathcal{L}}={\mathcal{L}}_{0,\alpha}$ with ${\mathcal{L}}_{t,\alpha}$
given by (6.1). Furthermore, define $\mathcal{L}^{(J)}$

$\mathcal{L}^{(J)}\phi({\bf y})=\int_{X^{k}}\phi({\bf
x})\prod_{i=1}^{k}\sqrt{F(x_{i},\alpha)F(y_{i},\alpha)}J({\bf x})e^{-W({\bf
y},{\bf x})}={\mathcal{L}}(J\phi)({\bf y})\,,$

and $\mathcal{L}^{(I)}$ analogously. With $G^{(I)}_{\bf x}({\bf y})=I({\bf
x})G_{\bf x}({\bf y})$, $G_{\bf x}({\bf y})$ given in (6.12), we have for
$j\geq 3$

$\displaystyle{\mathbb{E}}_{1}\left[I({\bf x}_{1})J({\bf x}_{j})\right]$
$\displaystyle=\frac{\mathcal{S}_{0}\left(\left(\mathcal{L}^{M-j}\mathcal{L}^{(J)}\mathcal{L}^{j-3}G^{(I)}_{{\bf
x}_{M}}\right)({\bf x}_{M})\right)}{\mathcal{S}_{0}((\mathcal{L}^{M-2}G_{{\bf
x}_{M}})({\bf x}_{M}))}$
$\displaystyle=\frac{{\widetilde{\lambda}}^{M-j}(\alpha,0)\mathcal{S}_{0}((\pi_{0}\mathcal{L}^{(J)}\mathcal{L}^{j-3}G^{(I)}_{{\bf
x}_{M}})({\bf
x}_{M}))+O({\widetilde{\lambda}}^{j-3}r^{M-j}))}{{\widetilde{\lambda}}^{M-2}(\alpha,0)\mathcal{S}_{0}((\pi_{0}G_{{\bf
x}_{M}})({\bf x}_{M}))+O(r^{M-2})}\,,$

where ${\mathcal{S}}_{t}$ is defined in (6.13), and we used the decomposition

$\mathcal{L}^{k}_{0}={\widetilde{\lambda}}^{k}(\alpha,0)\pi_{0}+\mathcal{Q}_{0}^{k}\,,$

where $\pi_{0}$ is the orthogonal projection on the (one dimensional)
eigenspace associated with ${\widetilde{\lambda}}(\alpha,0)$, and
${\mathcal{Q}}_{0}$ is an operator such that $\|\mathcal{Q}_{0}^{k}\|\leq
Cr^{k}$ for some $0<r<{\widetilde{\lambda}}$. Similarly,

$\mathcal{S}_{0}((\pi_{0}\mathcal{L}^{(J)}\mathcal{L}^{j-3}G^{(I)}_{{\bf
x}_{M}})({\bf
x}_{M}))={\widetilde{\lambda}}^{j-3}(\alpha,0)\mathcal{S}_{0}((\pi_{0}\mathcal{L}^{(J)}\pi_{0}G^{(I)}_{{\bf
x}_{M}})({\bf x}_{M}))+O(r^{j-3}).$

We deduce

${\mathbb{E}}_{1}\left[I({\bf x}_{1})J({\bf
x}_{j})\right]=\frac{\mathcal{S}_{0}((\pi_{0}\mathcal{L}^{(J)}\pi_{0}G^{(I)}_{{\bf
x}_{M}})({\bf
x}_{M}))+O((r/{\widetilde{\lambda}})^{M-j}+(r/{\widetilde{\lambda}})^{j-3})}{{\widetilde{\lambda}}\left(\mathcal{S}_{0}((\pi_{0}G_{{\bf
x}_{M}})({\bf x}_{M}))+O((r/{\widetilde{\lambda}})^{M-2})\right)}\,.$

Similarly, we deduce

${\mathbb{E}}_{1}[I({\bf x}_{1})]{\mathbb{E}}_{1}[J({\bf
x}_{j})]=\frac{\mathcal{S}_{0}((\pi_{0}G^{(I)}_{{\bf x}_{M}})({\bf
x}_{M})){\mathcal{S}}_{0}\left((\pi_{0}{\mathcal{L}}^{(J)}\pi_{0}G_{{\bf
x}_{M}})({\bf
x}_{M})\right)+O((r/{\widetilde{\lambda}})^{M-j}+(r/{\widetilde{\lambda}})^{j-3})}{{\widetilde{\lambda}}\left(\mathcal{S}_{0}((\pi_{0}G_{{\bf
x}_{M}})({\bf x}_{M}))^{2}+O((r/{\widetilde{\lambda}})^{M-2})\right)}\,.$
(6.43)

By a direct computation, recalling that
$\pi_{0}\phi=\braket{\varphi_{1},\phi}\varphi_{1}$ where $\varphi_{1}$ is the
eigenfunction associated with ${\widetilde{\lambda}}$, we deduce the following

${\mathcal{S}}_{0}\left((\pi_{0}{\mathcal{L}}^{(J)}\pi_{0}G_{{\bf
x}_{M}}^{(I)})({\bf
x}_{M})\right)=\braket{{\mathcal{L}}^{(J)}\varphi_{1},\varphi_{1}}\int\braket{G_{{\bf
x}},\varphi_{1}}I({\bf x})F({\bf x})\varphi_{1}({\bf x}){\rm d}{\bf x}\,,$

