# Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble

Quentin François, David García-Zelada

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Quentin François, David García-Zelada. Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble. 2023. hal-04143518v2

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# Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble

Quentin François, David García-Zelada

<figure><img src="image_1.png" /></figure>

## To cite this version:

Quentin François, David García-Zelada. Asymptotic analysis of the characteristic polynomial for the Elliptic Ginibre Ensemble. 2023. hal-04143518v3

HAL Id: hal-04143518

https://hal.science/hal-04143518v3

Preprint submitted on 11 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# ASYMPTOTIC ANALYSIS OF THE CHARACTERISTIC POLYNOMIAL FOR THE ELLIPTIC GINIBRE ENSEMBLE

QUENTIN FRANÇOIS AND DAVID GARCÍA-ZELADA

ABSTRACT. We consider the Elliptic Ginibre Ensemble, a family of random matrix models interpolating between the Ginibre Ensemble and the Gaussian Unitary Ensemble and such that its empirical spectral measure converges to the uniform measure on an ellipse. We show the convergence in law of its normalised characteristic polynomial outside of this ellipse. Our proof contains two main steps. We first show the tightness of the normalised characteristic polynomial as a random holomorphic function using the link between the Elliptic Ginibre Ensemble and Hermite polynomials. This part relies on the uniform control of the Hermite kernel which is derived from the recent work of Akemann, Duits and Molag. In the second step, we identify the limiting object as the exponential of a Gaussian analytic function. The limit expression is derived from the convergence of traces of random matrices, based on an adaptation of techniques that were used to study fluctuations of Wigner and deterministic matrices by Male, Mingo, Péché and Speicher. This work answers the interpolation problem raised in the work of Bordenave, Chafaï and the second author of this paper for the integrable case of the Elliptic Ginibre Ensemble and is therefore a first step towards the conjectured universality of this result.

## CONTENTS

<table><tr><td>1. Introduction and main result</td><td>1</td></tr><tr><td>1.1. The model of the Elliptic Ginibre Ensemble (EGE)</td><td>1</td></tr><tr><td>1.2. Main result</td><td>2</td></tr><tr><td>1.3. Method of proof</td><td>4</td></tr><tr><td>1.4. Open questions and comments</td><td>5</td></tr><tr><td>2. Tightness of the normalised characteristic polynomial</td><td>6</td></tr><tr><td>3. Convergence of the coefficients</td><td>7</td></tr><tr><td>4. Proofs of statements used for tightness</td><td>8</td></tr><tr><td>4.1. Proof of Lemma 2.5</td><td>8</td></tr><tr><td>4.2. Proof of Lemma 2.6</td><td>8</td></tr><tr><td>4.3. Proof of Proposition 2.7</td><td>9</td></tr><tr><td>5. Proofs of statements used for the convergence of coefficients</td><td>10</td></tr><tr><td>5.1. Proof of Proposition 3.2</td><td>10</td></tr><tr><td>5.2. Graph encoding of traces</td><td>10</td></tr><tr><td>5.3. Proof of Lemma 3.3</td><td>12</td></tr><tr><td>5.4. Proof of Proposition 3.4</td><td>16</td></tr><tr><td>References</td><td>25</td></tr></table>

## 1. INTRODUCTION AND MAIN RESULT

1.1. The model of the Elliptic Ginibre Ensemble (EGE). The random matrices that we consider in this paper are called Elliptic Ginibre Ensemble. This model is parameterized by $t \in [0, 1]$ and interpolates between the Ginibre Ensemble and the Gaussian Unitary Ensemble (GUE) for $t = 0$ and $t = 1$ respectively. This ensemble has been studied extensively; see for instance [2], [22]. The generic law for any $t \in [0, 1]$ is induced by the following construction.

Consider $X_n$ and $Y_n$ independent random matrices sampled from the Gaussian Unitary Ensemble of size $n \ge 1$. The law of the Elliptic Ginibre Ensemble at $t \in [0, 1]$ is the law of the matrix

$$ (1.1) \qquad A_{n,t} = \sqrt{\frac{1+t}{2}} X_n + i \sqrt{\frac{1-t}{2}} Y_n, $$
ASYMPTOTIC ANALYSIS OF THE CHARACTERISTIC POLYNOMIAL
FOR THE ELLIPTIC GINIBRE ENSEMBLE

QUENTIN FRANÇOIS, DAVID GARCÍA-ZELADA

ABSTRACT. We consider the complex Elliptic Ginibre Ensemble, a family of random matrix models introduced by Girko that interpolates between the Ginibre Ensemble and the Gaussian Unitary Ensemble and such that its empirical spectral measure converges to the uniform measure on an ellipse. We show the convergence in law of its normalised characteristic polynomial outside of this ellipse. Our proof contains two main steps. We first show the tightness of the normalised characteristic polynomial using the link between the Elliptic Ginibre Ensemble and Hermite polynomials. This part relies on the uniform control of the Hermite kernel which is derived from the recent work of Akemann, Duits and Molag. In the second step, we identify the limiting object as the exponential of a Gaussian analytic function. The limit expression is derived from the convergence of traces of Chebyshev polynomials of random matrices by the method of moments. These traces of Chebyshev polynomials appear naturally as a kind of centered version, or normal ordering, of the traces of the monomials. This work answers the interpolation problem raised in the work of Bordenave, Chafaï and the second author of this paper for the integrable case of the Elliptic Ginibre Ensemble and is therefore a first step towards the conjectured universality of this result.

# 1. INTRODUCTION AND MAIN RESULT

## 1.1. The model of the Elliptic Ginibre Ensemble (EGE).
The random matrices that we consider in this paper are sampled from the complex Elliptic Ginibre Ensemble introduced by Girko in [Gir85]. This model is parameterized by $t \in [0, 1]$ and interpolates between the Ginibre Ensemble and the Gaussian Unitary Ensemble (GUE) for $t = 0$ and $t = 1$ respectively. A concise review of this model can be found in [KS11]. Its law is the one of a random matrix given by the following construction.

Recall that a matrix sampled from the Gaussian Unitary Ensemble is a Hermitian random matrix whose density is proportional to $e^{-\text{Tr}(M^2)/2}$. Consider $X_n$ and $Y_n$ independent random matrices sampled from the Gaussian Unitary Ensemble of size $n \ge 1$. The law of the Elliptic Ginibre Ensemble at $t \in [0, 1]$ is the law of the matrix

$$ (1.1) \qquad A_{n,t} = \sqrt{\frac{1+t}{2}} X_n + i \sqrt{\frac{1-t}{2}} Y_n, $$

where $i$ is the imaginary unit. Equivalently, $A_{n,t}$ has a law proportional to

$$ (1.2) \qquad \exp \left( - \frac{1}{1-t^2} \text{Tr} \left[ M^* M - \frac{t}{2} (M^2 + (M^*)^2) \right] \right) dM, $$

where $dM = \prod_{1 \le i,j \le n} dM_{ij}$ is the product Lebesgue measure on the entries of the matrix, see [ADM23, Eq. (4)]. Notice that $A_{n,t}$ could also be defined as a centered complex Gaussian matrix whose entries $a_{ij}$ satisfy $E[|a_{ij}|^2] = 1$, $E[a_{ij}a_{ji}] = t$ for every $i, j$, and for $i \ne j$ $E[a_{ij}\bar{a}_{ji}] = E[a_{ij}^2] = 0$ while covariance between $a_{ij}$ and $a_{i'j'}$ are zero if $\{i, j\} \ne \{i', j'\}$. Moreover, for $i \ne j$, $E[(a_{ij}a_{ji} - t)^2] = t^2$ and $E[|a_{ij}a_{ji} - t|^2] = 1$. Many results are known for EGE matrices. In particular, the limiting eigenvalue distribution has been proved by Girko to be the uniform law on the ellipse centered at the origin with half long axis $1+t$ and short axis $1-t$. We refer to [Gir85, Theorem 7] and [Som+88] for the first instances of this result.

In the recent work [BCG22], it has been proved that the spectral radius of matrices with i.i.d. centered entries, called Girko matrices, converges in probability to 1 under the minimal assumption of a second moment on its entries. In order to derive this result, the authors considered the reciprocal characteristic polynomial associated to such matrices defined by $q_n(z) = z^n p_n(\frac{1}{z})$
where $i^2 = -1$. Equivalently, the density of $A_{n,t}$ is proportional to the function, see [1, eq. (4)],

$$ (1.2) \qquad \exp\left(-\frac{1}{1-t^2}\mathrm{Tr}\left[M^*M - \frac{t}{2}(M^2 + (M^*)^2)\right]\right) dM $$

where $dM = \prod_{1<i,j\le n} dM_{i,j}$ is the product Lebesgue measure on the entries of the matrix. Many results are known for EGE matrices. In particular, the limiting eigenvalue distribution has been proved to be the uniform law on the ellipse centered at the origin with half long axis $1+t$ and short axis $1-t$. We refer to [16] and [34] for the study of this model.

In the recent work [12], it has been proved that the spectral radius of matrices with i.i.d coefficients, called Girko matrices, converges in probability to 1 under the minimal assumption of a second moment on its entries. In order to derive this result, the authors considered the reciprocal characteristic polynomial associated to such matrices defined by $q_n(z) = z^n p_n(\frac{1}{z})$ for $z \in \mathbb{D}$ where $p_n$ is the characteristic polynomial. The main result of [12] is the convergence in law, for the topology of local uniform convergence, of the sequence of functions $\{q_n\}_{n \ge 1}$ to a random function which is universal, in the sense that its expression involves only the second moment of the entries of the matrix. Our result aims at deriving the convergence of the normalised characteristic polynomial in the case of the EGE (1.1) at each $t \in [0, 1]$ and at identifying the limiting object in the conjectured universality. The same result holds for Haar unitary matrices, which is implied by the normal asymptotics of the traces of the powers of Haar unitary matrices [15] and by the asymptotics of the kernel which has a simple form in this case. Moreover, it can be even seen that the characteristic polynomial inside and outside exhibit the same but independent behavior. More interestingly, the scaling limit around a point at the unit circle has been studied in [13] by showing a convergence towards a random analytic function whose zeros form a determinantal point process on the real line.

### 1.2. Main result.
Let $n \ge 1$ and $t \in [0, 1]$. Consider $p_{n,t}(z) = \det(z - \frac{1}{\sqrt{n}}A_{n,t})$, the characteristic polynomial of a scaling by $\frac{1}{\sqrt{n}}$ of a matrix $A_{n,t}$ from the Elliptic Ginibre Ensemble (1.1). Define the normalised characteristic polynomial of $A_{n,t}$ by

$$ (1.3) \qquad f_{n,t}: \mathbb{D} \to \mathbb{C} \\ z \mapsto \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}}A_{n,t} \right) e^{-\frac{ntz^2}{2}}. $$

Our main result is the following convergence.

**Theorem 1.1** (Convergence of the normalised characteristic polynomial for the EGE). Let $t \in [0, 1]$ and consider the sequence of random holomorphic functions $\{f_{n,t}\}_{n \ge 1}$ as in (1.3). Then, as $n \to \infty$,

$$ f_{n,t} \xrightarrow{\text{law}} f_t = \kappa_t e^{-F_t} $$

for the topology of uniform convergence on compact subsets of $\mathbb{D}$, where the function $F_t$ is a Gaussian analytic function given by

$$ (1.4) \qquad F_t(z) = \sum_{k \ge 1} X_{k,t} \frac{z^k}{\sqrt{k}} $$

for a family $(X_{k,t})_{k \ge 1}$ of independent complex Gaussian random variables such that $\mathbb{E}[X_{k,t}] = 0$, $\mathbb{E}[X_{k,t}^2] = t^k$ and $\mathbb{E}[|X_{k,t}|^2] = 1$, and the function $\kappa_t$ is given by

$$ (1.5) \qquad \kappa_t(z) = \exp\left(-\frac{1}{2}\sum_{k \ge 1} \frac{h_{k,t}}{k} z^{2k}\right) \cdot \exp\left(\frac{tz^2}{2(1 - tz^2)}\right), $$

where the real coefficients $\{h_{k,t}\}_{k \ge 1}$ are given in (5.10).

Let us give some explanations on (1.3). In the case $t = 0$, the matrix $A_{n,0}$ belongs to the Ginibre Ensemble, which is a particular case of Girko matrices. For such matrices, we have exactly the reciprocal characteristic polynomial of [12],

$$ q_{n,0}: \mathbb{D} \to \mathbb{C} \\ z \mapsto q_{n,0}(z) = z^n p_{n,0}(g_0(z)) $$
for $z \in \mathbb{D} = \{y \in \mathbb{C} : |y| < 1\}$, where $p_n$ is the characteristic polynomial. The main result of [BCG22] is the convergence in law, for the topology of local uniform convergence, of the sequence of functions $\{q_n\}_{n \ge 1}$ to a random function which is universal, in the sense that its expression involves only the second moment of the entries of the matrix. Our result aims at deriving the convergence of the normalised characteristic polynomial in the case of the EGE (1.1) at each $t \in [0, 1]$ and at identifying the limiting object in the conjectured universality. In particular, for $t=1$ our result gives the convergence of the characteristic polynomial for GUE matrices to a random analytic function.

Characteristic polynomials of random matrices have been studied extensively. For Haar unitary matrices or, more generally, for Circular $\beta$-Ensembles (C$\beta$E), the characteristic polynomial outside the unit disk behaves in a similar way as in Theorem 1.1 for $t=0$ as is stated in [NPS23, Theorem 1.3]. Moreover, the characteristic polynomial inside and outside the unit disk exhibit the same but independent limiting behavior. More interestingly, the scaling limit around a point at the unit circle has been studied in [CNN17] by showing a convergence towards a random analytic function whose zeros form a determinantal point process on the real line. Limit expressions for the characteristic polynomial of C$\beta$E matrices are furthermore related to the Gaussian multiplicative chaos and to the theory of orthogonal polynomials on the unit circle, see [LN24]. In the case of Gaussian $\beta$-Ensembles, approximations of the characteristic polynomial in the complex plane where found in terms of log-correlated Gaussian fields, see [LP23]. For Haar random matrices, asymptotics for moments of derivatives of the characteristic polynomial have also been computed in [SW24]. They derive limits both inside the unit disk and for mesoscopic and microscopic regimes when $z$ approaches the unit circle. The cases of orthogonal, symplectic and GUE random matrices have been studied in [Chh+19] where ratios of characteristic polynomials are shown to converge to a random entire function which was constructed in [CNN17] and related to Haar random matrices.

The study of the reciprocal characteristic polynomial for Girko matrices in [BCG22] was partially inspired from the work [BZ20] on Toeplitz matrices. The same object was studied for other models. The case of sparse matrix models having i.i.d. non-centered Bernoulli entries was treated in [Cos23]. The reciprocal characteristic polynomial of such matrices converges to a random function expressed using Poisson series [Cos23]. In [CLZ24], the same type of convergence was obtained for sums of random uniform permutation matrices. For a fixed number of random matrices in the sum, the limit has the same form given by the exponential of a Poisson series, whereas for a number of terms going to infinity in a prescribed way, the limit has the form given by the exponential of a Gaussian series as in [BCG22]. Exponential of Poisson series were also obtained as the limit of characteristic polynomials for permutation matrices in [Bah19a; Bah19b].

In relation to elliptic matrices, the uniform law on the ellipse can be obtained as the asymptotic distribution of zeroes of random polynomials which are related to Weyl polynomials, see [Hal+].

The motivation from the work [BCG22] was to obtain the convergence of the spectral radius for Girko matrices. One could ask for a study of the fluctuations around the limit. For the Ginibre Ensemble, one has Gumbel fluctuations for the maximum modulus around 1, see [Rid03]. The Gumbel distribution also appears as the limit fluctuation for the largest real part of either real of complex Ginibre matrices [Cip+22]. For the GUE, one has Tracy-Widom fluctuations for the maximum eigenvalue around 2, see [TW02] and references therein. In [Joh07], we may find a family of determinantal processes that interpolates between a Poisson process with intensity $e^{-x}$ and the Airy process. The distribution function of its last particle interpolates between the Gumbel and Tracy-Widom distributions, see [Joh07, Theorem 1.3]. As a two-dimensional version, [Ben10] considered the Elliptic Ginibre Ensemble and an interpolating determinantal processes to prove scaling limits for the eigenvalue point process.

1.2. Main result. Let $n \ge 1$ and $t \in [0, 1]$. Consider $p_{n,t}(z) = \det(z - \frac{1}{\sqrt{n}}A_{n,t})$, the scaled characteristic polynomial of a matrix $A_{n,t}$ sampled from the Elliptic Ginibre Ensemble (1.1). Define $f_{n,t}: \mathbb{D} \to \mathbb{C}$ as the normalised characteristic polynomial of $A_{n,t}$,
where $g_0 : z \mapsto \frac{1}{z}$ is the conformal mapping that sends the unit disk to its complementary. One can think of the choice of $g_0$ as the following. We know that the empirical measure of eigenvalues of a Girko matrix converges weakly to the uniform measure on the unit disk $D$, see [11] for complements on this universal result known as the circular law. One therefore chooses a mapping $g_0$ that sends $D$ to the complementary of the support of the limiting eigenvalue measure so that $p_{n,0} \circ g_0$ does not vanish on the unit disk. Driven by this intuition, we construct $f_{n,t}$ by composing the characteristic polynomial by a function that sends $D \setminus \{0\}$ to $C \setminus E_t$ where $E_t$ is the support of the limiting eigenvalue distribution for the EGE with parameter $t$.

For $t \in [0, 1]$, it has been proven in [34] that the average eigenvalue distribution for the Elliptic Ginibre Ensemble converges to the uniform distribution on the ellipsoid $E_t$ given by the equation

$$ (1.6) \qquad E_t = \left\{ x + iy \in \mathbb{C} : \left(\frac{x}{1+t}\right)^2 + \left(\frac{y}{1-t}\right)^2 \le 1 \right\}. $$

For $t = 1$, we define $E_1$ to be the interval $[-2, 2]$ and the corresponding limiting distribution to be the semi-circle distribution in accordance with Wigner's theorem. Since the function $g_t : z \mapsto \frac{1}{z} + tz$ maps $D \setminus \{0\}$ to $C \setminus E_t$, a natural candidate for the normalised characteristic polynomial would be the mapping $z \mapsto z^n p_{n,t} \circ g_t(z)$. To have a convergence, it is necessary to add the factor $\exp(-ntz^2/2)$ which gives our expression of $f_{n,t}$ in (1.3). We refer the reader to Remark 5.13 for an explanation on this factor. Using these notations, from Theorem 1.1 we obtain the convergence of the normalised characteristic polynomial $\tilde{p}_{n,t}(u) = (g_t^{-1}(u))^n e^{-nt(g_t^{-1}(u))^2/2} p_{n,t}(u)$

$$ \tilde{p}_{n,t} \xrightarrow{\text{law}} (\kappa_t \circ g_t^{-1}) e^{-F_t \circ g_t^{-1}} $$

in the topology of uniform convergence on compact sets of $C \setminus E_t$. This is, in fact, equivalent to Theorem 1.1 due to the holomorphicity of $f_{n,t}$ at zero. It explains the notation “normalised characteristic polynomial” since $f_{n,t}$ and $\tilde{p}_{n,t}$ are the same functions in different coordinate systems.

From Theorem 1.1, one can derive the following result which was given in [30, Theorem 2.2] for a class of elliptic matrices that includes our Gaussian case. Nevertheless, since an explicit density can be written for the eigenvalues in the Gaussian case, we may also use large deviation arguments to obtain the lack of outliers.

**Corollary 1.2 (Lack of outliers).** Let $C \subset \mathbb{C}$ be a closed set disjoint from $E_t$ and define $N_n(C) = \#\{i \in [n] : \frac{\lambda_i}{\sqrt{n}} \in C\}$. Then,

$$ (1.7) \qquad N_n(C) \xrightarrow{\mathbb{P}} 0. $$

*Proof.* Let $C \subset \mathbb{C}$ be a closed set disjoint from $E_t$. Recall that $g_t(z) = \frac{1}{z} + tz$ and consider $\tilde{C} = g_t^{-1}(C)$ which is closed in $D \setminus \{0\}$ so that its closure $K$ on $D$ is compact. Then, we have that

$$ \begin{aligned} \mathbb{P}[|N_n(C)| > 0] &= \mathbb{P}[\inf_{z \in C} |p_{n,t}(z)| = 0] \\ &= \mathbb{P}[\inf_{u \in K} |f_{n,t}(u)| = 0] \\ &\to \mathbb{P}[\inf_{u \in K} |f_t(u)| = 0] = 0. \end{aligned} $$

□
$$ (1.3) \qquad f_{n,t}(z) := \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}} A_{n,t} \right) e^{-\frac{ntz^2}{2}}. $$

We endow the space of holomorphic functions on $\mathbb{D}$ with the topology of uniform convergence on compact sets and state our main result as follows.

**Theorem 1.1** (Convergence of the normalised characteristic polynomial). As $n \to \infty$,

$$ f_{n,t} \xrightarrow{\text{law}} \exp(-F_t) $$

where $F_t$ is the Gaussian holomorphic function on $\mathbb{D}$ defined by

$$ (1.4) \qquad F_t(z) := \sum_{k \ge 1} X_k \frac{z^k}{\sqrt{k}} $$

for a family $(X_k)_{k \ge 1}$ of independent Gaussian random variables on $\mathbb{C}$ satisfying

$$ \mathbb{E}[X_k] = 0, \quad \mathbb{E}[X_k^2] = t^k \quad \text{and} \quad \mathbb{E}[|X_k|^2] = 1. $$

Let us give some intuition for the choice in (1.3). Since the empirical measure of eigenvalues of $A_{n,t}$ converges to the uniform measure $\sigma_t$ on the ellipse

$$ (1.5) \qquad \mathcal{E}_t = \left\{ x + iy \in \mathbb{C} : \left( \frac{x}{1+t} \right)^2 + \left( \frac{y}{1-t} \right)^2 \le 1 \right\}, $$

to study the behavior of the characteristic polynomial $p_{n,t}$ on $\mathbb{C} \setminus \mathcal{E}_t$, we send this set to the unit disk in the simplest holomorphic way, namely, by using the map $g_t: \mathbb{D} \setminus \{0\} \to \mathbb{C} \setminus \mathcal{E}_t$,

$$ g_t(z) = \frac{1}{z} + tz. $$

In the case of $t = 1$, we should define $\mathcal{E}_1 = [-2, 2]$ and $\sigma_1 = \lim_{t \to 1} \sigma_t$ is the semi-circular law, which is consistent with Wigner's semicircular law. In this case, $g_1$ is the so-called Joukowsky transform and, moreover, we have the simple relation $g_t(z) = \sqrt{t}g_1(\sqrt{tz})$ whenever $t \ne 0$. Under this change of variables, the characteristic polynomial $p_{n,t}$ is

$$ p_{n,t} \circ g_t(z) = \det \left( g_t(z) - \frac{A_{n,t}}{\sqrt{n}} \right) = \frac{1}{z^n} \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}} A_{n,t} \right) $$

Finally, $n^{-1} \log \det (1 + tz^2 - zn^{-1/2}A_{n,t})$ should converge to $\int \log(1 + tz^2 - zw)d\sigma_t(w)$ which is $tz^2/2$, as seen in the proof of Lemma 2.14, suggesting the exponential factor in $f_{n,t}$.

Using these notations, from Theorem 1.1 we obtain the convergence of the normalised characteristic polynomial $\tilde{p}_{n,t}(u) = (g_t^{-1}(u))^n e^{-nt(g_t^{-1}(u))^2/2} p_{n,t}(u)$

$$ \tilde{p}_{n,t} \xrightarrow{\text{law}} \exp(-F_t \circ g_t^{-1}) $$

for the topology of uniform convergence on compact sets of $\mathbb{C} \setminus \mathcal{E}_t$. This is, in fact, equivalent to Theorem 1.1 due to the holomorphicity of $f_{n,t}$ at zero. It explains the notation “normalised characteristic polynomial” since $f_{n,t}$ and $\tilde{p}_{n,t}$ are the same written in different coordinate systems.

From Theorem 1.1, one can derive the following result given in [OR14, Theorem 2.2] for a class of elliptic matrices that includes our Gaussian case. Nevertheless, since an explicit density can be written for the eigenvalues in the Gaussian case, we may also use large deviation arguments to obtain the lack of outliers.

**Corollary 1.2** (Lack of outliers). Let $C \subset \mathbb{C}$ be a closed set disjoint from $\mathcal{E}_t$. Then, as $n \to \infty$,

$$ (1.6) \qquad N_n(C) := \# \left\{ i \in [n] : \frac{\lambda_i}{\sqrt{n}} \in C \right\} \xrightarrow{\mathbb{P}} 0. $$
<figure><img src="image_3.png" /><figcaption>FIGURE 1. Illustration of Theorem 1.1. Phase portrait of the normalised characteristic polynomial of an EGE matrix of size 250 for different values of $t$: 0 (top left), 0.3 (top right), 0.6 (bottom left) and 1 (bottom right).</figcaption></figure>

**1.3. Method of proof.** We will follow the same proof structure as for [12, Theorem 1.2], which was inspired by [9] and [18]. In order to prove the convergence in law for the topology of local uniform convergence, we will use [12, Lemma 3.2] which is close to [32, Proposition 2.5].

**Lemma 1.3** (Tightness and convergence of coefficients imply convergence of functions). Let $\{f_n\}_{n \ge 1}$ be a sequence of random elements in $\mathcal{H}(\mathbb{D})$ and denote the coefficients of $f_n$ by $(\xi_k^{(n)})_{k \ge 0}$ so that for all $z \in \mathbb{D}$, $f_n(z) = \sum_{k \ge 0} \xi_k^{(n)} z^k$. Suppose also that the following conditions hold.

(a) The sequence $\{f_n\}_{n \ge 1}$ is tight in $\mathcal{H}(\mathbb{D})$.

(b) For every $m \ge 0$, the vector $(\xi_0^{(n)}, \dots, \xi_m^{(n)})$ converges in law as $n \to \infty$ to $(\xi_0, \dots, \xi_m)$ for a common sequence of random variables $(\xi_k)_{k \ge 0}$.

Then, $f(z) = \sum_{k \ge 0} \xi_k z^k$ is a well-defined function in $\mathcal{H}(\mathbb{D})$ and $f_n$ converges in law towards $f$ in $\mathcal{H}(\mathbb{D})$ for the topology of local uniform convergence.

The first part of our proof of Theorem 1.1 is to show, for every fixed value of $t \in [0, 1]$, the tightness (a) of the family of random functions $\{f_{n,t}\}_{n \ge 1}$. To do so, we will use the known properties of the Elliptic Ginibre Ensemble and its relation to scaled Hermite polynomials. A local uniform control is derived from the recent work [1]. This local uniform control allows us to derive tightness thanks to Lemma 2.2 below.

The second part of the proof consists in proving a convergence in distribution of the coefficients appearing in the functions $\{f_{n,t}\}_{n \ge 1}$. We reduce the former to proving convergence of traces of polynomials in $A_{n,t}$, which is classic in random matrix theory and is done by a combinatorial argument using the method of moments. The study we conduct will follow the lines developed in [23] in which the authors studied the asymptotic fluctuations of Wigner and deterministic matrices. As in [23], we will not use the Gaussian nature of $A_{n,t}$ but only the fact that all its moments are finite.
Proof. Let $C \subset \mathbb{C}$ be a closed set disjoint from $\mathcal{E}_t$. Recall that $g_t(z) = \frac{1}{z} + tz$ and consider $\tilde{C} = g_t^{-1}(C)$ which is closed in $\mathbb{D} \setminus \{0\}$ so that its closure $K$ on $\mathbb{D}$ is compact. Then,

$$ \begin{aligned} \mathbb{P}[|N_n(C)| > 0] &= \mathbb{P}[\inf_{z \in C} |p_{n,t}(z)| = 0] \\ &= \mathbb{P}[\inf_{u \in K} |f_{n,t}(u)| = 0] \\ &\rightarrow \mathbb{P}[\inf_{u \in K} |e^{-F_t(u)}| = 0] = 0. \end{aligned} $$

□

<figure><img src="image_3.png" /><figcaption>FIGURE 1. Illustration of Theorem 1.1. Phase portrait of the normalised characteristic polynomial of an EGE matrix of size 250 for different values of $t$: 0 (top left), 0.3 (top right), 0.6 (bottom left) and 1 (bottom right). The unit circle is represented in red.</figcaption></figure>

In fact, we expect an analogue of Theorem 1.1 to hold in a much more general setting as conjectured in [BCG22]. The limit would only depend on some of the first four moments of the coefficients of the random matrix as suggested in Section 1.3.2. A glimpse of this universality can be seen, for instance, when calculating the expected value of the characteristic polynomial. This depends only on $t = \mathbb{E}[a_{12}a_{21}]$ as explained in the proof of the following theorem where a simple expression for its limit is stated.

**Theorem 1.3 (Average characteristic polynomial).** For each $n$, let $A_{n,t} = (a_{ij}, 1 \le i, j \le n)$ be a random matrix such that $\{(a_{ij}, a_{ji}), 1 \le i < j \le n\}$ are i.i.d. centered pairs which are independent of the i.i.d. centered family $\{a_{ii}, 1 \le i \le n\}$ with $\mathbb{E}[|a_{ij}|^2] < \infty$ for all $1 \le i, j \le n$ and $\mathbb{E}[a_{12}a_{21}] = t \in [0, 1]$. Then, for $z$ uniformly in $\mathbb{D}$,

$$ (1.7) \qquad \lim_{n \to +\infty} \mathbb{E} \left[ \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}} A_{n,t} \right) e^{-\frac{ntz^2}{2}} \right] = \frac{1}{\sqrt{1 - tz^2}}. $$

1.3. Open questions and comments.

1.3.1. Extension via matching moments. Since the way we show tightness is by controlling the second moment of $f_{n,t}$ and since this second moment depends only on the first four moments of $A_{n,t}$, tightness of $f_{n,t}$ still holds for the model described in Section 1.3.2 for coefficients $(a_{ij})_{i,j \ge 1}$ whose first four moments coincide with those of the EGE. Moreover, the proof of convergence
The use of the method of moments to derive a CLT for traces of random matrices was initiated by the work [21] in the context of Gaussian Wishart matrices. Using similar techniques, the authors in [33] derived a CLT for traces of Wigner matrices, together with a universality result on the limiting Gaussian distribution. Another approach leading to CLT for traces of random matrices is the resolvent method. For instance, one is able to prove CLT for functions that are analytic inside the support of the limiting eigenvalue distribution. We refer the reader to [8] for the case of Wigner matrices and [7] for Wishart matrices for a use of these techniques. For a more complete review on techniques leading to CLT, see also the introduction of [5] and references therein.

Finally, since the way we show tightness is by controlling the second moment of $f_{n,t}$ and since this second moment depends only on the first four moments of $A_{n,t}$, tightness of $f_{n,t}$ still holds for the model described in Subsection 1.4.1 for coefficients $(a_{ij})_{i,j \ge 1}$ whose first four moments coincide with those of the EGE. Moreover, the method of moments proof of the convergence of traces can be adapted to the case where the coefficients have all moments finite and Theorem 1.1 holds for coefficients $(a_{ij})_{i,j \ge 1}$ with all moments finite and whose first four moments coincide with those of the EGE.

## 1.4. Open questions and comments.

1.4.1. Minimal moment condition and universality. As conjectured in [12], the convergence in Theorem 1.1 of the normalised characteristic polynomial is believed to hold under the minimal moment condition

$$ (1.8) \qquad \mathbb{E}[|a_{1,2}|^2|a_{2,1}|^2] < \infty $$

on the entries $(a_{i,j})_{i,j \ge 1}$, which gives a condition of a fourth order moment for Wigner matrices and second order moment for Girko matrices. The more general context of condition (1.8) seems adapted to the model of elliptic random matrices. This model was introduced by Girko in [16] and [17]. A version of this consists of matrices having the following dependence relations be found in [28, Definition 1.3]. Fix some random vector $(\xi_1, \xi_2)$ in $\mathbb{C}^2$ with zero mean such that $\mathbb{E}[|\xi_1|^2] = \mathbb{E}[|\xi_2|^2] = 1$. A matrix $A_n = (a_{i,j})_{1 \le i,j \le n}$ is said to be elliptic with atom distribution $(\xi_1, \xi_2)$ if

* $(a_{i,i}, 1 \le i)$, $((a_{i,j}, a_{j,i}), 1 \le i < j)$ are independent families.

* $((a_{i,j}, a_{j,i}), 1 \le i < j)$ consists of i.i.d copies of $(\xi_1, \xi_2)$.

* $(a_{i,i}, 1 \le i)$ are i.i.d with zero mean and finite variance.

Convergence of the average eigenvalue distribution towards the uniform distribution on a rotated version of the ellipse (1.6) where $t = |\mathbb{E}[\xi_1\xi_2]|$ has been proved, under different conditions on the variables, in [28, 30, 27]. We expect a version of Theorem 1.1 to hold for the general elliptic matrices described above. This is work in progress.

1.4.2. Matrices with entries in $\{0, 1\}$. In the recent work [14], a convergence of the reciprocal characteristic polynomial for matrices with independent Bernoulli entries with non-zero expectation has been proved. The limiting random function can be expressed using Poisson random variables, see [14, Theorem 2.3]. One could ask for an analogue of the Elliptic Ginibre Ensemble for such matrices and the convergence of its normalised characteristic polynomial.

1.4.3. Fluctuations of the extreme eigenvalue. For both Ginibre and GUE cases, one knows the law of the fluctuations of the largest eigenvalue around its limit. For the Ginibre Ensemble, one has Gumbel fluctuations for the maximum modulus around 1, see [31], whereas for the GUE, one has Tracy-Widom fluctuations for the maximum eigenvalue around 2, see [36] and references therein. In [19], we may find a family of determinantal processes that interpolates between a Poisson process with intensity $e^{-x}$ and the Airy process. The distribution function of its last particle interpolates between the Gumbel and Tracy-Widom distributions, see [19, Theorem 1.3]. As a two-dimensional version, [10] considered the Elliptic Ginibre Ensemble and an interpolating determinantal processes to prove scaling limits for the eigenvalue point process.