${\mathcal{S}}\left((\pi_{0}G_{{\bf x}_{M}})({\bf
x}_{M})\right)=\int\braket{G_{{\bf x}},\varphi_{1}}\varphi_{1}({\bf x})F({\bf
x}){\rm d}{\bf x}\,,\quad{\mathcal{S}}_{0}\left((\pi_{0}G^{(I)}_{{\bf
x}_{M}})({\bf x}_{M})\right)=\int\braket{G_{\bf x},\varphi_{1}}I({\bf
x})F({\bf x})\varphi_{1}({\bf x}){\rm d}{\bf x}\,,$

and

${\mathcal{S}}_{0}\left((\pi_{0}{\mathcal{L}}^{(J)}\pi_{0}G_{{\bf
x}_{M}})({\bf
x}_{M})\right)=\braket{{\mathcal{L}}^{(J)}\varphi_{1},\varphi_{1}}\int\braket{G_{{\bf
x}},\varphi_{1}}F({\bf x})\varphi_{1}({\bf x}){\rm d}{\bf x}\,.$

These formulas imply that

$\mathcal{S}_{0}((\pi_{0}\mathcal{L}^{(J)}\pi_{0}G^{(I)}_{{\bf x}_{M}})({\bf
x}_{M}))=\frac{\mathcal{S}_{0}((\pi_{0}G^{(I)}_{{\bf x}_{M}})({\bf
x}_{M})){\mathcal{S}}_{0}\left((\pi_{0}{\mathcal{L}}^{(J)}\pi_{0}G_{{\bf
x}_{M}})({\bf x}_{M})\right)}{{\mathcal{S}}_{0}\left((\pi_{0}G_{{\bf
x}_{M}})({\bf x}_{M})\right)}\,,$

and so

${\mathbb{E}}_{1}\left[I({\bf x}_{1})J({\bf
x}_{j})\right]-{\mathbb{E}}_{1}\left[I({\bf
x}_{1})\right]{\mathbb{E}}_{1}\left[J({\bf
x}_{j})\right]=O((r/{\widetilde{\lambda}})^{M-j}+(r/{\widetilde{\lambda}})^{j-3})\,.$

∎

Finally, we prove a Berry-Esseen bound type theorem for the measure
$\mu_{N}^{(1)}$:

######  Theorem 6.10.

Under Hypotheses 1.2. Consider $s\in{\mathbb{N}}$ and the measure
$\mu_{N}^{(1)}$ (1.31), let $W,h$ be the common seeds of $\mathrm{Tr}\left(\Re
P(L)\right)$ and $\mathrm{Tr}\left(\Re L^{s}\right)$ respectively and let
${\widetilde{W}},{\widetilde{h}}$ be their weeds. Assume that for some $d\geq
3$, the functions $|u|^{a}e^{-h}$, $1\leq a\leq d$ are bounded. 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

$\left|\mathbb{P}\left(\left[\mathrm{Tr}\left(\Re
P(L)\right)-NA\right]/\sqrt{N}\in J\right)-\mathbb{P}\left(Y\in
J\right)\right|\leq\frac{C}{\sqrt{N}}\,.$ (6.44)

###### Proof.

We adapt the arguments of [32, Theorem 3.7]. By [24, Lemma XVI.3.2], there
exists a constant $C$ such that for any $X$ real random variable, and $Y$
Gaussian random variable, for any interval $J\subset{\mathbb{R}}$ and for any
$T>0$, we have

$\left|\mathbb{P}\left(X\in J\right)-\mathbb{P}\left(Y\in J\right)\right|\leq
C\int_{0}^{T}\frac{|\mathbb{E}[e^{-itX}]-e^{-\sigma^{2}t^{2}/2}|}{t}{\rm
d}t+\frac{C}{T}.$

We take $X=\left(\mathrm{Tr}\left(\Re(L^{s})\right)-NA\right)/\sqrt{N}$. We
are going to show that, taking $T=\varepsilon\sqrt{N}$ for some small enough
$\varepsilon$, the last integral remains bounded by
$\frac{C_{k,\ell}}{\sqrt{N}}$, where $C_{k,\ell}$ is a constant depending on
$k,\ell$. Recall $N=kM+\ell$. By Theorem 1.5, there exists an
$A\in{\mathbb{R}},\sigma>0$ such that as $N$ goes to infinity $X$ converges to
${\mathcal{N}}(0,\sigma^{2})$. Since $t^{-1}$ is not integrable at $0$, we
consider the special interval $[0,N^{-1}]$. In this interval, we have the
following estimate:

$\begin{split}\left|\mathbb{E}\left[e^{-itX}\right]-e^{-it\sqrt{N}A}\right|&\stackrel{{\scriptstyle\eqref{eq:ration_BE}}}{{=}}\frac{\lvert
Z_{N}^{(1)}\left(\alpha,\Re P+i\frac{t}{\sqrt{N}}\Re
z^{s}\right)-Z_{N}^{(1)}\left(\alpha,\Re
P\right)\rvert}{Z_{N}^{(1)}\left(\alpha,\Re P\right)}\\\
&=\frac{1}{Z_{N}^{(1)}\left(\alpha,\Re
P\right)}\Big{|}\sum_{p=1}^{M-1}\int_{X^{kM+\ell}}\mathfrak{F}({\bf
x})\prod_{j=1}^{p-1}e^{i\frac{t}{\sqrt{N}}h({\bf x}_{j},{\bf
x}_{j+1})}\left(e^{i\frac{t}{\sqrt{N}}h({\bf x}_{p},{\bf
x}_{p+1})}-1\right)\prod_{j=1}^{N}{\rm d}x_{j}\\\
&+\int_{X^{kM+\ell}}\mathfrak{F}({\bf
x})\left(e^{i\frac{t}{\sqrt{N}}{\widetilde{h}}({\bf
x}_{M},x_{kM+1},\ldots,x_{kM+\ell},{\bf
x}_{1})}-1\right)\Big{|}\,,\end{split}$ (6.45)

with the convention that the empty product is equal to one. Here we defined

$\begin{split}\mathfrak{F}({\bf
x})=&\prod_{j=1}^{N}F(x_{j},\alpha)\exp\left(-\sum_{j=1}^{M-1}W({\bf
x}_{j},{\bf x}_{j+1})-{\widetilde{W}}({\bf
x}_{M},x_{kM+1},\ldots,x_{kM+\ell},{\bf x}_{1})\right)\\\
&\times\exp\left(-\frac{it}{\sqrt{N}}{\widetilde{h}}({\bf
x}_{M},x_{kM+1},\ldots,x_{kM+\ell},{\bf x}_{1})\right)\,,\end{split}$ (6.46)

Thus, since $\lvert e^{i\frac{t}{\sqrt{N}}h({\bf x}_{p},{\bf
x}_{p+1})}-1\rvert\leq|h({\bf x}_{p},{\bf x}_{p+1})|N^{-1/2}t$, we deduce the
following inequality

$\left|\mathbb{E}_{1}[e^{-itX}]-e^{-it\sqrt{N}A}\right|\leq\mathbb{E}_{1}\left[|h({\bf
x}_{1},{\bf
x}_{2})|\right]t\sqrt{N}+\frac{t}{\sqrt{N}}\mathbb{E}_{1}\left[|{\widetilde{h}}({\bf
x}_{M},x_{kM+1},\ldots,x_{kM+\ell},{\bf x}_{1})|\right]\,,$ (6.47)

and this last term is by assumption bounded by $Ct\sqrt{N}$ for some $C$
independent of $N$ and $t$. Thus integrating for $t\in[0,N^{-1}]$ we deduce
the following

$\begin{split}\int_{0}^{\frac{1}{N}}&\frac{|\mathbb{E}_{1}[e^{-itX}]-e^{-\sigma^{2}t^{2}/2}|}{t}{\rm
d}t\\\
&\leq\int_{0}^{\frac{1}{N}}\frac{\left|\mathbb{E}_{1}\left[e^{-itX}\right]-e^{-it\sqrt{N}A}\right|+\left|e^{-it\sqrt{N}A}-1\right|+\left|1-e^{-\sigma^{2}t^{2}/2}\right|}{t}{\rm
d}t\\\
&\stackrel{{\scriptstyle\eqref{eq:small_set_2}}}{{\leq}}\int_{0}^{\frac{1}{N}}\frac{C\sqrt{N}t+t\sqrt{N}A+\sigma^{2}t^{2}/2}{t}{\rm
d}t\leq\frac{C_{1}}{\sqrt{N}}\,,\end{split}$ (6.48)

for some constant $C_{1}$.

We now consider the integral on $[1/N,\varepsilon\sqrt{N}]$. Here we use the
spectral decomposition of $\mathbb{E}_{1}[e^{itX}]$. Since $|h|^{a}e^{-W}$ is
bounded for $1\leq a\leq 3$, we deduce by Proposition 6.3, Remark 1.8 and
Theorem 1.5 the following third order expansion and that there exist two
continuous functions $p(t)\in C^{0}([-\varepsilon,\varepsilon])$ and
$c_{k,\ell}(y,t)\in C^{1,d}({\mathbb{R}}\times[-\varepsilon,\varepsilon])$ for
some $\varepsilon>0$, such that $c_{k,\ell}(y,0)=1$ and
$\|p\|_{\infty}<+\infty$, and for $q\geq 1$

$\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{|\mathbb{E}_{1}[e^{-itX}]-e^{-\sigma^{2}t^{2}/2}|}{t}{\rm
d}t=\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|c_{k,\ell}(\alpha,t/\sqrt{N})e^{-\sigma^{2}t^{2}/2+t^{3}p(t/\sqrt{N})/\sqrt{N}}(1+o(N^{-q}))-e^{-\sigma^{2}t^{2}/2}\right|}{t}{\rm
d}t\,,$ (6.49)

thus we have the following estimate

$\begin{split}\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|\mathbb{E}_{1}[e^{-itX}]-e^{-\sigma^{2}t^{2}/2}\right|}{t}{\rm
d}t&\leq\left|\left|c_{k,\ell}(\alpha,\cdot)(1+o(M^{-q}))\right|\right|_{\infty,[0,\varepsilon]}\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|\left(1-e^{t^{3}p(t/\sqrt{N})/\sqrt{N}}\right)e^{-\sigma^{2}t^{2}/2}\right|}{t}{\rm
d}t\\\
&+\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|1-c_{k,\ell}(\alpha,t/\sqrt{N})(1+o(M^{-q}))\right|e^{-\sigma^{2}t/2}}{t}{\rm
d}t\,,\end{split}$ (6.50)

where $||\cdot||_{\infty,[0,\varepsilon]}$ in the $L^{\infty}$ norm on
$[0,\varepsilon]$.

We notice that
$\left|\left|c_{k,\ell}(\alpha,\cdot)(1+o(M^{-q}))\right|\right|_{\infty,[0,\varepsilon]}$
is uniformly bounded in $N$. Moreover,

$\begin{split}\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{e^{-\sigma^{2}t^{2}/2}}{t}\left|e^{t^{3}p(t/\sqrt{N})/\sqrt{N}}-1\right|{\rm
d}t&\leq\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{e^{-\sigma^{2}t^{2}/2}}{t\sqrt{N}}e^{t^{3}\|p\|_{\infty,[0,\varepsilon]}/\sqrt{N}}t^{3}\|p\|_{\infty,[0,\varepsilon]}{\rm
d}t\\\
&\leq\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{e^{-\sigma^{2}t^{2}/2}}{\sqrt{N}}e^{t^{2}\varepsilon\|p\|_{\infty,[0,\varepsilon]}}t^{2}\|p\|_{\infty,[0,\varepsilon]}{\rm
d}t\,,\end{split}$ (6.51)

where in the first inequality we used the bound $|e^{x}-1|\leq|x|e^{|x|}$.
Since for $\varepsilon$ small enough $||p||_{\infty}\varepsilon<\sigma^{2}/4$,
thus integrating, we deduce that