1.4.4. Determinantal Coulomb gases. As explained in 1.4.1, this work can be thought of as a first step towards the convergence of the characteristic polynomial outside the support of the equilibrium measure for general elliptic random matrices. Nevertheless, we could have followed a different path, which is to look the Elliptic Ginibre Ensembles as a particular case of a determinantal Coulomb gas. In this vein, it may be possible to show the convergence of the traces by adapting results from [4] and
of the coefficients of $f_{n,t}$ also works in the case where the coefficients have all moments finite so that Theorem 1.1 holds for coefficients $(a_{ij})_{i,j \ge 1}$ with all moments finite and whose first four moments coincide with those of the EGE.

### 1.3.2. Minimal moment condition and universality.
As conjectured in [BCG22], the convergence in Theorem 1.1 of the normalised characteristic polynomial is believed to hold under the minimal moment condition

$$ (1.8) \qquad \mathbb{E}[|a_{12}a_{21}|^2] < \infty $$

on the entries $(a_{ij})_{i,j \ge 1}$, which gives a condition of a fourth order moment for Wigner matrices and second order moment for Girko matrices. The context adapted to this conjecture is the one of elliptic random matrices [NO15, Definition 1.3]. This model was introduced by Girko in [Gir85] and [Gir95] and a version of this model consists of the following random matrices. Consider a family of square-integrable centered random variables $(a_{ij})_{i,j \ge 1}$ satisfying that the family of random elements $\{(a_{ij}, a_{ji}) : i < j\} \cup \{a_{ii} : i \ge 1\}$ is independent and whose law is invariant under any permutation of the indices or, equivalently, the law of $(a_{ij}, a_{ji})$ coincides with the law of $(a_{i'j'}, a_{j'i'})$ whenever $|\{i, j\}| = |\{i', j'\}|$. If

$$ \mathbb{E}[|a_{12}|^2] = 1 \quad \text{and} \quad \mathbb{E}[a_{12}a_{21}] = t, $$

the matrix $A_n = (a_{ij})_{1 \le i,j \le n}$ is said to be t-Girko. The convergence of the average eigenvalue distribution towards the uniform distribution on the ellipse (1.5) has been proved under different conditions on the variables, see [NO15; OR14; Nau12]. We expect the following version of Theorem 1.1 to hold for the general t-Girko matrices described above. Denoting $\tau = \mathbb{E}[a_{12}^2]$, $s = \mathbb{E}[a_{11}^2] - t - \tau$ and $q = \mathbb{E}[(a_{12}a_{21}-t)^2] - t^2 - \tau^2$, the limit of $\det(1+tz^2-z\frac{A_{n,t}}{\sqrt{n}})\exp(-ntz^2/2)$ is expected to be given by

$$ (1.9) \qquad \sqrt{1 - \tau z^2} e^{-sz^2/2} e^{-qz^4/4} e^{-\sum_{k \ge 1} Y_k \frac{z^k}{\sqrt{k}}} $$

where $(Y_k)_{k \ge 1}$ are independent centered complex Gaussians such that $Y_1$ has the same variance as $a_{11}$, $Y_2$ has the same variance as $a_{12}a_{21}$ and, for $k \ge 3$, the variance of $Y_k$ is the sum of the k-th power of the variance of $a_{12}$ and the k-th power of the covariance of $a_{12}$ and $a_{21}$ or, more explicitly, $\mathbb{E}[Y_k^2] = \mathbb{E}[a_{12}^2]^k + \mathbb{E}[a_{12}a_{21}]^k = \tau^k + t^k$ and $\mathbb{E}[|Y_k|^2] = \mathbb{E}[|a_{12}|^2]^k + \mathbb{E}[a_{12}\bar{a}_{21}]^k = 1 + \mathbb{E}[a_{12}\bar{a}_{21}]^k$.

### 1.3.3. Matrices with entries in $\{0, 1\}$.
As described above, a convergence of the reciprocal characteristic polynomial for matrices with independent Bernoulli entries with non-zero expectation has been proved in [Cos23]. The limiting random holomorphic function can be expressed using Poisson random variables, see [Cos23, Theorem 2.3]. One could ask for an analogue of the Elliptic Ginibre Ensemble for such matrices and for the convergence of its normalised characteristic polynomial.

Extension of the work [CLZ24] for Ewens distributed random permutations can be found in [Fra]. The convergence of traces of such random matrices were understood from the work of Nikeghbali and Zeindler [NZ13], while tightness uses asymptotics of Hwang [Hwa94].

### 1.3.4. Determinantal Coulomb gases.
As explained in 1.3.2, this work can be thought of as a first step towards the convergence of the characteristic polynomial outside the support of the equilibrium measure for general elliptic random matrices. Nevertheless, we could have followed a different path, which is to look the Elliptic Ginibre Ensembles as a particular case of a determinantal Coulomb gas. In this vein, it may be possible to show the convergence of the traces by adapting results from [AHM15] and to show tightness of the characteristic polynomial outside the support of the equilibrium measure for more general determinantal Coulomb gases by using, for instance, the results from [AC23].

## 2. PROOF OF THEOREM 1.1

From now on, given that the case $t = 0$ is already treated in [BCG22], we assume $t \ne 0$ and omit the index $t$ since it is considered fixed for the rest of the article.

As is standard, to show that ${f_n}_{n \ge 1}$ converges we show that ${f_n}_{n \ge 1}$ is tight and that the
to show tightness of the characteristic polynomial outside the support of the equilibrium measure for more general determinantal Coulomb gases by using, for instance, the results from [3].

## 2. TIGHTNESS OF THE NORMALISED CHARACTERISTIC POLYNOMIAL

This section is devoted to the proof of the following theorem.

**Theorem 2.1 (Tightness).** For every $t \in [0, 1]$, the sequence $\{f_{n,t}\}_{n \ge 1}$ is tight, viewed as random elements of $\mathcal{H}(\mathbb{D})$, the set of holomorphic functions on $\mathbb{D}$.

Since the case $t = 0$ is treated in [12], we will assume for the rest of this section that $t \in (0, 1]$. Recall that for $z \in \mathbb{D}$, $n \ge 1$ and $t \in [0, 1]$,

$$f_{n,t}(z) = \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}} A_{n,t} \right) e^{-\frac{ntz^2}{2}}.$$

For our purposes, we will only be interested in the holomorphic function $f_{n,t}$ from $\mathbb{D}$ to $\mathbb{C}$. Equip the set $\mathcal{H}(\mathbb{D})$ with the topology of uniform convergence on compact sets. Lemma 2.2 below reduces the proof of tightness to a uniform control on compact subsets of $\mathbb{D}$.

**Lemma 2.2 (Reduction to uniform control).** Fix $t \in [0, 1]$. Suppose that for every compact $K \subset \mathbb{D}$, the sequence $\|f_{n,t}\|_K$ is tight, where $\|f_{n,t}\|_K = \max_{z \in K} |f_{n,t}(z)|$. Then, $\{f_{n,t}\}_{n \ge 1}$ is tight.

*Proof.* It is a consequence of Montel's theorem. See, for instance, [32, Proposition 2.5]. □

**Remark 2.3.** By the subharmonicity of $|f_{n,t}(z)|^2$, saying that $\{\mathbb{E}[\|f_{n,t}\|_K^2]\}_{n \ge 1}$ is a bounded sequence for every compact $K \subset \mathbb{D}$ is equivalent to say that $\{\sup_{z \in K} \mathbb{E}[\|f_{n,t}(z)\|^2]\}_{n \ge 1}$ is a bounded sequence for every compact $K \subset \mathbb{D}$. See, for instance, [32, Lemma 2.6]. In the Girko case of [12], one had a remarkable orthogonality of the sub-determinants which led to an upper bound on the desired quantity. As we no longer have this property, our proof is based on the article [2], which exploits the integrability of the Elliptic Ginibre Ensemble.

The main result of this subsection is Proposition 2.7, proved in Section 4.3. Its proof is based on Lemma 2.5 below which expresses the quantity $\mathbb{E}[|f_{n,t}(z)|^2]$ in terms of Hermite polynomials. This lemma is proved in Section 4.1.

**Definition 2.4 (Hermite polynomials).** The Hermite polynomials $\{He_n\}_{n \ge 0}$ are the monic orthogonal polynomials with respect to the measure $e^{-x^2/2}dx$ on $\mathbb{R}$ so that

$$\int_{\mathbb{R}} He_n(x) He_m(x) e^{-\frac{x^2}{2}} dx = \sqrt{2\pi n!} \delta_{n,m}.$$

Recall that for $t \in [0, 1]$ and $z \ne 0$, we have $g_t(z) = z^{-1} + tz$.

**Lemma 2.5 (Hermite expression of the characteristic polynomial).** For $n \ge 1$, $t \in (0, 1]$ and $z \in \mathbb{D} \setminus \{0\}$, one has the following expression

$$(2.1) \qquad \mathbb{E}[|f_{n,t}(z)|^2] = \frac{n!|z|^{2n}}{n^n} \left| e^{-ntz^2} \left| \sum_{k=0}^{n} \frac{t^k}{k!} \left| He_k\left(\sqrt{\frac{n}{t}}g_t(z)\right) \right| \right|^2.$$

With the help of the expression (2.1) and using the results from [1], we will control $\mathbb{E}[|f_{n,t}(z)|^2]$ uniformly on bounded sets from above and from below. In fact, [1] allows us to give an explicit expression for the limit of $\mathbb{E}[|f_{n,t}(z)|^2]$ but since we do not need it, we will only state the following.

**Lemma 2.6 (Convergence of the second moment).** Fix $t \in (0, 1]$. There exists a continuous function $\mathcal{F}: \mathbb{D} \setminus \{0\} \to (0, \infty)$ such that, uniformly on compact sets,

$$\mathbb{E}[|f_{n,t}|^2] \xrightarrow{n \to \infty} \mathcal{F}.$$

But, since $f_{n,t}$ is holomorphic on the whole disk $\mathbb{D}$, the origin was never actually an issue so that one can extend the control on any $\mathbb{D}_r$ for $r \in (0, 1)$. This is written in the next proposition.

**Proposition 2.7 (Uniform control).** Fix $t \in (0, 1]$. Then, for every $r \in (0, 1)$ there exists $C_r > 0$ such that, for every $n \ge 1$,

$$\mathbb{E}[\|f_{n,t}\|_{D_r}^2] \le C_r.$$

Proposition 2.7 proves Theorem 2.1 thanks to Lemma 2.2 and Remark 2.3. The proof of the tightness (a) in Lemma 1.3 is thus complete.
coefficients in its power-series expansion around the origin converge in law. We state this classical fact as Lemma 2.1 below and we refer to [BCG22, Section 4.2] for a proof. Recall that $\mathcal{H}(\mathbb{D})$ is the space of holomorphic functions on the unit disk $\mathbb{D}$ endowed with the topology of uniform convergence on compact sets.

**Lemma 2.1** (Tightness and convergence of coefficients imply convergence of functions). Let $\{h_n\}_{n \ge 1}$ be a sequence of random elements in $\mathcal{H}(\mathbb{D})$ and denote the coefficients of $h_n$ by $(\xi_k^{(n)})_{k \ge 0}$ so that for all $z \in \mathbb{D}$, $h_n(z) = \sum_{k \ge 0} \xi_k^{(n)} z^k$. Suppose also that the following conditions hold.

(a) The sequence $\{h_n\}_{n \ge 1}$ is a tight sequence of random elements of $\mathcal{H}(\mathbb{D})$.

(b) There exists a sequence $(\xi_k)_{k \ge 0}$ of random variables such that, for every $m \ge 0$, the vector $(\xi_0^{(m)}, \dots, \xi_m^{(m)})$ converges in law as $n \to \infty$ to $(\xi_0, \dots, \xi_m)$.

Then, $h(z) = \sum_{k \ge 0} \xi_k z^k$ is a well-defined function in $\mathcal{H}(\mathbb{D})$ and $h_n$ converges in law towards $h$ in $\mathcal{H}(\mathbb{D})$ for the topology of local uniform convergence.

The first step is to show the following theorem.

**Theorem 2.2** (Tightness). The sequence $\{f_n\}_{n \ge 1}$ is tight.

The proof uses known properties of the Elliptic Ginibre Ensemble and its relation to scaled Hermite polynomials. In particular, it relies on the determinantal aspect of its eigenvalue point process. A local uniform control is derived from [ADM23]. This control allows us to derive tightness thanks to Montel's theorem as stated in Lemma 2.5 below. This ensures that Condition (a) of Lemma 2.1 is satisfied for $\{f_n\}_{n \ge 1}$.

To guarantee Condition (b) of Lemma 2.1, we express the coefficients of $\{f_n\}_{n \ge 1}$ using a family of polynomials that we call the modified Chebyshev polynomials.

**Definition 2.3** (Modified Chebyshev polynomials). The modified Chebyshev polynomials are the polynomials $\{P_k\}_{k \ge 1}$ satisfying the recurrence relation

$$ (2.1) \qquad P_{k+1} = XP_k - tP_{k-1}, \quad P_1 = X, \quad P_2 = X^2 - 2t. $$

We may also give the modified Chebyshev polynomials more explicitly by their coefficients

$$ (2.2) \qquad \alpha_{k-2j}^{(k)} := (-t)^j \frac{k}{k-j} \binom{k-j}{j} $$

so that $P_k = \sum_{j \ge 0} \alpha_{k-2j}^{(k)} X^{k-2j}$ or by its generating function

$$ (2.3) \qquad \sum_{k \ge 1} P_k(w) \frac{z^k}{k} = -\log(1 + tz^2 - zw). $$

The latter expression can be obtained either by noticing that $P_k(w) = 2\sqrt{t}^k T_k(w/(2\sqrt{t}))$ for $T_k$ the classical Chebyshev polynomials of the first kind or by taking the derivative in $z$ of (2.3), which is $(w - 2zt)(1 + tz^2 - zw)^{-1}$, so that $(\sum_{k \ge 0} P_{k+1}(w)z^k)(1 + tz^2 - zw) = w - 2zt$ which is the recurrence (2.1).

Then, for $z \in \mathbb{D}$ small enough using (2.3)

$$ 1 + tz^2 - \left(\frac{A_n}{\sqrt{n}}\right) z = \exp\left(-\sum_{k \ge 1} P_k\left(\frac{A_n}{\sqrt{n}}\right) \frac{z^k}{k}\right), $$

by, for instance, showing this for diagonalizable matrices and the using density of this set of matrices. Finally, taking the determinant we obtain that, for $z \in \mathbb{D}$ small enough,

$$ (2.4) \qquad f_n(z) := \det \left(1 + tz^2 - \frac{z}{\sqrt{n}}A_n\right) e^{-\frac{ntz^2}{2}} = \exp\left(-\sum_{k \ge 1} U_k^{(n)} \frac{z^k}{k}\right), $$

where

$$ (2.5) \qquad U_k^{(n)} := \mathrm{Tr} \left[ P_k \left( \frac{A_n}{\sqrt{n}} \right) \right] + nt\delta_{k=2}. $$
### 3. CONVERGENCE OF THE COEFFICIENTS

In this section, we will prove the (b) part of Lemma 1.3. We thus have to study the convergence in law of the coefficients appearing in $f_{n,t}$. We will give a new expression of these coefficients, using a family of polynomials that we call the modified Chebyshev polynomials introduced in Definition 3.1.

**Definition 3.1 (Modified Chebyshev polynomials).** For $t \in [0, 1]$, the modified Chebyshev polynomials are the polynomials $\{P_k^{(t)}\}_{k \ge 0}$ satisfying the recurrence relation

$$ (3.1) \qquad P_{k+1}^{(t)} = XP_k^{(t)} - tP_{k-1}^{(t)}, \quad P_0^{(t)} = 2, \quad P_1^{(t)} = X. $$

One has an alternative expression for $f_{n,t}$ given by the following proposition, proved in Section 5.1.

**Proposition 3.2 (Trace expression).** For all $n \ge 1$, $t \in [0, 1]$ and $z \in \mathbb{D}$ close to the origin,

$$ (3.2) \qquad f_{n,t}(z) = \exp \left( - \sum_{k \ge 1} U_{k,t}^{(n)} \frac{z^k}{k} \right), $$

where

$$ (3.3) \qquad U_{k,t}^{(n)} = \mathrm{Tr} \left[ P_k^{(t)} \left( \frac{A_{n,t}}{\sqrt{n}} \right) \right] + nt\delta_{k=2}. $$

By Proposition 3.2, the coefficients $(\xi_{0,t}^{(n)}, \dots, \xi_{m,t}^{(n)})$ of Lemma 1.3 (b) associated to $f_{n,t}$ can be expressed as polynomials (which do not depend on $n$) of coefficients $(U_{0,t}^{(n)}, \dots, U_{m,t}^{(n)})$ and vice versa. Thus, showing the convergence in law of $\{\xi_{k,t}^{(n)}\}_k$ is equivalent to showing the convergence in law of $\{U_{k,t}^{(n)}\}_k$. Since it is easier to deal with traces, we will study $U_{k,t}^{(n)}$. This is done in two steps, we study the convergence of the expectation in Lemma 3.3 and the convergence of fluctuations in Proposition 3.4 below. These statements are proved in Sections 5.3 and 5.4 respectively.