$\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|e^{-\sigma^{2}t^{2}/2+t^{3}p(t/\sqrt{N})/\sqrt{N}}-e^{-\sigma^{2}t^{2}/2}\right|}{t}{\rm
d}t=O\left(\frac{1}{\sqrt{N}}\right)\,.$ (6.52)

To conclude, we have to show that the last integral is of order $N^{-1/2}$.
Since $c_{k,\ell}(\alpha,t)$ is $C^{1}$ in $t$, and $c_{k,\ell}(\alpha,0)=1$,
it is easy to deduce that there exists a constant $C$ such that

$\int_{\frac{1}{N}}^{\varepsilon\sqrt{N}}\frac{\left|1-c_{k,\ell}(\alpha,t/\sqrt{N})(1+o(M^{-q}))\right|e^{-\sigma^{2}t/2}}{t}{\rm
d}t\leq\frac{C}{\sqrt{N}}\,$ (6.53)

so we conclude. ∎

##  7 Conclusion and Outlooks

In this paper, we proved a general Central Limit Theorem type result and we
apply it to several models in random matrix theory and integrable systems. By
doing this, we strengthen the connection between these two subjects.
Specifically, we could connect the expected values and the variances of the
moments of each classical $\beta$-ensemble in the high-temperature regime with
one specific integrable model, see Table 1.

The results that we have obtained are relevant for two main reasons. Under the
random matrix theory perspective, we were able to develop a general framework
to prove polynomial central limit theorems for the classical $\beta$-ensemble
in the high-temperature regime, based on their band matrix representation and
on the transfer operator technique. Under the integrable systems’ theory point
of view, our result enables the explicit computation of the so-called
susceptibility matrix, which is a fundamental object in the theory of
Generalized Hydrodynamics to compute the correlation functions for integrable
models. Furthermore, we can prove rigorously the exponential decay of
correlation for short-range interacting systems with polynomial potential.

It would be fascinating to generalize our result to a wider class of potential
and functions and to obtain a Berry-Esseen bound for the classical
$\beta$-ensemble in the high-temperature regime. Furthermore, defining a new
$\beta$-ensemble related to the INB lattice would be interesting. Finally, we
point out that it would be interesting to obtain large deviation principles
for the Exponential Toda lattice and the Volterra one in the spirit of [36,
53].

##### Acknowledgements

The authors warmly thank Alice Guionnet for giving the initial idea of the
proof of the polynomial CLT, and Herbert Spohn for the useful discussions
about the applications of our result. We also aknowledge Sébastien Gouëzel and
two anonymous referees for useful remarks.

R.M. is supported by ERC Project LDRAM : ERC-2019-ADG Project 884584.

G.M. is financed by the KAW grant number 2018.0344

##  Appendix A Proof of Lemma 1.3

In this section we prove Lemma 1.3, we report here the statement for reader
convenience:

######  Lemma A.1.

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

To prove this lemma, we need some preparation. First, we introduce the family
of shift operators $S_{\ell}$, for any $\ell\in{\mathbb{Z}}$, and ${\bf
x}=(x_{1},x_{2},\ldots,x_{N})\in X^{N}$ we define the cyclic shift of order
$\ell$ as the map

$S_{\ell}\colon X^{N}\to X^{N},\qquad(S_{\ell}x)_{j}:=x_{((j+\ell-1){\,{\rm
mod}\,}N)+1}.$ (A.1)

For example $S_{1}$ and $S_{-1}$ are the left respectively right shifts:

$S_{1}(x_{1},x_{2},\ldots,x_{N}):=(x_{2},\ldots,x_{N},x_{1}),\qquad
S_{-1}(x_{1},x_{2},\ldots,x_{N}):=(x_{N},x_{1},\ldots,x_{N-1}).$

It is immediate to check that for any $\ell,\ell^{\prime}\in{\mathbb{Z}}$,
cyclic shifts fulfills:

$S_{\ell}\circ S_{\ell^{\prime}}=S_{\ell+\ell^{\prime}},\qquad
S_{\ell}^{-1}=S_{-\ell},\qquad S_{0}=\mathds{1},\qquad S_{\ell+2N}=S_{\ell}.$
(A.2)

Consider now a a function $H\colon X^{N}\to{\mathbb{C}}$; we shall denote
$S_{\ell}H\colon X^{N}\to{\mathbb{C}}$ as the operator

$(S_{\ell}H)({\bf x}):=H(S_{\ell}{\bf x}),\qquad\forall{\bf x}\in X^{N}\,.$
(A.3)

Clearly, $S_{\ell}$ is a linear operator. We can now define cyclic functions:

######  Definition A.2.

A function $H\colon X^{N}\to{\mathbb{C}}$ is called cyclic if $S_{1}H=H$.

It follows from the definition that a cyclic function fulfils $S_{\ell}H=H$
$\,\forall\ell\in{\mathbb{Z}}$.
On can construct cyclic functions as follows: given a function $h\colon
X^{N}\to{\mathbb{C}}$ we define the new function $H$ by

$H({\bf x}):=\sum_{\ell=0}^{2N-1}(S_{\ell}h)({\bf x}).$ (A.4)

$H$ is clearly cyclic, and we say that $H$ is generated  by $h$. We notice
that these definitions were previously introduced in [30, 34]

Our first aim is to show that the trace of powers of the matrices of type $1$
are cyclic functions and that the ones of the matrices of type 2 are cyclic up
to a certain function. To this end, we introduce the companion matrices $P,T$
as

$P=\begin{pmatrix}0&1&\\\ 0&0&1&\\\ \vdots&\ddots&\ddots&\ddots\\\
0&&\ddots&\ddots&1\\\ 1&0&\ldots&\ldots&0\end{pmatrix}\,\qquad
T=\begin{pmatrix}0&1&\\\ 0&0&1\\\ &\ddots&\ddots&\ddots\end{pmatrix}\,,$ (A.5)

so $T$ is a sparse matrix, with just the first upper diagonal filled with
ones, and $P$ is similar to $T$, except that there is an extra $1$ in the
bottom left corner.

Using this notation, we can prove the following lemma Toeplitz-like and
circular-like matrices as follows

######  Lemma A.3.

Fix $m,N\in\mathbb{N}$. Given the $2m+1$ sequences
$\\{a_{j}^{(k)}\\}_{j=1}^{N}$ for $k=-m,m$ and defining the two matrices

$\begin{split}&A_{1}=\sum_{j=1}^{m}\left(\textbf{diag}\left({\bf
a}^{(j)}\right)P^{j}+\textbf{diag}\left({\bf
a}^{(-j)}\right)P^{N-j}\right)+\textbf{diag}\left({\bf a}^{(0)}\right)\,,\\\
&A_{2}=\sum_{j=1}^{m}\left(\textbf{diag}\left({\bf
a}^{(j)}\right)T^{j}+\textbf{diag}\left({\bf
a}^{(-j)}\right)(T^{\intercal})^{j}\right)+\textbf{diag}\left({\bf
a}^{(0)}\right)\,.\end{split}$ (A.6)

Here we denoted by $\textbf{diag}\left({\bf a}\right)$ the diagonal matrices
with entries $a_{1},\ldots,a_{N}$. Then for all $\ell\in\mathbb{N}$,
$\mathrm{Tr}\left(A_{1}^{\ell}\right)$ is cyclic, and
$\mathrm{Tr}\left(A_{2}^{\ell}\right)=\mathrm{Tr}\left(A_{1}^{\ell}\right)+K({\bf
a})$, where $K({\bf a})$ depends on a number of variables which is uniformly
bounded in $N$. We call the matrices having the shape of $A_{1}$ circular-like
matrices, and the one having the shape of $A_{2}$ Toeplitz-like matrices

###### Proof.

First, we focus on the matrix $A_{1}$, we expand the trace as

$\mathrm{Tr}\left(A_{1}^{\ell}\right)=\sum_{i_{1},\ldots,i_{\ell}=1}^{N}\prod_{j=1}^{\ell-1}(A_{1})_{i_{j},i_{j+1}}(A_{1})_{i_{\ell},i_{1}}=\sum_{i_{1},\ldots,i_{\ell}=1}^{N}\prod_{j=1}^{\ell-1}a^{(i_{j}-i_{j+1})}_{i_{j}}a^{(i_{\ell}-i_{1})}_{i_{\ell}}\,,$
(A.7)

where in the second equality, we used the structure of $A_{1}$, and we use the
convention that if $|i_{j}-i_{j+1}|>m$ then $a^{(i_{j}-i_{j+1})}_{i_{j}}=0$;
from the previous expression is easy to see that
$\mathrm{Tr}\left(A_{1}^{\ell}\right)$ is cyclic. Regarding the matrix
$A_{2}$, we notice that $T$ is a rank $1$ perturbation of $P$, thus by
developing the trace as before, one can easily obtain the claim. ∎

######  Remark A.4.

We notice that the matrices of type $1-2$ are Toeplitz-like or circular-like.

Using this Lemma, we can prove the following Corollary

######  Corollary A.5.

Fix $m,N\in\mathbb{N}$. Let ${\bf x}$ be the coordinates of the manifold
$X^{N}$. Given the $2m+1$ sequences $\\{a_{j}^{(k)}({\bf x})\\}_{j=1}^{N}$ for
$k=-m,\ldots,m$ such that $a_{j}^{(k)}({\bf x})$ is local for all $j,k$, then
the trace of powers of Toepliz-like or circular-like matrices are circular in
the sense of Definition 1.1

###### Proof.

From Lemma A.3 we know that the trace of powers of Circular-like matrices are
cyclic, and they can be written as

$\mathrm{Tr}\left(A_{1}^{\ell}\right)=\sum_{i_{1},\ldots,i_{\ell}=1}^{N}\prod_{j=1}^{\ell-1}a^{(i_{j}-i_{j+1})}_{i_{j}}({\bf
x})a^{(i_{\ell}-i_{1})}_{i_{\ell}}({\bf x})\,.$ (A.8)

In the previous expression, gather all the terms involving $x_{1}$, and
compute the diameter $d$ of the resulting function, given the locality
property of $a_{j}^{(k)}({\bf x})$ and the structure of the single terms of
the product in (A.8), we deduce that $0\leq d\leq D(m,\ell)$, thus do not
depend on $N$, moreover, it is an even number because of the cyclicity
property.

Consider the function
$h^{(\ell)}(x_{1},\ldots,x_{d+1})=\frac{1}{2}h^{(\ell)}_{1}(x_{1},\ldots,x_{d/2+1})+\frac{1}{2}h^{(\ell)}_{1}(x_{d/2+1},\ldots,x_{d})+h^{(\ell)}_{2}(x_{1},\ldots,x_{d+1})+h_{3}(x_{d/2+1})$
where

  * •

$h^{(\ell)}_{1}(x_{1},\ldots,x_{d/2+1})$ contains all the terms in (A.8) that
involves just $x_{1},\ldots,x_{d/2+1}$

  * •

$h^{(\ell)}_{2}(x_{1},\ldots,x_{d+1})$ contains all the terms in (A.8) that
involves just $x_{1},\ldots,x_{d+1}$, at least one variable among
$x_{1},\ldots,x_{d/2}$, and at least one variable among
$x_{d/2+2},\ldots,x_{d+1}$

  * •

$h_{3}(x_{d/2+1})$ contains terms involving just $x_{d/2+1}$

Given this construction, it is easy to see that
$h^{(\ell)}(x_{1},\ldots,x_{d+1})$ is the seed of
$\mathrm{Tr}\left(A_{1}^{\ell}\right)$.