**Lemma 3.3 (Convergence of the expectation).** For all $k \ge 1$, denote $e_{k,t}^{(n)} = \mathbb{E}[U_{k,t}^{(n)}]$. Then,

$$ (3.4) \qquad e_{2k,t}^{(n)} = -kt^k + h_{k,t} + O\left(\frac{1}{n}\right) $$

$$ (3.5) \qquad e_{2k+1,t}^{(n)} = o(1) $$

as $n \to \infty$. The explicit expression of $h_{k,t}$ can be found in (5.10).

Proposition 5.4 below shows that the variables $\{V_{k,t}^{(n)} = U_{k,t}^{(n)} - \mathbb{E}[U_{k,t}^{(n)}]\}_{k \ge 0}$ converge in law to a Gaussian family.

**Proposition 3.4 (Convergence of the fluctuations).** For every $t \in [0, 1]$, the family $\{V_{k,t}^{(n)}\}_{k \ge 0}$ converges in law to a family $\{V_{k,t}\}_{k \ge 0}$ of centered and independent complex Gaussians such that $\mathbb{E}[V_{k,t}^2] = kt^k$ and $\mathbb{E}[|V_{k,t}|^2] = k$.

Proposition 3.4 together with Lemma 3.3 show the convergence in distribution of $(U_{0,t}^{(n)}, \dots, U_{m,t}^{(n)})$ to $(V_{0,t}, \dots, V_{m,t}) + (e_{0,t}^{(\infty)}, \dots, e_{m,t}^{(\infty)})$ where $e_{2k,t}^{(\infty)} = -kt^k + h_{k,t}$ and $e_{2k+1,t}^{(\infty)} = 0$ for $k \ge 1$. By Lemma 1.3, the limit of $f_{n,t}$ is the well-defined function $f_t \in \mathcal{H}(\mathbb{D})$ given, for $z$ small, by

$$ (3.6) \qquad f_t(z) = \exp \left( - \sum_{k \ge 1} V_{k,t} \frac{z^k}{k} \right) \exp \left( - \sum_{k \ge 1} e_{k,t}^{(\infty)} \frac{z^k}{k} \right) = \kappa_t(z) e^{-F_t(z)}, $$

where

$$ (3.7) \qquad F_t(z) = \sum_{k \ge 1} V_{k,t} \frac{z^k}{k} \quad \text{and} $$

$$ (3.8) \qquad \kappa_t(z) = \exp \left( - \frac{1}{2} \sum_{k \ge 1} h_{k,t} \frac{z^{2k}}{k} \right) \cdot \exp \left( \frac{tz^2}{2(1 - tz^2)} \right) $$
In particular, the first $m$ coefficients of $f_n$ can be expressed as polynomials which are independent of $n$ of $U_0^{(n)}, \dots, U_m^{(n)}$ and vice versa. Thus, showing the convergence in law of the variables $(U_k^{(n)})_{k \ge 1}$ is equivalent to showing the convergence in law of the coefficients of $f_n$, which is our way of characterizing the limit in Condition (b) of Lemma 2.1. Since it is easier to deal with traces, we will study $U_k^{(n)}$ and prove the convergence stated in the following theorem.

**Theorem 2.4 (Convergence of the traces of Chebyshev polynomials).**

$$ (U_k^{(n)})_{k \ge 1} \xrightarrow{n \to \infty} (\sqrt{k} X_k)_{k \ge 1}, $$

where $(X_k)_{k \ge 1}$ is a family of independent centered complex Gaussian random variables such that $\mathbb{E}[X_k^2] = t^k$ and $\mathbb{E}[|X_k|^2] = 1$.

*Conclusion of the proof of Theorem 1.1.* We use Lemma 2.1 with $h_n = f_n$ and $\xi_k$ being the coefficients of $h(z) = e^{-F_t(z)} = \exp(-\sum_{k \ge 1} X_k \frac{z^k}{\sqrt{k}})$. Theorem 2.2 ensures that Condition (a) is satisfied while Theorem 2.4 ensures that Condition (b) is satisfied. □

**2.1. Tightness: Proof of Theorem 2.2.** Recall that $f_n : \mathbb{D} \to \mathbb{C}$ is given by

$$ f_n(z) = \det \left( 1 + tz^2 - \frac{z}{\sqrt{n}} A_n \right) e^{-\frac{ntz^2}{2}}. $$

and the space $\mathcal{H}(\mathbb{D})$ of holomorphic functions on $\mathbb{D}$ is endowed with the topology of uniform convergence on compact sets. Lemma 2.5 below is the stochastic version of Montel's theorem which reduces the proof of tightness to a control on compact sets.

**Lemma 2.5 (Montel's theorem).** Suppose that for every compact $K \subset \mathbb{D}$, the sequence $(||f_n||_K)_{n \ge 1}$ is tight, where $||f_n||_K = \max_{z \in K} |f_n(z)|$. Then, $\{f_n\}_{n \ge 1}$ is tight.

*Proof.* It is a consequence of the classical Montel's theorem of complex analysis. See, for instance, [Shi12, Proposition 2.5]. □

**Remark 2.6.** By the subharmonicity of $|f_n(z)|^2$, saying that $(\mathbb{E}[||f_n||_K^2])_{n \ge 1}$ is a bounded sequence for every compact $K \subset \mathbb{D}$ is equivalent to saying that $(\sup_{z \in K} \mathbb{E}[|f_n(z)|^2])_{n \ge 1}$ is a bounded sequence for every compact $K \subset \mathbb{D}$. See, for instance, [Shi12, Lemma 2.6]. In the Girko case of [BCG22], one had a remarkable orthogonality of the sub-determinants which led to an upper bound on the desired quantity. As we no longer have this property, our proof is based on [AV03] which exploits the integrability of the Elliptic Ginibre Ensemble.

Our goal is to control $\mathbb{E}[|f_n(z)|^2]$ and, for this, we will begin by giving an explicit expression of the second moment using Hermite polynomials.

**Definition 2.7 (Hermite polynomials).** The Hermite polynomials $\{H_n\}_{n \ge 0}$ are the monic orthonogonal polynomials with respect to the measure $e^{-x^2/2}dx$ on $\mathbb{R}$ so that

$$ \int_{\mathbb{R}} H_n(x) H_m(x) e^{-\frac{x^2}{2}} dx = \sqrt{2\pi n!} \delta_{n,m}. $$

Recall that $g_t(z) = z^{-1} + tz$.

**Lemma 2.8 (Hermite expression of the characteristic polynomial).** For $n \ge 1$ and $z \in \mathbb{D} \setminus \{0\}$, one has the following expression

$$ (2.6) \qquad \mathbb{E}[|f_n(z)|^2] = \frac{n!|z|^{2n}}{n^n} \left| e^{-ntz^2} \sum_{k=0}^{n} \frac{t^k}{k!} \left| H_k \left( \sqrt{\frac{n}{t}} g_t(z) \right) \right| \right|^2. $$

*Proof.* In the case of the Elliptic Ginibre Ensemble given by (1.1), the matrix $A_{n,t}$ has the following density, which can be found in [ADM23, eq. (4)].

$$ (2.7) \qquad d\mathbb{P}_t(M) = \left(\frac{1}{\pi\sqrt{1-t^2}}\right)^{n^2} \exp\left(-\frac{1}{1-t^2}\text{Tr}\left[MM^* - \frac{t}{2}(M^2 + (M^*)^2)\right]\right) dM $$
which is the limit function in Theorem 1.1 with $X_{k,t} = \frac{1}{\sqrt{k}}V_{k,t}$. The series for $F_t$ defines a holomorphic function on the whole disk but it may not be clear that $\sum_{k \ge 1} e_{k,t}^{(\infty)} \frac{z^k}{k}$ also does. Since $f_t(0) = 1$, we may take a holomorphic logarithm and notice that $\sum_{k \ge 1} e_{k,t}^{(\infty)} \frac{z^k}{k}$ converges for $z$ small, but we may have a problem for $z$ far from the origin. Nevertheless, $\kappa_t$ is well defined as $f_t e^{F_t}$ and it is deterministic on $\mathbb{D}$ since we have the deterministic expression near the origin given by (3.8). To show that $\sum_{k \ge 1} e_{k,t}^{(\infty)} \frac{z^k}{k}$ converges for $z \in \mathbb{D}$, it would be enough to show that $f_t e^{F_t}$ has no zeros since, in that case, we can take a holomorphic logarithm of $f_t e^{F_t}$ which would coincide, up to an additive constant, with $\sum_{k \ge 1} e_{k,t}^{(\infty)} \frac{z^k}{k}$ near the origin. That $\kappa_t = f_t e^{F_t}$ has no zeros can be seen by using Lemma 2.6 which implies that

$$0 < \mathbb{E}[|f_t(z)|^2] = |\kappa_t(z)|^2 \mathbb{E}[|e^{-F_t(z)}|^2]$$

for $z \in \mathbb{D} \setminus \{0\}$.

## 4. PROOFS OF STATEMENTS USED FOR TIGHTNESS

**4.1. Proof of Lemma 2.5.** In the case of the Elliptic Ginibre Ensemble given by (1.1), the matrix $A_{n,t}$ has the following density, which can be found in [1, eq. (4)].

$$(4.1) \qquad \mathrm{d}\mathbb{P}_t(M) = \left(\frac{1}{\pi\sqrt{1-t^2}}\right)^{n^2} \exp\left(-\frac{1}{1-t^2}\mathrm{Tr}\left[MM^* - \frac{t}{2}(M^2 + (M^*)^2)\right]\right) \mathrm{d}M$$

which has the form $\mathrm{d}\mathbb{P}_t(M) = w_t(M, M^*)\mathrm{d}M$ associated to the weight function

$$w_t(z) = \frac{1}{\pi\sqrt{1-t^2}} \exp\left(-\frac{1}{1-t^2}\left(|z|^2 - \frac{t}{2}(z^2 + \bar{z}^2)\right)\right) = \frac{1}{\pi\sqrt{1-t^2}} \exp\left(-\left(\frac{x^2}{1+t} + \frac{y^2}{1-t}\right)\right)$$

with $x = \mathrm{Re}(z)$ and $y = \mathrm{Im}(z)$. In order to use the main theorem of [2], we should compute the orthonormal polynomials with respect to $w_t(z)\mathrm{d}z$. Using results in [1, eq. (3)], these polynomials are $\{P_n\}_{n \ge 0}$ given by

$$(4.2) \qquad P_n(z) = \frac{\sqrt{t^n}}{\sqrt{n!}} He_n\left(\frac{z}{\sqrt{t}}\right).$$

Define $R_n(z) = \sqrt{t^n}He_n\left(\frac{z}{\sqrt{t}}\right)$ and $c_n = n!$. The family $\{R_n\}_{n \ge 0}$ are the monic orthogonal polynomials with respect to $w_t(z)\mathrm{d}z$. Using results of [2], one has for $M$ sampled from (4.1) and $u, v \in \mathbb{C}$, one has

$$(4.3) \qquad \mathbb{E} \left[ \det(u - M) \overline{\det(v - M)} \right] = c_N \sum_{k=0}^{N} P_k(u) \overline{P_k(v)}.$$

Since $f_{n,t}(z) = e^{-\frac{ntz^2}{2}} \left(\frac{z}{\sqrt{n}}\right)^n \det\left(\sqrt{n}(z^{-1} + tz) - A_{n,t}\right)$, setting $u = v = g_t(z) = \sqrt{n}(z^{-1} + tz)$ gives

$$\mathbb{E}|f_{n,t}(z)|^2 = \frac{n!|z|^{2n}}{n^n} \left|e^{-ntz^2}\right| \sum_{k=0}^{n} \frac{t^k}{k!} \left|\mathrm{He}_k\left(\sqrt{\frac{n}{t}}g_t(z)\right)\right|^2$$

which is the desired expression of $\mathbb{E}|f_{n,t}(z)|^2$ in terms of Hermite polynomials.

**4.2. Proof of Lemma 2.6.** Recall the function $g_t: \mathbb{D} \to \mathbb{C} \setminus E_t$ given by $g_t(z) = \frac{1}{z} + tz$ and define $L_n: \mathbb{C} \setminus E_t \to [0, \infty)$ by

$$L_n(u) = \sum_{k=0}^{n-1} \frac{t^k}{k!} \left| \mathrm{He}_k \left( \sqrt{\frac{n}{t}} u \right) \right|^2.$$

By using the contour integral representation around a small loop enclosing the origin,

$$L_n(u) = \frac{1}{2\pi i} \oint_0 \frac{e^{nF_u(s)}}{t-s} \frac{ds}{\sqrt{1-s^2}}, \quad \text{with} \quad F_u(s) = \frac{s}{t} \left( \frac{\mathrm{Re}(z)^2}{1+s} + \frac{\mathrm{Im}(z)^2}{1-s} \right) - \log s + \log t,$$
which has the form $d\mathbb{P}_t(M) = w_t(M)dM$ associated to the weight function

$$ 
\begin{aligned}
w_t(z) &= \frac{1}{\pi\sqrt{1-t^2}} \exp\left(-\frac{1}{1-t^2}\left(|z|^2 - \frac{t}{2}(z^2 + \bar{z}^2)\right)\right) \\
&= \frac{1}{\pi\sqrt{1-t^2}} \exp\left(-\left(\frac{x^2}{1+t} + \frac{y^2}{1-t}\right)\right)
\end{aligned}
$$

with $x = \mathrm{Re}(z)$ and $y = \mathrm{Im}(z)$. In order to use the main theorem of [AV03], we should compute the orthonormal polynomials with respect to $w_t(z)dz$. Using [ADM23, eq. (3)], these polynomials are $\{P_n\}_{n\ge0}$ given by

$$ (2.8) \qquad P_n(z) = \frac{\sqrt{t^n}}{\sqrt{n!}} H_n\left(\frac{z}{\sqrt{t}}\right). $$

For $M$ sampled from (2.7) and any $u, v \in \mathbb{C}$, we may use [AV03, eq. (2.11)] to get

$$ (2.9) \qquad \mathbb{E}\left[\det(u - M)\overline{\det(v - M)}\right] = n! \sum_{k=0}^{n} P_k(u)\overline{P_k(v)}, $$

where the global factor $n!$ is the square inverse of the dominant coefficient of $P_n$. Finally, setting $u = v = g_t(z) = z^{-1} + tz$ gives

$$ 
\begin{aligned}
\mathbb{E}[|f_n(z)|^2] &= \mathbb{E}\left[\left|e^{-\frac{ntz^2}{2}}\left(\frac{z}{\sqrt{n}}\right)^n \det\left(\sqrt{n}(z^{-1} + tz) - A_{n,t}\right)\right|^2\right] \\
&= \frac{n!|z|^{2n}}{n^n} \left|e^{-ntz^2}\right| \sum_{k=0}^{n} \frac{t^k}{k!} \left|H_k\left(\sqrt{\frac{n}{t}}g_t(z)\right)\right|^2
\end{aligned}
$$

which is the desired expression of $\mathbb{E}[|f_{n,t}(z)|^2]$ in terms of Hermite polynomials.

□

With the help of the expression (2.6) and using the results from [ADM23], we will control $\mathbb{E}[|f_n(z)|^2]$ uniformly on bounded sets. In fact, [ADM23] allows us to give an explicit expression for the limit of $\mathbb{E}[|f_n(z)|^2]$. Since we do not need an explicit expression, we will only state the following.

**Lemma 2.9 (Convergence of the second moment).** There exists a strictly positive continuous function $\mathcal{F}: \mathbb{D} \setminus \{0\} \to (0, \infty)$ such that, uniformly on compact sets,

$$ \mathbb{E}[|f_n|^2] \xrightarrow{n \to \infty} \mathcal{F}. $$

Since $f_n$ is holomorphic on $\mathbb{D}$, the function $|f_n|^2$ on the circle of radius $r \in (0, 1)$ controls $|f_n|^2$ on the disk $\mathbb{D}_r = \{z \in \mathbb{C} : |z| < r\}$ for $r \in (0, 1)$. This is written in the next proposition.