Regarding the Toeplitz-like matrices, we get the claim because of their
relation with the circular-like proved in Lemma A.3.

∎

######  Remark A.6.

The previous lemma implies that the trace of powers of type $1-2$ matrices are
circular.

As an example, we consider $L$ to be a periodic Jacobi matrix, then

$\mathrm{Tr}\left(L^{4}\right)=\sum_{i=1}^{N}\left[a_{i}^{4}+4(a_{i}^{2}+a_{i}a_{i+1}+a_{i+1}^{2})b_{i}^{2}+2b^{4}_{i}+4b_{i}^{2}b_{i+1}^{2}\right]\,,$

then $d=2$ and

$\begin{split}&h_{1}(a_{1},b_{1},a_{2},b_{2})=4a_{1}a_{2}b_{1}^{2}+4a_{2}^{2}b_{1}^{2}+4b^{2}_{1}b^{2}_{2}\,,\\\
&h_{1}(a_{N},b_{N},a_{1},b_{1},a_{2},a_{3})=0\,,\\\
&h_{3}(a_{1},b_{1})=a_{1}^{4}+4a_{1}^{2}b_{1}^{2}+2b_{1}^{4}\,.\end{split}$
(A.9)

##  Appendix B Proof of Lemma 6.6

######  Lemma B.1.

Under assumptions 3.1. For $\alpha>0$, $t\in{\mathbb{R}}$, consider the
operators ${\mathcal{L}}_{\alpha,t},{\mathcal{L}}_{t,\alpha}^{(j)}$, $1\leq
j\leq M-1$ given by Definition 6.5. Assume that for some $d\geq 2$,
$|u|^{a}\exp\left(-h\right)$ is bounded for any $1\leq a\leq d$, then

  * 1)

Both ${\mathcal{L}}_{\alpha,t}$ and ${\mathcal{L}}_{t,\alpha}^{(j)}$ are
Hilbert-Schmidt.

  * 2)

For any fixed $t\in{\mathbb{R}}$, the $\mathbf{L}(L^{2}(X^{K}))$-valued
functions $\alpha\mapsto{\mathcal{L}}_{\alpha,t}$ and
$\alpha\mapsto{\mathcal{L}}_{\alpha,t}^{(j)}$ are continuously differentiable.

  * 3)

For any fixed $\alpha>0$, $t\mapsto{\mathcal{L}}_{\alpha,t}$ and
$\alpha\mapsto{\mathcal{L}}_{\alpha,t}^{(j)}$ are $d$-times continuously
differentiable.

  * 4)

For all $\alpha>0$, define

${\mathcal{B}}_{\alpha}=\frac{1}{\|{\mathcal{L}}_{\alpha,0}\|_{\mathrm{HS}}}{\mathcal{L}}_{\alpha,0}$

and

${\mathcal{B}}_{\alpha}^{(j)}=\frac{1}{\|{\mathcal{L}}_{\alpha,0}^{(j)}\|_{\mathrm{HS}}}{\mathcal{L}}_{\alpha,0}^{(j)},\
1\leq j\leq M-1\,.$

Then, ${\mathcal{B}}_{\alpha}$ (resp. ${\mathcal{B}}_{\alpha}^{(j)}$) has a
dominant, simple eigenvalue $\mu(\alpha)>0$ (resp. $\mu^{(j)}(\alpha)$).
Furthermore, for all $A>0$ there exists $\delta>0$ such that and all
$\alpha\in(0,A]$, any other eigenvalue $\mu$ satisfies
$|\mu-\mu(\alpha)|\geq\delta$ (resp. $|\mu-\mu^{(j)}(\alpha)|\geq\delta$). We
say that those operators have a uniform spectral gap in $\alpha\in(0,A]$.

######  Remark B.2.

We notice that $1)$ and $2)$ imply that
${\mathcal{L}}_{\alpha,t},{\mathcal{L}}_{t,\alpha}^{(j)}\in
C^{1,d}((0,+\infty)\times{\mathbb{R}},\mathbf{L}(L^{2}(X^{k})))$

###### Proof.

Proof of 1) Since $F\in L^{1}(X)$ and that $h$ is bounded away from $-\infty$,
we easily see that $k_{\alpha,t}$ is $L^{2}(X^{k}\times X^{k})$. Therefore
${\mathcal{L}}_{\alpha,t}$ is Hilbert Schmidt, and the same goes for
${\mathcal{L}}_{\alpha,t}^{(j)}$, $1\leq j\leq M-1$.