**Proposition 2.10 (Uniform control).** For every $r \in (0, 1)$ there exists $C_r > 0$ such that

$$ \mathbb{E}[\|f_n\|_{\mathbb{D}_r}^2] \le C_r \text{ for every } n \ge 1. $$

*Proof.* By Lemma 2.9, we have a bound for $\mathbb{E}[|f_n(z)|^2]$ on compact sets of $\mathbb{D} \setminus \{0\}$. This is the same as a bound for $\mathbb{E}[\|f_n\|_K^2]$ for compact sets $K \subset \mathbb{D} \setminus \{0\}$ by Remark 2.6. We may obtain a bound for $\mathbb{E}[\|f_n\|_{\mathbb{D}_r}^2]$ for $r \in (0, 1)$ by using that $\|f_n\|_{\mathbb{D}_r} \le \|f_n\|_{\partial\mathbb{D}_r}$ thanks to the maximum modulus principle. □

*Proof of Lemma 2.9.* Recall the function $g_t: \mathbb{D} \to \mathbb{C} \setminus \mathcal{E}_t$ given by $g_t(z) = \frac{1}{z} + tz$ and define $L_n: \mathbb{C} \setminus \mathcal{E}_t \to [0, \infty)$ by

$$ L_n(u) = \sum_{k=0}^{n-1} \frac{t^k}{k!} \left| H_k \left( \sqrt{\frac{n}{t}} u \right) \right|^2. $$

By using the contour integral representation around a small loop enclosing the origin,

$$ L_n(u) = \frac{1}{2\pi i} \oint_0^1 \frac{e^{nF_u(s)}}{t-s} \frac{ds}{\sqrt{1-s^2}}, \quad \text{with} \quad F_u(s) = \frac{s}{t} \left( \frac{\mathrm{Re}(u)^2}{1+s} + \frac{\mathrm{Im}(u)^2}{1-s} \right) - \log s + \log t, $$
the following has been proved in [1, Theorem II.12, (i)] and [1, Theorem II.13, (i)] in the case $t = 1$ for $u \in \mathbb{C} \setminus E_t$ and $z = g_t^{-1}(u)$,

$$ (4.4) \qquad L_n(u) = \frac{1}{2\pi} \sqrt{\frac{2\pi}{nF_u''(t|z|^2)}} \frac{e^{ntF_u(t|z|^2)}}{\sqrt{1-t^2|z|^4}} \frac{1}{t(1-|z|^2)} \left(1+O\left(\frac{1}{n}\right)\right), $$

where the error term is uniform on compact sets of $\mathbb{C} \setminus E_t$. In our case we need to control

$$ \begin{aligned} \mathbb{E}[|f_{n,t}(z)|^2] &= \frac{n!|z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} \sum_{k=0}^{n} \frac{t^k}{k!} \left| \mathrm{He}_k \left( \sqrt{\frac{n}{t}} \left( \frac{1}{z} + tz \right) \right) \right|^2 \\ &= \frac{n!|z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} L_{n+1} \left( \sqrt{\frac{n}{n+1}} g_t(z) \right). \end{aligned} $$

The term $F_u(t|z|^2)$ in (4.4) can be explicitly calculated by using [1, eq. (32)] or, in a more direct way,

$$ \begin{aligned} F_u(t|z|^2) &= |z|^2 \left( \frac{\mathrm{Re}\left(\frac{1}{z} + tz\right)^2}{1 + t|z|^2} + \frac{\mathrm{Im}\left(\frac{1}{z} + tz\right)^2}{1 - t|z|^2} \right) - \log(t|z|^2) + \log t \\ &= |z|^2 \left( \frac{\mathrm{Re}\left(\frac{1}{z} + t\bar{z}\right)^2}{1 + t|z|^2} + \frac{\mathrm{Im}\left(\frac{1}{z} - t\bar{z}\right)^2}{1 - t|z|^2} \right) - \log(|z|^2) \\ &= |z|^2 \left( (1 + t|z|^2) \mathrm{Re}(1/z)^2 + (1 - t|z|^2) \mathrm{Im}(1/z)^2 \right) - \log(|z|^2) \\ &= 1 + t(z\bar{z})^2 \mathrm{Re}(1/z)^2 - t(z\bar{z})^2 \mathrm{Im}(1/z)^2 - \log(|z|^2) \\ &= 1 + t\mathrm{Re}(\bar{z})^2 - t\mathrm{Im}(\bar{z})^2 - \log(|z|^2) \\ &= 1 + t\mathrm{Re}(z^2) - \log(|z|^2). \end{aligned} $$

By (4.4) and Stirling's formula, we immediately notice that

$$ \frac{n!|z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} L_n(g_t(z)) $$

converges uniformly on compact sets of $\mathbb{D} \setminus \{0\}$ towards

$$ \frac{1}{\sqrt{2\pi F_u''(t|z|^2)(1-t^2|z|^4)t(1-|z|^2)}}. $$

It is now enough to notice that

$$ \frac{L_{n+1}\left(\sqrt{\frac{n}{n+1}} g_t(z)\right)}{L_n(g_t(z))} $$

converges uniformly on compact sets towards a nowhere zero function, the only possible problem being the exponential term $\exp((n+1)G(\sqrt{\frac{n}{n+1}}u) - nG(u))$, where $G(w) = F_w(t|g_t^{-1}(w)|^2)$. But we have the convergence for $u$ uniformly on compact sets of $\mathbb{C} \setminus E_t$

$$ e^{(n+1)G(\sqrt{\frac{n}{n+1}}u) - nG(u)} \underset{n \to \infty}{\longrightarrow} e^{G(u) - \frac{1}{2}\langle \nabla G(u), u \rangle} $$

so that the proof is complete.

**4.3. Proof of Proposition 2.7.** By Lemma 2.6, we have a bound for $\mathbb{E}[|f_{n,t}(z)|^2]$ on compact sets of $\mathbb{D} \setminus \{0\}$. This is the same as a bound for $\mathbb{E}[\|f_{n,t}\|_K^2]$ for compact sets $K \subset \mathbb{D} \setminus \{0\}$ by Remark 2.3. We may obtain a bound for $\mathbb{E}[\|f_{n,t}\|_{D_r}^2]$ for $r \in (0,1)$ by using that $\|f_{n,t}\|_{D_r} \le \|f_{n,t}\|_{\partial D_r}$ by the maximum modulus principle.
the following has been proved in [ADM23, Theorem II.12, (i)] and [ADM23, Theorem II.13, (i)]
for $u \in \mathbb{C} \setminus \mathcal{E}_t$ and $z = g_t^{-1}(u)$,

$$
(2.10) \qquad L_n(u) = \frac{1}{2\pi} \sqrt{\frac{2\pi}{nF_u''(t|z|^2)}} \frac{e^{nF_u(t|z|^2)}}{\sqrt{1-t^2|z|^4} t(1-|z|^2)} \left(1+O\left(\frac{1}{n}\right)\right),
$$

where the error term is uniform on compact sets of $\mathbb{C} \setminus \mathcal{E}_t$. Here $s = t|z|^2$ is a critical point of $F_u$ which can be seen by first calculating

$$
\begin{align*}
tF_u'(s) &= \frac{\mathrm{Re}(u)^2}{1+s} + \frac{\mathrm{Im}(u)^2}{1-s} + s\left(-\frac{\mathrm{Re}(u)^2}{(1+s)^2} + \frac{\mathrm{Im}(u)^2}{(1-s)^2}\right) - \frac{t}{s} \\
&= \frac{\mathrm{Re}(u)^2}{(1+s)^2} + \frac{\mathrm{Im}(u)^2}{(1-s)^2} - \frac{t}{s} \\
&= \frac{t}{s} \left[ \frac{\mathrm{Re}(u)^2}{\left(\frac{1}{\sqrt{s/t}} + t\sqrt{s/t}\right)^2} + \frac{\mathrm{Im}(u)^2}{\left(\frac{1}{\sqrt{s/t}} - t\sqrt{s/t}\right)^2} - 1 \right]
\end{align*}
$$

and then noticing that $g_t$ sends $\{z \in \mathbb{C} : |z| = r\}$ to $\{u \in \mathbb{C} : \frac{\mathrm{Re}(u)^2}{\left(\frac{1}{r} + tr\right)^2} + \frac{\mathrm{Im}(u)^2}{\left(\frac{1}{r} - tr\right)^2} = 1\}$. Moreover, we can find

$$
\begin{align*}
F_u(t|z|^2) &= |z|^2 \left( \frac{\mathrm{Re}(u)^2}{1+t|z|^2} + \frac{\mathrm{Im}(u)^2}{1-t|z|^2} \right) - \log(t|z|^2) + \log t \\
&= 1 + t|z|^4 \left( \frac{\mathrm{Re}(u)^2}{(1+t|z|^2)^2} - \frac{\mathrm{Im}(u)^2}{(1-t|z|^2)^2} \right) - \log(|z|^2) \\
&= 1 + t\mathrm{Re}(z^2) - \log(|z|^2),
\end{align*}
$$

where for the second equality we have used that $F'_u(t|z|^2) = 0$ to simplify the calculation and
for the third equality we have used that $u = z^{-1} + tz$ allows us to relate the real parts $(t|z|^2 +
1)\mathrm{Re}(z) = |z|^2\mathrm{Re}(u)$ and the imaginary parts $(t|z|^2 - 1)\mathrm{Im}(z) = |z|^2\mathrm{Im}(u)$.
In our case we need
to control the second moment

$$
\begin{align*}
\mathbb{E} [|f_n(z)|^2] &= \frac{n! |z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} \sum_{k=0}^{n} \frac{t^k}{k!} \left| H_k \left( \sqrt{\frac{n}{t}} \left( \frac{1}{z} + tz \right) \right) \right|^2 \\
&= \frac{n! |z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} L_{n+1} \left( \sqrt{\frac{n}{n+1}} g_t(z) \right).
\end{align*}
$$

By (2.10) and Stirling's formula, we immediately notice that

$$
\frac{n!|z|^{2n}}{n^n} e^{-nt\mathrm{Re}(z^2)} L_n(g_t(z)) \xrightarrow{n\to\infty} \frac{1}{\sqrt{F_u''(t|z|^2)(1-t^2|z|^4)t(1-|z|^2)}}
$$

uniformly on compact sets of $\mathbb{D} \setminus \{0\}$. It is now enough to notice that the quotient

$$
L_{n+1}(\sqrt{n/(n+1)}g_t(z))/L_n(g_t(z))
$$

converges uniformly on compact sets towards a nowhere zero function. To this end, we must
compute the limit of exp((n+1)G(√n/(n+1)u) - nG(u)), where G(w) = Fw(t|gt⁻¹(w)|²). But,
for u uniformly on compact sets of C \ E,

$$
e^{(n+1)G(\sqrt{\frac{n}{n+1}}u) - nG(u)} \xrightarrow{n \to \infty} e^{G(u) - \frac{1}{2} \langle \nabla G(u), u \rangle}
$$

so that the proof is complete with $\mathcal{F}(z) = \frac{e^{G(u)} - \frac{1}{2} \langle \nabla G(u), u \rangle}{\sqrt{F_u''(t|z|^2)(1-t^2|z|^4)t(1-|z|^2)}}$, $u = g_t(z)$. $\square$

Conclusion of the proof of Theorem 2.2. For every compact subset $K \subset \mathbb{D}$, Proposition 2.10 implies that $(||f_n||_K)_{n \ge 1}$ has its second moment uniformly bounded. The sequence is therefore tight from which one derives the tightness of $\{f_n\}_{n \ge 1}$ using Lemma 2.5 and Remark 2.6. $\square$

<figure><img src="image_4.png" /></figure>
5. PROOFS OF STATEMENTS USED FOR THE CONVERGENCE OF COEFFICIENTS

This section aims at proving the results stated in Section 3. One first proves that the contributions can be parameterised by families of graphs defined in Section 5.2 below. To prove the convergence to a Gaussian family, we will show that asymptotic contributions come from pairings of specific graphs hence the Gaussian aspect using Wick's formula. Furthermore, we compute the limiting covariance function and prove that it is diagonal for the family of modified Chebyshev polynomials defined in (3.1).

5.1. **Proof of Proposition 3.2.** We begin by the following lemma which expresses the modified Chebyshev polynomials in terms of the usual Chebyshev polynomials.

**Lemma 5.1** (Scaling relations for Chebyshev polynomials). Let {$T_k$}$_{k\ge 0}$ be the Chebyshev polynomials of the first kind, i.e., polynomials satisfying the recurrence relation

$$ (5.1) \qquad T_{k+1} = 2XT_k - T_{k-1} $$

with $T_0 = 1$ and $T_1 = X$. For $t \in (0, 1]$ and $w \in \mathbb{C}$, one has

$$ (5.2) \qquad P_k^{(t)}(w) = \sqrt{t} P_k^{(1)}\left(\frac{w}{\sqrt{t}}\right) = 2\sqrt{t} T_k\left(\frac{w}{2\sqrt{t}}\right). $$

*Proof.* The sequences {$P_k^{(t)}$}$_{k\ge 0}$, {$\sqrt{t} P_k^{(1)}\left(\frac{\cdot}{\sqrt{t}}\right)$}$_{k\ge 0}$ and {$2\sqrt{t} T_k\left(\frac{\cdot}{2\sqrt{t}}\right)$}$_{k\ge 0}$ satisfy the same recurrence relation (3.1). □

From the generating function of the Chebyshev polynomials {$T_k$}$_{k\ge 0}$ one has, see [35, eq. (4.7.25)],

$$ (5.3) \qquad \sum_{k \ge 1} 2T_k(w) \frac{z^k}{k} = -\log(1 + w^2 - 2zw). $$

Using (5.2), one gets

$$ \sum_{k \ge 1} P_k^{(t)}(w) \frac{z^k}{k} = \sum_{k \ge 1} 2T_k \left( \frac{w}{2\sqrt{t}} \right) \frac{(\sqrt{t}z)^k}{k} = -\log(1 + tz^2 - zw). $$

Therefore, for $n \ge 1$ and $z$ close enough to the origin, one can write

$$ \left(1 - \left(\frac{A_{n,t}}{\sqrt{n}}\right) z + t z^2\right) = \exp\left(-\sum_{k \ge 1} P_k^{(t)}\left(\frac{A_{n,t}}{\sqrt{n}}\right) \frac{z^k}{k}\right), $$

which is valid for diagonalizable matrices and extended to any matrix by continuity. Since $\det \exp M = \exp \text{Tr}[M]$, the proof of Proposition 3.2 is complete.

5.2. **Graph encoding of traces.** For an integer $m$, we use the notation $[m] = \{1, \dots, m\}$. For a square matrix $A$ of order $n$ and for $k \ge 1$ one has

$$ \text{Tr}[A^k] = \sum_{(i_1, \dots, i_k) \in [n]^k} a_{i_1, i_2} a_{i_2, i_3} \dots a_{i_{k-1}, i_k} a_{i_k, i_1}. $$

Each tuple $i = (i_1, \dots, i_k) \in [n]^k$ can be viewed as a function $i : [k] \to [n]$ defined by $i(j) = i_j$ for every $j \in [k]$. To a function $\psi : [k] \to [n]$, we associate the directed multigraph $G_\psi$ with vertex set $V = \psi([k]) = \text{Im}(\psi)$ and edge multiset $E = (\psi(1), \psi(2)), \dots, (\psi(k), \psi(1))$. There might be loops or multiple edges between vertices. For a directed graph $G = (V, E)$ with vertex $V \subset [n]$, we associate its weight $a_G = \prod_{e \in E} a_{s(e), t(e)}$, where $s(e)$ (respectively $t(e)$) denote the source (respectively the target) of the directed edge $e \in E$. Thus,

$$ \text{Tr}[A^k] = \sum_{i=(i_1, \dots, i_k)} a_{i_1, i_2} a_{i_2, i_3} \dots a_{i_{k-1}, i_k} a_{i_k, i_1} = \sum_{i:[k] \to [n]} a_i $$

where for a tuple $i = (i_1, \dots, i_k)$, we denote by $a_i$ the weight of the graph $G_i$. Thus, the trace of $A^k$ can be seen as a graph-indexed sum of random variables induced by $k$-tuples. We now give some definitions on directed graphs that were introduced in [23].
## 2.2. Convergence of the coefficients: Proof of Theorem 2.4.
We will divide the proof in three parts. First, we show that the expected value $E[U_k^{(n)}]$ converge to zero. As a second step we show that the fluctuations are Gaussian. Finally, we identify the covariance. We will deal with traces of polynomials of $A_n$. Recall that

$$ \mathrm{Tr}[A_n^k] = \sum_{(i_1, \dots, i_k) \in \{1, \dots, n\}^k} a_{i_1 i_2} a_{i_2 i_3} \dots a_{i_{k-1} i_k} a_{i_k, i_1}. $$

It is convenient to think of $(i_1, \dots, i_k)$ as a path of length $k$ with values in $\{1, \dots, n\}$. We use the notation $[m] = \{1, \dots, m\}$ which will be thought of as the abelian group $\mathbb{Z}/m\mathbb{Z}$ when performing additions, and for $\psi : [k] \to [n]$ we denote $a_\psi = \prod_{i=1}^k a_{\psi(i)}\psi(i+1)$. Due to the invariance under permutations of the law of the entries in the matrix $A_n$, quantities such as $E[a_\psi]$ or $E[a_\psi a_\varphi]$ are invariant under $\psi \mapsto T \circ \psi$ for any permutation $T : [n] \to [n]$, i.e., $E[a_\psi] = E[a_{T \circ \psi}]$ or $E[a_\psi a_\varphi] = E[a_{T \circ \psi} a_{T \circ \varphi}]$. In the following, we may identify $\psi$ and $T \circ \psi$ or $(\psi, \varphi)$ and $(T \circ \psi, T \circ \varphi)$ for some purposes.

One can describe these equivalence classes by considering the partition induced by $\psi$ or by $(\psi, \varphi)$ as follows. Let $k_1, \dots, k_\ell$ be positive integers and let us use the notation $\mathbf{k} = (k_1, \dots, k_\ell)$. First recall that the information of $\ell$ maps $(\psi_j : [k_j] \to [n])_{1 \le j \le \ell}$ is contained in the map $\psi = \sqcup \psi_j : \sqcup_{j=1}^\ell [k_j] \to [n]$. For simplicity, we denote $[\mathbf{k}] = \sqcup_{j=1}^\ell [k_j]$.

**Definition 2.11** (Path pattern). Let $\psi : [\mathbf{k}] \to [n]$. The path pattern traced by $\psi$, or the partition induced by $\psi$ is the partition of $[k]$ given by

$$ \pi_{\psi} := \{\psi^{-1}(x) : x \in \psi([\mathbf{k}])\}. $$

We denote by $\mathcal{P}(\mathbf{k})$ the set of path patterns, or partitions, of $[\mathbf{k}]$. We use these path patterns to write, for instance,

$$ (2.11) \qquad \mathrm{Tr}[A_n^k] = \sum_{\pi \in \mathcal{P}(k)} A_{\pi}^{(n)}, \quad \text{where } A_{\pi}^{(n)} := \sum_{\substack{\psi : [k] \to [n] \\ \pi_{\psi} = \pi}} a_{\psi}. $$

Since the set $\mathcal{P}(k)$ of possible path patterns is finite, it will be convenient to study the limit of $A_{\pi}^{(n)}$ for a given $\pi \in \mathcal{P}(\mathbf{k})$. We chose to consider path patterns because they are convenient to count. We proceed to define the directed multigraph $G_{\psi}$ associated to a map $\psi = \sqcup_{j=1}^\ell \psi_j : [\mathbf{k}] \to [n]$ in a more explicit way in Definition 2.12. The motivation is that $E[a_\psi]$ depends only on isomorphism class of the graph $G_{\psi}$.

**Definition 2.12** (Graph of $\psi$). Let $\psi = \sqcup_{j=1}^\ell \psi_j : [\mathbf{k}] \to [n]$. The graph $G_{\psi} = (V, E, s, t)$ is the directed multigraph having vertex set $V = \psi([\mathbf{k}])$, edge set $E = \sqcup_{j=1}^\ell E_j$ where we defined $E_j = \{(i, i+1) : i \in [k_j]\}$, source map $s = \sqcup_{j=1}^\ell s_j$ with $s_j : E_j \to V$ given by $s(i, i+1) = \psi_j(i)$ and target map $t = \sqcup_{j=1}^\ell t_j$ with $t_j : E_j \to V$ given by $t(i, i+1) = \psi_j(i+1)$.

More concisely but equivalently for our purposes, if $I_j : [k_j] \to [\mathbf{k}]$ are the canonical inclusion maps, we may say that $G_{\psi}$ is the directed multigraph with vertex set $V = \psi([\mathbf{k}])$ and edge multiset $\{(I_j(i), I_j(i+1)) : j \in [\ell], i \in [k_j]\}$. For a directed multigraph $G = (V, E, s, t)$, denote by $\bar{G} = (V, \bar{E})$ the undirected simple graph with edge set $\bar{E} = \{\{s(e), t(e)\} : e \in E\}$. So, $\bar{G}_{\psi}$ is the undirected simple graph (with possible loops) induced by $\psi$.

Notice that the isomorphism class of $G_{\psi}$ depends only on the path pattern $\pi \in \mathcal{P}(\mathbf{k})$ induced by $\psi$. We denote by $G_{\pi}$ this isomorphism class. When dealing with $G_{\pi}$ for purposes where the quantities only depend on the path pattern, we may write $V_{\pi}$ and $E_{\pi}$ for its set of vertices and edges viewed as $V_{\psi}$ and $E_{\psi}$ for some $\psi$ inducing the partition $\pi$. The same goes for $\bar{G}_{\pi}$. Moreover, if $\psi = \sqcup_{i=1}^\ell \psi_j$ and $\pi = \pi_{\psi}$, we can consider $\pi_j$, the path pattern induced by $\psi_j$ or, what is the same, the partition induced by $\pi$ via the canonical inclusion $I_j : [k_j] \to [\mathbf{k}]$. It will then be convenient to see $G_{\pi_i}$ as a subgraph of $G_{\pi}$ and $\bar{G}_{\pi_i}$ as a subgraph of $\bar{G}_{\pi}$. This suggests
**Definition 5.2.** Let $G = (V, E)$ be a directed multigraph. For vertices $u \neq v$, we say that two distinct directed edges with both endpoints in $\{u, v\}$ are **twins**. If two twin edges have the same source (or equivalently the same target), then they are called **parallel**. Otherwise, they are called **opposite**. If the number of edges between $u$ and $v$ counted with multiplicities is two, the edges $(u, v)$ and $(v, u)$ are both called double, double parallel or double opposite if one wants to make the distinction. An edge is called **simple** if it has no twin edge and multiple otherwise.

**Definition 5.3.** Let $G = (V, E)$ be a directed multigraph. We associate the undirected graph $\bar{G}$ with same vertex set $V$ and edge set $\bar{E}$ such that for $u, v \in V, \{u, v\} \in \bar{E}$ $\iff$ $(u, v)$ or $(v, u) \in E$. Furthermore, to the graph $G$, we associate the pair $(q_1, q_2)$ defined by

$$ (5.4) \qquad q_1 = |\bar{E}| - \frac{|E|}{2} \quad \text{and} $$

$$ (5.5) \qquad q_2 = |V| - |\bar{E}|. $$

The construction $G \mapsto \bar{G}$ turns a directed graph into an undirected graph where edge multiplicities and orientations are forgotten. Values of $q_2$ in $\{0, 1\}$ characterise the graph $\bar{G} = (V, \bar{E})$ by the following proposition. We refer the reader to [29] for the proofs of these characterisations.

**Proposition 5.4.** Consider an undirected graph $G = (V, E)$. Then,

(i) $G$ is a tree if and only if $|V| = |E| + 1$.

(ii) $G$ is unicyclic (i.e has only one cycle) if and only if $|V| = |E|$.

We now introduce three types of graphs that will play a fundamental role in our analysis.

**Definition 5.5 (Types of graphs).** Let $G = (V, E)$ be a connected directed graph. One says that

* $G$ is of **double tree type** whenever $(q_1, q_2) = (0, 1)$.

* $G$ is of **double unicyclic type** whenever $(q_1, q_2) = (0, 0)$.

* $G$ is of **2-4 tree type** whenever $(q_1, q_2) = (-1, 1)$.

We denote by $\mathcal{T}_k^{(n)}$ the set of double tree type graphs $G$ with $k$ edges on vertex set $V \subset [n]$. Note that $\mathcal{T}_k^{(n)}$ is empty if $k$ is odd. Define $C_k^{(n)}$ the directed cycles on $k$ vertices. Finally, we say that $G$ is a **double unicyclic tree** if

* $q_2 = 0$, which is equivalent to say that $\bar{G}$ is unicyclic.

* Edges of the cycle of $\bar{G}$ form a directed cycle of simple edges in $G$.

* Edges outside the cycle of $\bar{G}$ are double opposite edges in $G$.

We say that a double unicyclic tree has parameters $(k, q)$ if its cycle has length $k - 2q \geq 1$ and has $q$ double opposite edges. Let $\mathrm{UC}^{(n)}(k, q)$ be the set of double unicyclic trees with parameters $(k, q)$.

In the case of even multiplicities, one has the following descriptions of the graphs in Definition 5.5.

**Proposition 5.6 (Characterisation of graphs).** Let $G = (V, E)$ be a connected directed graph. Assume that each edge of $E$ has multiplicity at least two, so that $G$ has no simple edge. Then,

(i) $G$ is of double tree type if and only if $\bar{G}$ is a tree and each edge of $E$ has multiplicity two.

(ii) $G$ is of double unicyclic type if and only if $\bar{G}$ is unicyclic and each edge of $E$ has multiplicity two.

Furthermore, if edges of $E$ have even multiplicities, then

(iii) $G$ is of 2-4 tree type if and only if $\bar{G}$ is a tree and each edge of $E$ has multiplicity two, except for one edge with multiplicity four.

*Proof.* The statements about $\bar{G}$ being a tree or a unicyclic graph do not require the multiplicity assumption and are consequences of Proposition 5.4 together with the definition of $q_2$. For $e \in \bar{E}$, denote $m_e$ the multiplicity of $e$ in $E$ which is at least two by assumption. If $q_1 = 0$, then $|E| = 2|\bar{E}| = \sum_{e \in \bar{E}} m_e$ so that $m_e = 2$ for all $e \in \bar{E}$ which proves (i) and (ii). In the case of even multiplicities and $q_1 = -1$ one has $|E| = 2|\bar{E}| + 2$. For $k \geq 1$, let $x_{2k} \geq 0$ be the number of edges in $\bar{E}$ with multiplicity $2k$ in $E$. Then, $|\bar{E}| = \sum_{k \geq 1} x_{2k}$ and $|E| = \sum_{k \geq 1} 2kx_{2k}$ so that

$$ (5.6) \qquad 2 \sum_{k \geq 1} x_{2k} + 2 = \sum_{k \geq 1} 2kx_{2k} $$
to consider the equivalence class of $(G_\psi, G_{\psi_1}, \dots, G_{\psi_l})$ and viewing $G_{\pi_i}$ as a subgraph of $G_\pi$ by taking representatives of this equivalence class.

We will see below in the proof of Lemma 2.15 and in Proposition 2.17 the following behaviors for $n^{-k/2}A_{\pi}^{(n)}$. The first two cases correspond to the random part in the limit whereas the last two cases give deterministic contributions.

(1) If $\bar{G}_{\pi}$ is unicyclic, i.e., if it contains one and only one cycle (that may be a loop), with each edge of the cycle being traversed just once and every edge outside the cycle being traversed once in each direction in $G_{\pi}$, then $n^{-k/2}A_{\pi}^{(n)}$ converges to a centered Gaussian.

(2) If $\bar{G}_{\pi}$ is a tree where each edge is traversed twice in $G_{\pi}$, once in each direction ($\pi$ is called a rooted plane tree in this case, see Definition 2.16), then $n^{-k/2}A_{\pi}^{(n)} - n$ converges to a non-centered Gaussian. This case can be thought of as a unicyclic graph with a cycle of length two.

(3) If $\bar{G}_{\pi}$ is a tree but there is an edge traversed four times in $G_{\pi}$, two in each direction, then $n^{-k/2}A_{\pi}^{(n)}$ converges to a constant. Those $\pi$ can be thought of as rooted plane trees with two vertices at distance two identified.

(4) If $\bar{G}_{\pi}$ is unicyclic but with each edge traversed two times, once in each direction, then $n^{-k/2}A_{\pi}^{(n)}$ also converges to a constant. Those $\pi$ can be thought of as rooted plane trees with two vertices at distance different from two identified.

(5) All other cases converge to zero.

<figure><img src="image_5.png" /><figcaption>FIGURE 2. Graphs for which $n^{-k/2}A_{\pi}^{(n)}$ has a non-trivial limit. Top left, top right, bottom left, bottom right correspond respectively to cases 1, 2, 3 and 4 above. Green dots denote the vertices and red stars denote initial points.</figcaption></figure>

Before turning to the convergence of the expectation and the fluctuations of traces, we state a general lemma which will be used frequently.

**Lemma 2.13** (Even multiplicities in trees). Let $\psi : [k] \to [n]$ be such that $\bar{G}_{\psi}$ is a tree. Then, each edge of $\bar{G}_{\psi}$ has an even multiplicity in $G_{\psi}$ with an equal number of oriented edges in each direction.

*Proof.* Let $e = (i, i+1) \in E_{\psi}$ be an edge. Denote $u = \psi(i)$ and $v = \psi(i+1)$ its endpoints. Since $\psi$ traces a cycle, there exists a sequence of edges $(i+1, i+2), \dots, (i+r, i+r+1)$ such that $s((i+1, i+2)) = v$ and $t((i+r, i+r+1)) = u$ for some $1 \le r \le k-1$. Take such $r$ minimal. Since $\bar{G}_{\psi}$ is a tree, one must have $s((i+r, i+r+1)) = v$ otherwise the graph $\bar{G}_{\psi}$ would have a
which gives

$$ (5.7) \qquad \sum_{k \ge 2} (k-1)x_{2k} = 1 $$

and therefore, $x_{2k} = 0$ if $k \ge 3$ and $x_4 = 1$. There is exactly one edge with multiplicity four and all other edges have multiplicity two which proves (iii). $\square$

The following proposition asserts that if the graph $G_i$ associated to a $k$-tuple $i = (i_1, \dots, i_k)$ is of double tree type, then every edge of $G_i$ is double opposite.

**Proposition 5.7 (Double tree types have opposite branches).** Let $i = (i_1, \dots, i_k)$ be a $k$-tuple such that $G_i$ is of double tree type. Then, each edge of $G_i$ is double opposite.

*Proof.* Let us prove that each edge of $G_i$ has multiplicity at least two. Denote $\bar{G} = (V, \bar{E})$ the associated graph from Definition 5.3.

Let $e = (i_r, i_{r+1})$ be a directed edge in $G_i$. Then, as $(i_r, i_{r+1}, \dots, i_k, i_1, i_2, \dots, i_{r-1})$ forms a directed cycle in $G_i$, there must exists an edge $(i_s, i_{s+1} = i_r)$ for some $s \in [k]$. Consider the first such $s$ in the ordered set $\{r+1, \dots, k, 1, 2, \dots, r-1\}$. If $i_s \neq i_{r+1}$, then there is a cycle $(i_r, i_{r+1}, \dots, i_s)$ in $G_i$ which would give a cycle in $\bar{G}$ which would not be a tree. This would contradict the assumption that $q_2 = 1$ and by extension that $G_i$ is of double tree type. Therefore, $i_s = i_{r+1}$ which implies that every edge of $G_i$ has multiplicity at least two. Since $q_1 = 0$, each edge has exactly multiplicity two by Proposition 5.6 (i) and the two edges twin edges are $e = (i_r, i_{r+1})$, $e' = (i_s = i_{r+1}; i_r)$ which are opposite. $\square$

**Remark 5.8.** By the same argument, one can prove that if $G_i$ is of 2-4 tree type, then double edges are double opposite and the quadruple edge consists of two pairs of opposite edges.

By extension of the mapping $i = (i_1, \dots, i_k) \mapsto G_i$, we say that a $k$-tuple $i$ has double tree type (respectively a double unicyclic type, 2-4 tree type) if the corresponding graph $G_i$ has. The same applies to being a double unicyclic tree. We identify the directed $k$-cycles with $UC^{(n)}(k, 0)$ the set of double unicyclic trees having no tree branches outside its cycle. Likewise, for even values of $k$, we identify $\mathcal{T}_k^{(n)}$ with $UC^{(n)}(k, \frac{k}{2} - 1)$ thanks to Proposition 5.7.

For future asymptotics, we are interested in computing the number of $k$-tuples $i$ such that $G_i \in \mathcal{T}_k^{(n)}$.

**Lemma 5.9 (Double tree type enumeration).** Let $k = 2m$ be an even integer. Then,

$$ (5.8) \qquad \mathrm{Card} \left( \left\{ i : [k] \to [n] \mid G_i \in \mathcal{T}_k^{(n)} \right\} \right) = n(n-1)\dots(n-m)C_m := (n)_{m+1}C_m. $$

where $C_m = \frac{1}{m+1} \binom{2m}{m}$ is the $m$-th Catalan number.

*Proof.* The result is a direct consequence of [6, Lemma 2.2 and Lemma 2.4], as graphs $G_i$ of double tree type are exactly the $\Gamma_1(2m)$ graphs considered in [6]. $\square$

**5.3. Proof of Lemma 3.3.** This section is dedicated to the proof of Lemma 3.3 using the graph encoding of the previous section. Before we compute the expectation of traces involving the modified Chebyshev polynomials, let us prove the convergence for the expectation of monomials. This is the purpose of Lemma 5.10 proved below.

**Lemma 5.10 (Monomial expectation).** Let $m \ge 1$. Then, as $n \to \infty$,

$$ (i) \quad \mathrm{E} \left[ \mathrm{Tr} \left[ \left( \frac{A}{\sqrt{n}} \right)^{2m} \right] \right] = n^{-m} (n)_{m+1} C_m t^m + l_{m,t}^{(U)} + O\left(\frac{1}{n}\right). $$

$$ (ii) \quad \mathrm{E} \left[ \mathrm{Tr} \left[ \left( \frac{A}{\sqrt{n}} \right)^{2m+1} \right] \right] = o(1). $$

where $l_{m,t}^{(U)}$ is a constant that only depends on $m$ and $t$.

From the recurrence relation (3.1), one notices that the polynomials $(P_{2k}^{(t)})_k$ are even and $(P_{2k+1}^{(t)})_k$ are odd. We give an explicit formula for the coefficients that will be helpful later.
cycle. Removing the edges $(i, i+1), \dots, (i+r, i+r+1)$ from $G_\psi$ shows the result by induction on $|\{e \in E_\psi : \{s(e), t(e)\} = \{u, v\}\}|$. $\square$

**2.2.1. Convergence of expected value.** Recall that

$$U_k^{(n)} = \mathrm{Tr} \left[ P_k \left( \frac{A_n}{\sqrt{n}} \right) \right] + nt\delta_{k=2}.$$

as defined previously in (2.5). The main result of this section is Proposition 2.14 which shows the convergence of $E[U_k^{(n)}]$ for each $k \ge 1$.

**Proposition 2.14 (Vanishing Chebyshev expectation).** For every $k \ge 1$,

$$\lim_{n \to \infty} \mathbb{E}[U_k^{(n)}] = 0.$$

The proof is based on the convergence of traces of monomials which is Lemma 2.15. As for Wigner matrices, Catalan numbers $C_p = \frac{1}{p+1} \binom{2p}{p}$ are involved.