Proof of 2) & 3) We show these points for ${\mathcal{L}}_{\alpha,t}$, the
proofs for ${\mathcal{L}}_{t,\alpha}^{(j)}$ being similar. Let us show that
for any fixed $t$, $\alpha\mapsto{\mathcal{L}}_{\alpha,t}$ is continuously
differentiable with derivative given by

$\left(\partial_{\alpha}{\mathcal{L}}_{\alpha,t}\right)u({\bf
y})=\int_{X^{k}}\partial_{\alpha}k_{\alpha,t}({\bf x},{\bf y})u({\bf x}){\rm
d}{\bf x}\,.$

Let $\alpha>0$ and $h\in{\mathbb{R}}$ such that $\alpha+h>0$. For $\varphi\in
L^{2}(X^{k})$ we have

$\displaystyle\left\|\frac{1}{h}\left({\mathcal{L}}_{\alpha+h,t}-{\mathcal{L}}_{\alpha,t}\right)\varphi-\left(\partial_{\alpha}{\mathcal{L}}_{\alpha,t}\right)\varphi\right\|_{2}^{2}$
$\displaystyle\leq\int_{X^{k}}\left(\int_{X^{k}}\left|\frac{1}{h}\big{(}k_{\alpha+h,t}({\bf
x},{\bf y})-k_{\alpha,t}({\bf x},{\bf
y})\big{)}-\partial_{\alpha}k_{\alpha}({\bf x},{\bf y})\right||\varphi({\bf
x})|{\rm d}{\bf x}\right)^{2}{\rm d}{\bf y}$
$\displaystyle\leq\|\varphi\|_{2}^{2}\int_{X^{k}\times
X^{k}}\left|\frac{1}{h}\big{(}k_{\alpha+h,t}({\bf x},{\bf
y})-k_{\alpha,t}({\bf x},{\bf y})\big{)}-\partial_{\alpha}k({\bf x},{\bf
y})\right|^{2}{\rm d}{\bf x}{\rm d}{\bf y}\,,$

where $\|\cdot\|_{2}$ is the standard $L^{2}$ norm, and we used Cauchy-
Schwartz inequality in the last line.

Now,

$\displaystyle\partial_{\alpha}k({\bf x},{\bf y})$ $\displaystyle=e^{-h({\bf
x},{\bf
y})}\partial_{\alpha}\prod_{q=1}^{k}\sqrt{F(x_{q},\alpha)F(y_{q},\alpha)}$
$\displaystyle=e^{-h({\bf x},{\bf
y})}\left(\left(\sum_{q=1}^{k}\partial_{\alpha}\sqrt{F(x_{q},\alpha)}\prod_{j\neq
q}F(x_{j}\alpha)\right)\prod_{q=1}^{k}\sqrt{F(y_{q},\alpha)}+\prod_{q=1}^{k}\sqrt{F(x_{q},\alpha)}\times\partial_{\alpha}\prod_{q=1}^{k}\sqrt{F(y_{q},\alpha)}\right)\,.$

Because of the domination hypothesis on
$\partial_{\alpha}\sqrt{F(\cdot,\alpha)}$ HP.3 and using that $e^{-h}\leq
c_{k}\in{\mathbb{R}}_{+}$ since $h$ is bounded away from $-\infty$ HP.4, we
can bound for $\alpha\in[c,d]\subset(0,+\infty)$

$|\partial_{\alpha}k({\bf x},{\bf y})|\leq
2c_{k}k\prod_{q=1}^{k}g(x_{q})\prod_{q=1}^{k}g(y_{q})\in L^{2}(X^{k}\times
X^{k})\,,$

where $g=g_{[c,d]}\in L^{2}(X)$ is a function given by HP.3, independent of
$\alpha$. We deduce by dominated convergence, taking $h\to 0$ that

$\frac{1}{h}\left({\mathcal{L}}_{\alpha+h,t}-{\mathcal{L}}_{\alpha,t}\right)=\partial_{\alpha}{\mathcal{L}}_{\alpha,t}+o_{h}(1)\,,$

where the $o(1)$ is with respect to the operator norm $\|\cdot\|$ on
$\mathbf{L}(L^{2}(X^{k}))$, establishing that
$\alpha\mapsto{\mathcal{L}}_{\alpha,t}$ is differentiable. By a similar
dominated convergence argument, one shows that
$\alpha\mapsto\partial_{\alpha}{\mathcal{L}}_{\alpha,t}$ is continuous,
establishing that ${\mathcal{L}}_{\alpha,t}$ is continuously differentiable
with respect to $\alpha$.
Similarly, let $\alpha>0$ be fixed and let $1\leq a\leq d$. Using that
$|u^{a}|e^{-h}$ is bounded by some $D_{k}\in{\mathbb{R}}_{+}$,

$\displaystyle\partial_{t}^{a}k_{\alpha,t}({\bf x},{\bf y})$
$\displaystyle=(-i)^{a}u({\bf x},{\bf
y})^{a}\prod_{q=1}^{k}\sqrt{F(x_{q},\alpha)F(y_{q},\alpha)}e^{-h({\bf y},{\bf
x})-itu({\bf y},{\bf x})}$ $\displaystyle\leq
D_{k}\prod_{q=1}^{k}\sqrt{F(x_{q},\alpha)F(y_{q},\alpha)}\in L^{2}(X^{k}\times
X^{k})\,,$

obtaining an $L^{2}$ bound independent of $t$ for
$\partial_{t}^{a}k_{\alpha,t}$. Therefore, by dominated convergence,
$t\mapsto{\mathcal{L}}_{\alpha,t}$ is at least $d$ times differentiable, with
continuous derivative of order $d$.

Proof of 4) We now establish the uniform spectral gap property for
${\mathcal{B}}_{\alpha}$. The proof for ${\mathcal{B}}_{\alpha}^{(j)}$ is
similar. First, by a generalization of Jentzsch’s theorem, because the kernel
$k_{\alpha}({\bf x},{\bf y})$ is positive, ${\mathcal{B}}_{\alpha,0}$ has a
simple, positive, dominant eigenvalue [72, Theorem 137.4], which we denote by
$\mu(\alpha).$ By continuity of $\alpha\mapsto{\mathcal{B}}_{\alpha}$, for
each $a>0$ we can find $\delta>0$ as in Proposition 6.3 applied to the family
$({\mathcal{B}}_{\alpha})_{\alpha>0}$ which is uniform in $\alpha\in[a,A]$.
Thus we only need to check that for some $a>0$, ${\mathcal{B}}_{\alpha}$ has a
uniform spectral gap in $\alpha\in(0,a]$.