**Lemma 2.15 (Monomial expectation).** Let $p \ge 1$. Then,

$$(i) \quad \mathbb{E} \left[ \mathrm{Tr} \left[ \left( \frac{A_n}{\sqrt{n}} \right)^{2p} \right] \right] - nC_p t^p \xrightarrow{n \to \infty} 0 \quad \text{and}$$

$$(ii) \quad \mathbb{E} \left[ \mathrm{Tr} \left[ \left( \frac{A_n}{\sqrt{n}} \right)^{2p+1} \right] \right] \xrightarrow{n \to \infty} 0.$$

We now prove Proposition 2.14, using the asymptotics of Lemma 2.15.

*Proof of Proposition 2.14.* We begin by noticing that, if $\sigma_t$ denotes the uniform probability measure on the ellipse $\mathcal{E}_t$ defined in (1.5),

$$\int_{\mathcal{E}_t} w^{2p} d\sigma_t(w) = \frac{1}{p+1} \binom{2p}{p} t^p = C_p t^p$$

which can be obtained by using elliptic coordinates. Then, from Lemma 2.15, (i), the asymptotic

$$\mathbb{E} \left[ \mathrm{Tr} \left[ \left( \frac{A_n}{\sqrt{n}} \right)^{2p} \right] \right] = nC_p t^p + o(1) = n \int_{\mathcal{E}_t} w^{2p} d\sigma_t(w) + o(1)$$

implies that

$$\mathbb{E} \left[ \mathrm{Tr} \left[ P_k^{(t)} \left( \frac{A_n}{\sqrt{n}} \right) \right] \right] = n \int_{\mathcal{E}_t} P_k^{(t)}(w) d\sigma_t(w) + o(1).$$

It remains to compute $\int_{\mathcal{E}_t} P_k^{(t)}(w) d\sigma_t(w)$. To achieve this, we define

$$\mathcal{M}(z) = \int_{\mathcal{E}_\rho} \log (1 + tz^2 - zw) \, d\sigma_t(w)$$

which is holomorphic for $z \in \mathbb{D}$ with

$$\frac{d^k}{dz^k} \mathcal{M}(z) = \int_{\mathcal{E}_t} \frac{d^k}{dz^k} \log (1 + tz^2 - zw) \, d\sigma_t(w).$$

In particular, the coefficient $z^k$ of $\mathcal{M}(z)$ is the integral of the coefficient $z^k$ of $\log (1 + tz^2 - zw)$,

$$[z^k] \mathcal{M}(z) = - \int_{\mathcal{E}_t} \frac{P_k^{(t)}(w)}{k} d\sigma_t(w).$$

To make a connection with classical Hermitian random matrix theory we can compute

$$S_t(z) = \int_{\mathcal{E}_t} \frac{d\sigma_t(w)}{z-w} = \frac{1}{z} \sum_{p=0}^{\infty} \left(\frac{\sqrt{t}}{z}\right)^{2p} C_p = \frac{1}{\sqrt{t}} h\left(\frac{z}{\sqrt{t}}\right),$$

where $h(z) = \frac{z-\sqrt{z^2-4}}{2}$ is the holomorphic solution to $z = h + \frac{1}{h}$ that goes to zero at infinity, which we may recognize as the Cauchy-Stieltjes transform of the semi-circle distribution (case
**Lemma 5.11** (Coefficients of modified Chebyshev polynomials). For $k \ge 1$, $t \in [0, 1]$ and $0 \le j \le k/2$, let $\alpha_{k-2j}^{(k,t)}$ be the coefficient of $X^{k-2j}$ in $P_k^{(t)}$, so that $P_k^{(t)} = \sum_{j \ge 0} \alpha_{k-2j}^{(k,t)} X^{k-2j}$. Then,

$$ (5.9) \qquad \alpha_{k-2j}^{(k,t)} = \begin{cases} (-t)^{\frac{j}{k-j}} \binom{k-j}{j} & \text{if } k-2j \ge 0 \\ 0 & \text{otherwise.} \end{cases} $$

*Proof.* The result is a direct consequence of the recurrence relation (3.1) since we have

$$ \forall 0 \le 2j \le k+1: \qquad \alpha_{k+1-2j}^{(k+1,t)} = \alpha_{k-2j}^{(k,t)} - t\alpha_{k+1-2j}^{(k-1,t)}. $$

□

Before turning to the proof of Lemma 3.3, we will need the following lemma which is proved in Section 5.3 that gives the leading term in the development of the modified Chebyshev polynomial. By Lemma 5.10, each even monomial has an asymptotic leading term of order $n$ which factors in $P_k^{(t)}$. However, Lemma 5.12 shows that algebraic relations in coefficients of the Chebyshev polynomials make this diverging contribution vanish.

**Lemma 5.12** (Double tree type contribution to expectation). Let $k = 2m$ be an even integer. Then,

$$ \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} n^{-m+q} (t^{m-q} (n)_{m-q+1} C_{m-q}) + nt\delta_{m=1} = l_{m,t}^{(T)} + O\left(\frac{1}{n}\right) $$

where for $m \ge 1$,

$$ l_{m,t}^{(T)} = -\frac{1}{2} \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} t^{m-q} C_{m-q}(m-q+1)(m-q) = -mt^m. $$

We now prove Lemma 3.3, using the Lemmas 5.10 and 5.12.

*Proof of Lemma 3.3.* By linearity, one has

$$ e_{k,t}^{(n)} = \sum_{q=0}^{[k/2]} \alpha_{k-2q}^{(k,t)} n^{-\frac{k}{2}+q} \mathbb{E} \left[ \mathrm{Tr} \left[ A_{n,t}^{k-2q} \right] \right] + nt\delta_{k=2}. $$

By Lemma 5.10 applied to each monomial $n^{-\frac{k}{2}+q} \mathbb{E} \left[ \mathrm{Tr} \left[ A_{n,t}^{k-2q} \right] \right]$, one has

$$ \begin{aligned} e_{2m+1,t}^{(n)} &= o(1) \\ e_{2m,t}^{(n)} &= \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} \left( n^{-m+q} t^{m-q} (n)_{m-q+1} C_{m-q} + l_{2m-2q,t}^{(U)} + O\left(\frac{1}{n}\right) \right) + nt\delta_{m=1} \\ &= \sum_{q=0}^{m} \left( \alpha_{2m-2q}^{(2m,t)} n^{-m+q} t^{m-q} (n)_{m-q+1} C_{m-q} \right) + nt\delta_{m=1} + h_{m,t} + O\left(\frac{1}{n}\right) \end{aligned} $$

where

$$ (5.10) \qquad h_{m,t} = \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} l_{2m-2q,t}^{(U)} $$

(we recall that $\alpha$, $l^{(U)}$ are defined respectively in (5.9) and (5.12)). By Lemma 5.12, we get

$$ (5.11) \qquad e_{2m,t}^{(n)} = l_{m}^{(T)} + h_{m,t} + O\left(\frac{1}{n}\right) = -mt^m + h_{m,t} + O\left(\frac{1}{n}\right) $$

from which one derives Lemma 3.3.

□

**Remark 5.13** (On the multiplicative factor $\exp(-ntz^2/2)$). Lemma 3.3 shows that in order to have convergence, one needs to consider the expectation of the random variables $\left(\mathrm{Tr} \left[ \left( \frac{A}{\sqrt{n}} \right)^{2m} \right] + nt\delta_{m=1} \right)$. The term $nt\delta_{m=1}$ corresponds to the additional factor $\exp(-ntz^2/2)$ in our definition of the normalised
t = 1). In particular, $S_t(z^{-1} + tz) = z$ and we may connect the function $S_t$ with $\mathcal{M}$ by taking its derivative

$$ 
\begin{aligned}
\frac{d}{dz}\mathcal{M}(z) &= \frac{1}{z} + \left(-\frac{1}{z^2} + t\right) \int_{\mathcal{E}_t} \frac{1}{\frac{1}{z} + tz - w} d\sigma_t(w) = \frac{1}{z} + \left(-\frac{1}{z^2} + t\right) S_t\left(\frac{1}{z} + tz\right) \\
&= \frac{1}{z} + \left(-\frac{1}{z^2} + t\right) z = tz.
\end{aligned}
$$

So, since we also know that $\mathcal{M}(0) = 0$, we obtain $\mathcal{M}(z) = \frac{tz^2}{2}$, which implies that

$$ \int_{\mathcal{E}_t} P_k^{(t)}(w) d\sigma_t(w) = 0 \text{ for } k \neq 2 \quad \text{while} \quad \int_{\mathcal{E}_t} P_2^{(t)}(w) d\sigma_t(w) = -t. $$

□

We now turn to the proof of Lemma 2.15.

*Proof of Lemma 2.15.* For $k \ge 1$, write

$$ \mathbb{E}\left[\mathrm{Tr}\left[\left(\frac{A_n}{\sqrt{n}}\right)^k\right]\right] = n^{-k/2} \sum_{\pi \in \mathcal{P}(k)} \mathbb{E}[A_{\pi}^{(n)}] = n^{-k/2} \sum_{\pi \in \mathcal{P}(k)} C_{\pi}^{(n)} \alpha_{\pi}, $$

with $\alpha_\pi$ being the common value of $\mathbb{E}[a_\psi]$ for $\psi$ inducing the partition $\pi$ and $C_\pi^{(n)}$ is the number of $\psi: [k] \to [n]$ that induce $\pi$. Since the choice of $\psi$ inducing $\pi$ amounts to choosing the image of each block of $\pi$, we find that, for $n \ge k$, $C_\pi^{(n)} = n^{|\pi|}$, where $|\pi|$ denotes the number of blocks of $\pi$ and $n^k = n(n-1)\dots(n-k+1)$ is the falling factorial. Let us investigate the cases where $\alpha_\pi \ne 0$.

As entries are centered, for $\alpha_\pi$ not to vanish, every edge of $G_\pi$ has to be at least double. Let us denote this set by $\mathcal{D}_k = \{\pi \in \mathcal{P}(k) : G_\pi$ has no simple edges$\}$. Since for $\pi \in \mathcal{D}_k$ we have $2|\bar{E}_\pi| \le |E_\pi|$, we obtain $|V_\pi| \le |\bar{E}_\pi| + 1 \le \frac{k}{2} + 1$ so that, for $n \ge k$,

$$ \mathbb{E}\left[\mathrm{Tr}\left[\left(\frac{A_n}{\sqrt{n}}\right)^k\right]\right] = n^{-k/2} \sum_{\ell=1}^{\lfloor\frac{k}{2}+1\rfloor} n^{\ell} \sum_{\pi \in \mathcal{D}_k, |\pi|=\ell} \alpha_\pi $$

For odd values of $k$, the only possible contribution to the limit comes from $\ell = (k+1)/2$. For this $\ell$, we would have $|V_\pi| = (k+1)/2$ and, since $|V_\pi|-1 \le |\bar{E}_\pi| \le k/2$, we must have $|\bar{E}_\pi| = (k-1)/2$ so that $\bar{G}_\pi$ is a tree. The fact that $|\bar{E}_\pi| = (k-1)/2$ and that each edge is at least double in $E_\pi$ tells us that there is exactly one edge that is triple in $E_\pi$. This would contradict the result of Lemma 2.13 so that there is no $\pi \in \mathcal{D}_k$ satisfying $|\pi| = (k+1)/2$. Then, for odd $k$ the expected value goes to zero.

For even values $k = 2p$,

$$ \mathbb{E}\left[\mathrm{Tr}\left[\left(\frac{A_n}{\sqrt{n}}\right)^{2p}\right]\right] = n \sum_{\pi \in \mathcal{D}_{2p}, |\pi|=p+1} \alpha_{\pi} + \left[-\binom{p+1}{2}\right] \sum_{\pi \in \mathcal{D}_{2p}, |\pi|=p+1} \alpha_{\pi} + \sum_{\pi \in \mathcal{D}_{2p}, |\pi|=p} \alpha_{\pi} + O\left(\frac{1}{n}\right). $$

For $\pi \in \mathcal{D}_{2p}$ satisfying $|\pi| = p+1$, the graph $\bar{G}_\pi$ has $p$ edges and $p+1$ vertices so that it is a tree and $\pi$ draws a path on $\bar{G}_\pi$ with exactly two edges passing through each edge in $\bar{E}_\pi$, necessarily once in each direction according to Lemma 2.13. Denote by $\mathcal{T}_p$ the set of these path patterns, also known as rooted plane trees. Then, $\alpha_\pi = t^p$ for every $\pi \in \mathcal{T}_p$ so that

$$ \sum_{\pi \in \mathcal{D}_{2p}, |\pi|=p+1} \alpha_{\pi} = |\mathcal{T}_p|t^p. $$

For $\pi \in \mathcal{D}_{2p}$ satisfying $|\pi| = p$, we can either have $|\bar{E}_\pi| = p$ or $p-1$.

* If $|\bar{E}_\pi| = p$, then $\bar{G}_\pi$ is a graph with $p$ edges and $p$ vertices so that it is a unicyclic graph. For $\alpha_\pi$ to be non-zero, the path drawn by $\pi$ on $\bar{G}_\pi$ has to have exactly two edges passing in opposite directions through each edge in $\bar{E}_\pi$ (its cycle can be a loop). The value of $\alpha_\pi$ in this case is $\mathbb{E}[a_{12}a_{21}]^p = t^p$ in the case of no loops and $\mathbb{E}[a_{12}a_{21}]^{p-1}\mathbb{E}[a_{11}^2] = t^{p-1}t$.
characteristic polynomial (1.3) that did not appear in the Girko setup of [12] since the latter corresponds
to the case $t = 0$. One could have seen this additional term for $k = 2$ by hand as $P_2^{(t)} = X^2 - 2t$ and

$$
\frac{1}{n} \mathrm{Tr}[A^2] = \frac{1}{n} \sum_{i=1}^{n} a_{i,i}^2 + \frac{2}{n} \sum_{i<j} a_{i,j} a_{j,i}.
$$

The first sum converges almost surely to $\mathbb{E}[a_{1,1}^2]$ by the law of large numbers. Since $\mathbb{E}[a_{i,j}a_{j,i}] = t$,

$$
\frac{2}{n} \sum_{i<j} a_{i,j} a_{j,i} = \frac{2}{n} \sum_{i<j} (a_{i,j} a_{j,i} - t) + \frac{2n(n-1)}{2} t.
$$

One sees the diverging term $\frac{2}{n} \frac{n(n-1)}{2} t \sim nt$ as $n \to \infty$. Thus,

$$
\mathrm{Tr} \left[ P_2^{(t)} \left( \frac{A_{n,t}}{\sqrt{n}} \right) \right] + nt = \frac{1}{n} \sum_{i=1}^{n} a_{i,i}^2 + \frac{2}{n} \sqrt{\binom{n}{2}} \cdot \frac{1}{\sqrt{\binom{n}{2}}} \sum_{i<j} (a_{i,j} a_{j,i} - t) + \frac{2n(n-1)}{2} t - 2nt + nt.
$$

The first right-hand side term converges almost surely to the constant $\mathbb{E}[a_{1,1}^2]$ while the sum of the three last terms is the constant $-t$. By the central limit theorem, the middle term converges in distribution to a normal distribution. Thus, $\mathrm{Tr}\left[P_2^{(t)}\left(\frac{A_{n,t}}{\sqrt{n}}\right)\right] + nt$ converges in distribution to a complex Gaussian random variable $\xi$ with parameters $\mathbb{E}[\xi] = \mathbb{E}[a_{1,1}^2] - t = 0$, $\mathbb{E}[\xi^2] = 2(\mathbb{E}[a_{1,2}^2 a_{2,1}^2] - t^2) = 2t^2$ and $\mathbb{E}[|\xi|^2] = 2(\mathbb{E}[|a_{1,2} a_{2,1}|^2] - t^2) = 2$. Note that this is exactly the result stated in Proposition 3.4 for $k=2$. Remark furthermore that one needs to have $\mathbb{E}[|a_{1,2}|^2|a_{2,1}|^2] < \infty$ for the variance of $\xi$ to be defined. This is the conjectured optimal moment condition for the normalised characteristic polynomial to converge given in Subsection (1.4.1).

We now turn to the proofs of Lemma 5.10 and Lemma 5.12.

*Proof of Lemma 5.10.* Let us take $k \ge 1$ and write

$$
\mathbb{E} \mathrm{Tr} \left[ \left( \frac{A}{\sqrt{n}} \right)^k \right] = n^{-k/2} \sum_{i:[k] \to [n]} \mathbb{E}[a_i].
$$

As coefficients are centered, the only $k$-tuples $i$ such that $\mathbb{E}[a_i]$ is non-vanishing are tuples for which the associated graph $G_i$ has no simple edge. Consider such a directed graph $G = (V, E)$ and denote $\bar{G} = (V, \bar{E})$ the corresponding undirected graph of Definition 5.3. The number of $k$-tuples for which $G_i$ is $G$ is of order $O(n^{|V|})$. To have a non-vanishing contribution, we should have $|V| \ge \frac{k}{2}$. As there is no simple edge, we have $|\bar{E}| \le \frac{k}{2}$. Thus,

$$
\frac{k}{2} - 1 \le |V| - 1 \le |\bar{E}| \le \frac{k}{2}.
$$

For odd values of $k$, there is only one integer between $\frac{k}{2}-1$ and $\frac{k}{2}$ so that $\frac{k-1}{2} = |\bar{E}| = |V|-1$ and $\bar{G}$
is a tree. Since edges are multiple and there are $k$ in total, $G_i$ necessarily has a triple edge while all
other edges are double. Since $\mathbb{E}[a_{1,2}^2 a_{2,1}] = 0$, this leads to a vanishing contribution. The next highest
order term is $O(n^{-\frac{1}{2}})$, which proves (ii). We now assume that $k=2m$ for some integer $m \ge 1$. Two
possibilities can happen, either $|\bar{E}| = m$ or $|\bar{E}| = m-1$.