Our strategy is based on the observation that for a kernel of the form $k({\bf
x},{\bf y})=f({\bf x})g({\bf y})$ for $0<f,g\in L^{2}(X^{k})$, the associated
kernel operator $T$ is of rank one, so it has a simple dominant eigenvalue
$\lambda=\int_{X^{k}}g({\bf x})f({\bf x}){\rm d}{\bf x}>0$ associated with the
eigenfunction $g$.

With this in mind, we want to use the fact that in
$\mathcal{O}_{\varepsilon}$, defined in 3.1 HP 4 the expression $\exp(-h({\bf
x},{\bf y}))$ factorizes into $\exp(-w({\bf x}))\exp(-w({\bf y}))$.

Step 1: Define, for $\varepsilon>0$,

${\mathcal{L}}^{\varepsilon}_{\alpha}u({\bf
y})=\int_{\mathcal{O}_{\varepsilon}}k_{\alpha,0}({\bf x},{\bf y})u({\bf
x}){\rm d}{\bf x}\,,$

where we recall that $k_{\alpha,0}$ is the kernel of ${\mathcal{L}}_{\alpha}$

Consider the following operator

${\mathcal{B}}_{\alpha}^{\varepsilon}=\frac{1}{\|{\mathcal{L}}_{\alpha}\|_{\mathrm{HS}}}{\mathcal{L}}^{\varepsilon}_{\alpha}\,.$

We first show that for all $\varepsilon>0$,
$\left\|{\mathcal{B}}_{\alpha}-{\mathcal{B}}_{\alpha}^{\varepsilon}\right\|_{\mathrm{HS}}\underset{\alpha\to
0}{\to}0\,.$

Indeed, with $c_{\alpha}=\|{\mathcal{L}}_{\alpha}\|_{\mathrm{HS}}$, and
denoting for simplicity

$\mathsf{F}_{\alpha}({\bf x})=\prod_{j=1}^{k}F(x_{j},\alpha)\,,$

we have

$\left\|{\mathcal{B}}_{\alpha}-{\mathcal{B}}_{\alpha}^{\varepsilon}\right\|_{\mathrm{HS}}\leq\left\|\frac{{\mathcal{L}}_{\alpha}}{c_{\alpha}}-\frac{{\mathcal{L}}_{\alpha}^{\varepsilon}}{c_{\alpha}}\right\|=\frac{1}{c_{\alpha}}\left(\iint_{\mathcal{O}_{\varepsilon}^{c}\times\mathcal{O}_{\varepsilon}^{c}}\mathsf{F}_{\alpha}({\bf
x})\mathsf{F}_{\alpha}({\bf y})e^{-2h({\bf y},{\bf x})}{\rm d}{\bf x}{\rm
d}{\bf y}\right)^{1/2}\,.$ (B.1)

Applying 1.2 HP 4, we can conclude that for any fixed $\varepsilon>0$,

$\lim_{\alpha\to
0}\left\|{\mathcal{B}}_{\alpha}-{\mathcal{B}}_{\alpha}^{\varepsilon}\right\|_{\mathrm{HS}}=0\,.$
(B.2)

Step 2: Define $\mathcal{T}_{\alpha}^{\varepsilon}$ as

$\mathcal{T}_{\alpha}^{\varepsilon}u({\bf
y})=\frac{1}{c_{\alpha}}\int_{\mathcal{O}_{\varepsilon}}\sqrt{\mathsf{F}_{\alpha}({\bf
x})}\sqrt{\mathsf{F}_{\alpha}({\bf y})}e^{-w({\bf x})-w({\bf y})}{\rm d}x\,.$

Then, by the previous remark, $\mathcal{T}_{\alpha}^{\varepsilon}$ is a rank
one operator with a leading simple eigenvalue, and it has a spectral gap
independent of $\alpha$. Meaning that there exists a $\delta$ independent of
$\alpha$ such that $\|\mathcal{T}_{\alpha}^{\varepsilon}\|>\delta$.
Furthermore, as in step 1, we show that
$\|{\mathcal{B}}_{\alpha}^{\varepsilon}-\mathcal{T}_{\alpha}^{\varepsilon}\|_{\mathrm{HS}}\underset{\varepsilon\to
0}{\to}0$ uniformly in $\alpha>0$. Therefore, proposition 6.3 applies with a
uniform $\delta>0$: i.e. there exists $\varepsilon_{0}>0$ such that for all
$\varepsilon<\varepsilon_{0}$ and for all $\alpha>0$,
${\mathcal{B}}_{\alpha}^{\varepsilon}$ has a simple, dominant eigenvalue
${\widetilde{\mu}}(\alpha)$, such that any other eigenvalue $\mu$ of
${\mathcal{B}}_{\alpha}^{\varepsilon}$ satisfies
$|\mu-{\widetilde{\mu}}(\alpha)|>2\delta$.

Conclusion: We now apply the result of step 1 with
$\varepsilon=\varepsilon_{0}$: we know that
$\|{\mathcal{B}}_{\alpha}-{\mathcal{B}}_{\alpha}^{\varepsilon_{0}}\|_{\mathrm{HS}}\underset{\alpha\to
0}{\to}0$. Therefore, by Proposition 6.3, there exists $\alpha_{0}$ such that
for all $\alpha<\alpha_{0}$, ${\mathcal{B}}_{\alpha}$ has a simple, dominant
eigenvalue separated from the rest of the spectrum of at least $\delta$.

∎

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