Suppose first that $|\bar{E}| = m$. Then, $q_1 = 0$ so that all edges in $G_i$ are double.

* If $|V| = m+1$ then $q_2 = 1$ which means that $\bar{G}_i$ is a tree. By Proposition 5.6, $G_i \in \mathcal{T}_k^{(n)}$ is of double tree type with $m$ opposite double edges. Each double tree gives a contribution of $\mathbb{E}[a_i] = t^m$.

* If $|V| = m$, then $q_2 = 0$ which means that $\bar{G}_i$ is unicyclic and $G_i$ is of double unicyclic type with opposite edges outside its cycle. The cycle $G_i$ can consist in either parallel or opposite edges. Denote

$\eta_m$ = number of non-isomorphic unicyclic graph on $m$ vertices.

The number of tuples $i : [k] \to [n]$ such that $G_i$ has double opposite edges and that $\bar{G}_i$ is unicyclic is $(n)_m \eta_m$. Since $\mathbb{E}[a_{1,2}^2] = 0$, the only non-vanishing contributions are those of graph having double opposite edges in their cycle. Each such graph gives a contribution of $t^m$ and therefore the next order non-vanishing contribution is $\eta_m t^m$.
in the case where there is a loop which turns out to be the same number. We denote by $\mathcal{N}_p$ the set of all these path patterns.

* If $|\overline{E}_\pi| = p-1$, then $\overline{G}_\pi$ is a graph with $p-1$ edges and $p$ vertices so that it is a tree. Since a closed path drawn on a tree should pass an even number of times by each edge with the same number of times in each direction by Lemma 2.13, every edge of $\overline{G}_\pi$ should be double in $G_\pi$ except only for one edge of multiplicity four. The value of $\alpha_\pi$ in this case is $\mathbb{E}[(a_{12}a_{21})^2]\mathbb{E}[a_{12}a_{21}]^{p-2} = (2t^2)t^{p-2} = 2t^p$ and we denote by $\mathcal{N}_p'$ the set of these path patterns.

We will now show that the $O(1)$ term in the asymptotic development of traces vanishes by showing the combinatorial equality

$$ (2.12) \qquad \binom{p+1}{2} |\mathcal{T}_p| = |\mathcal{N}_p| + 2|\mathcal{N}_p'|. $$

The idea behind (2.12) is that, given an element of $\mathcal{T}_p$, one can choose two vertices and identify them to obtain an element either of $\mathcal{N}_p$ if the vertices are not at distance two or of $\mathcal{N}_p'$ if the chosen vertices are at distance two. More precisely, consider

$$ \Phi: \bigsqcup_{\pi \in \mathcal{T}_p} \{\{a, b\} \subset \pi : a \neq b\} \to \mathcal{N}_p \cup \mathcal{N}_p' $$

defined by taking $\pi \in \mathcal{T}_p$ and a pair of blocs $a_1, a_2$ of $\pi = \{a_1, a_2, a_3, \dots, a_{p+1}\}$ to the partition $\{a_1 \cup a_2, a_3, \dots, a_{p+1}\}$. We may notice that an element of $\mathcal{N}_p$ has a unique preimage by $\Phi$. Here is a possible way to see this. Let $\pi \in \mathcal{N}_p$ and let $\psi: [k] \to [n]$ be a function which induces $\pi$. Consider the smallest index $\ell \in [k]$ such that

* edges $(\ell, \ell+1)$ and $(\ell+r, \ell+r+1)$ for some $r \ge 1$ are in opposite directions and belong to the cycle of $\overline{G}_\pi$, and

* edges $(\ell+1, \ell+2), \dots, (\ell+r-1, \ell+r)$ belong to a tree component in $\overline{G}_\pi$.

Then, as $\overline{G}_\pi$ has a cycle, there exists $s \in [k]$ such that $\psi(s) \neq \psi(\ell)$ and $\psi(s+1) = \psi(\ell+1)$. In $\pi$, there is the block

$$ \{t_1, \dots, t_p, \ell+1, \ell+r, s+1, t_{p+1}, \dots, t_q\} $$

for some $t_1, \dots, t_q \in [k]$ and $q \ge 0$. The preimage of $\pi$ is given by splitting the above block into two distinct blocks:

$$ \{t_1, \dots, t_p, \ell+1, \ell+r\} \sqcup \{s+1, t_{p+1}, \dots, t_q\}. $$

On the other hand, an element of $\mathcal{N}_p'$ has exactly two preimages by $\Phi$ each one being obtained by splitting blocks one of the two endpoints of the “quadruple” edge (see Figure 3). Then,

$$ \binom{p+1}{2} |\mathcal{T}_p| = |\mathcal{N}_p| + 2|\mathcal{N}_p'| $$

so that

$$ \mathbb{E} \left[ \mathrm{Tr} \left[ \left( \frac{A_n}{\sqrt{n}} \right)^{2p} \right] \right] = n |\mathcal{T}_p| t^p + \left[ \binom{p+1}{2} |\mathcal{T}_p| t^p - |\mathcal{N}_p| t^p - |\mathcal{N}_p'| 2t^p \right] + O\left(\frac{1}{n}\right) = nC_p t^p + O\left(\frac{1}{n}\right). \quad \square $$

2.2.2. *Convergence of fluctuations.* Recall the definition $\mathcal{A}_\pi^{(n)} = \sum_{\substack{\psi: [k] \to [n] \\ \pi_\psi = \pi}} a_\psi$ from (2.11) and let us use the following notation.

**Definition 2.16** (Rooted plane tree and uniclic graph). Let $k \ge 1$ and $\pi \in \mathcal{P}(k)$.

* We say that $\pi \in \mathcal{P}(k)$ is a *rooted plane tree* if $\overline{G}_\pi$ is a tree and each edge in $G_\pi$ is double, once in each direction.

* We say that $\pi \in \mathcal{P}(k)$ is a *rooted plane unicyclic graph* if $\overline{G}_\pi$ has a unique cycle, with each edge in $G_\pi$ being simple if it belongs to the unique cycle and double, once in each direction, if it does not belong to the unique cycle.
Suppose now that $|\bar{E}| = m - 1$. Then $q_1 = -1$. We must have $|V| = m$ to have a non-vanishing contribution so that $q_2 = 1$. Thus, $G$ is of 2-4 tree type so that $\bar{G}$ is a tree and the corresponding graph $G$ has double tree type except for two vertices between which there are two pairs of opposite edges forming a quadruple edge. Denote

$\eta'_m$ = number of non-isomorphic 2-4 trees on $m$ vertices.

Each such graph gives a contribution of $\mathbb{E}[a_i] = t^{m-2} \cdot \mathbb{E}[a_{1,2}^2 a_{2,1}^2] = 2t^m$. Thus, using Lemma 5.9,

$$ \mathbb{E}\mathrm{Tr}\left[\left(\frac{A}{\sqrt{n}}\right)^{2m}\right] = n^{-m}(n)_{m+1}C_m t^m + (\eta_m + 2\eta'_n)t^m + O\left(\frac{1}{n}\right) $$

which proves (i) with

$$ (5.12) \qquad l_{m,t}^{(U)} = (\eta_m + 2\eta'_n)t^m $$

and ends the proof of Lemma 5.10.

<figure><img src="image_4.png" /></figure>

*Proof of Lemma 5.12.* Let us take $k = 2m$ even. We want to compute

$$ \sum_{q=0}^{\frac{k}{2}} n^{-\frac{k}{2}+q} \alpha_{k-2q}^{(k,q)} \left( t^{\frac{k}{2}-q} (n)_{\frac{k}{2}-q+1} C_{\frac{k}{2}-q} \right) = \sum_{q=0}^{m} n^{-m+q} \alpha_{2m-2q}^{(2m,q)} \left( t^{m-q} (n)_{m-q+1} C_{m-q} \right) =: S_m. $$

One has

$$ (n)_{m-q+1} = n(n-1)\dots(n-(m-q)) = n^{m-q+1} - n^{m-q} \sum_{k=1}^{m-q} k + O(n^{m-q-1}), $$

so that

$$ S_m = n \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,q)} t^{m-q} C_{m-q} - \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,q)} t^{m-q} C_{m-q} \frac{(m-q+1)(m-q)}{2} + O\left(\frac{1}{n}\right). $$

Consider the first sum $S'_m = \sum_{q=0}^m \alpha_{2m-2q}^{(2m,q)} t^{m-q} C_{m-q}$. Recall that the Catalan number $C_l$ is the $2l$-th moment of the semi-circular distribution : $C_l = \int_{|x|<2} x^{2l} \frac{1}{2\pi} \sqrt{4-x^2} dx$. Denote $d\mu_{2\sqrt{t}}$ the probability distribution having density $\frac{1}{2t\pi}\sqrt{4t-x^2}dx$ on $[-2\sqrt{t}, 2\sqrt{t}]$. By a linear change of variables, one derives

$$ \int_{|x|<2\sqrt{t}} x^{k-2q} d\mu_{2\sqrt{t}}(x) = \int_{|x|<2\sqrt{t}} x^{k-2q} \frac{1}{2t\pi} \sqrt{4t-x^2} dx = C_{\frac{k}{2}-q} t^{\frac{k}{2}-q}. $$

One can then identify $S'_m$ as the integral of the $k = 2m$-th modified Chebyshev polynomial $P_k^{(t)}$ with respect to the distribution $\mu_{2\sqrt{t}}$. Thus, using that $P_{2m}^{(t)}$ is even,

$$ \begin{aligned} S'_m &= \int_{-2\sqrt{t}}^{2\sqrt{t}} P_{2m}^{(t)}(x) d\mu_{2\sqrt{t}}(x) \\ &= \frac{4}{\pi} \int_0^1 P_{2m}^{(t)}(2\sqrt{ty}) \sqrt{1-y^2} dy \\ &= \frac{4}{\pi} \int_0^{\frac{\pi}{2}} P_{2m}^{(t)}(2\sqrt{t} \cos(\theta)) \sin^2(\theta) d\theta. \end{aligned} $$

Inspired from the classic equality $T_k(\cos\theta) = \cos k\theta$ satisfied by the ordinary Chebyshev polynomials, let us prove that $P_k^{(t)}(2\sqrt{t} \cos(\theta)) = 2\sqrt{t}^k \cos k\theta$. One checks that the previous holds for $k = 0$ and $k = 1$. We prove it by induction for general $k$ using the recurrence relation (3.1) which gives

$$ \begin{aligned} P_k^{(t)}(2\sqrt{t} \cos(\theta)) &= 2\sqrt{t} \cos(\theta) P_{k-1}^{(t)}(2\sqrt{t} \cos(\theta)) - t P_{k-2}^{(t)}(2\sqrt{t} \cos(\theta)) \\ &= 2\sqrt{t} \cos(\theta) (2\sqrt{t}^{k-1} \cos(k-1)\theta) - 2t\sqrt{t}^{k-2} \cos(k-2)\theta \\ &= 2\sqrt{t}^k \cos k\theta. \end{aligned} $$

Therefore,
<figure><img src="image_6.png" /><figcaption>FIGURE 3. A tree with a quadruple edge with the two possible ways of ungluing it to obtain a planar tree.</figcaption></figure>

**Proposition 2.17** (Gaussian limit process). The family $(n^{-k/2}A_{\pi}^{(n)} - n^{-k/2}\mathbb{E}[A_{\pi}^{(n)}])_{\pi \in \mathcal{P}(k), k \ge 1}$ converges to a Gaussian process as $n$ goes to infinity. Moreover, the only $\pi$ that give non-trivial limits are the rooted plane trees and the rooted plane unicyclic graphs, i.e., if $\pi$ is neither of those, the sequence $n^{-k/2}A_{\pi}^{(n)} - n^{-k/2}\mathbb{E}[A_{\pi}^{(n)}]$ converges to zero in law.

*Proof.* Let $k_1, \dots, k_\ell \ge 1$ and denote $k = k_1 + \dots + k_\ell$. For each $i \in \{1, \dots, \ell\}$ choose $\pi_i \in \mathcal{P}(k_i)$ and an "exponent" $s_i \in \{\cdot, \cdot\}$, and use the convention $x^{(\cdot)} = x$ and $x^{(*)} = \bar{x}$. Our goal is to show the limit

$$ \lim_{n \to \infty} n^{-k/2} \mathbb{E} \left[ \left( \prod_{i=1}^{\ell} (A_{\pi_i} - \mathbb{E}[A_{\pi_i}])^{(s_i)} \right) \right] = \mathbb{E} \left[ \prod_{i=1}^{\ell} Y_i^{(s_i)} \right]. $$

where $(Y_i)_{1 \le i \le \ell}$ is a Gaussian vector on $\mathbb{C}^\ell$. Let us write

$$ n^{-k/2} \mathbb{E} \left[ \left( \prod_{i=1}^{\ell} (A_{\pi_i} - \mathbb{E}[A_{\pi_i}])^{(s_i)} \right) \right] = n^{-k/2} \sum_{\substack{\psi: [\mathbf{k}] \to [n] \\ \pi_{\psi_j} = \pi_j}} \mathbb{E} \left[ \left( \prod_{j=1}^{\ell} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right) \right], $$

where we recall that $[\mathbf{k}] = \bigsqcup_{j=1}^\ell [k_j]$ and $\psi = \bigsqcup_{j=1}^\ell \psi_j$ with $\psi_j : [k_j] \to [n]$. Since the expected value $\mathbb{E}[\prod_{j=1}^\ell (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)}]$ depends only on the partition induced by $\psi$, we can write

$$ \sum_{\substack{\psi: [\mathbf{k}] \to [n] \\ \pi_{\psi_j} = \pi_j}} \mathbb{E} \left[ \left( \prod_{j=1}^{\ell} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right) \right] = \sum_{\substack{\tau \in \mathcal{P}(\mathbf{k}) \\ \tau_j = \pi_j}} C_{\tau}^{(n)} \beta_{\tau} $$

with the following notation. The partition $\tau_j \in \mathcal{P}(k_j)$ is the one induced by $\tau$ via the canonical inclusion $I_i : [k_j] \to [\mathbf{k}]$, i.e., the part of the path pattern traced by the j-th path, $\beta_\tau = \mathbb{E}[\prod_{j=1}^\ell (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)}]$ for any $\psi$ inducing $\tau$ and $C_\tau^{(n)}$ is the cardinal of the set $\{\psi : [\mathbf{k}] \to [n] : \psi$ induces $\tau\}$ which equals $n^{|\tau|} = n(n-1)\dots(n-|\tau|+1)$ if $n \ge k$.

The question now reduces to understand that $n^{-k/2}C_\tau^{(n)}$ has a limit if $\beta_\tau \ne 0$ and to understand
$$S'_{m} = \frac{4}{\pi} \int_{0}^{\frac{\pi}{2}} 2t^{m} \cos(2m\theta) \sin^{2}(\theta) d\theta = \frac{8}{\pi} t^{m} \int_{0}^{\frac{\pi}{2}} \cos(2m\theta) \sin^{2}(\theta) d\theta.$$

Moreover,

$$\int_{0}^{\frac{\pi}{2}} \cos(kx) \sin^2(x) dx = \frac{(k^2 - 2)}{k(k^2 - 4)} \sin \frac{k\pi}{2}$$

which is zero for even values of $k$ greater or equal to 4. For $k \to 2$, the right hand side converges to $-\frac{\pi}{8}$ so that $S'_m = -t\delta_{1,m}$. Therefore,

$$\sum_{q=0}^{\frac{k}{2}} n^{-\frac{k}{2}+q} \alpha_{k-2q}^{(k,t)} \left( t^{\frac{k}{2}-q} (n)_{\frac{k}{2}-q+1} C_{\frac{k}{2}-q} \right) = -nt\delta_{2,k} + l_{\frac{k}{2}}^{(T)} + O\left(\frac{1}{n}\right)$$

where, for $m \ge 1$,

$$l_{\frac{k}{2}}^{(T)} = l_{m}^{(T)} = -\frac{1}{2} \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} t^{m-q} C_{m-q}(m-q+1)(m-q).$$

It remains to prove that $l_m^{(T)} = -mt^m$. Let $m \ge 1$.

$$\begin{aligned}
& \sum_{q=0}^{m} \alpha_{2m-2q}^{(2m,t)} t^{m-q} C_{m-q}(m-q+1)(m-q) \\
&= 2mt^m \sum_{q=0}^{m} (-1)^q \frac{1}{2m-q} \binom{2m-q}{q} \frac{1}{m-q+1} \binom{2m-2j}{m-j} (m-q+1)(m-q) \\
&= 2mt^m \sum_{q=0}^{m} (-1)^q \frac{m-q}{2m-q} \binom{2m-j}{j} \binom{2m-2q}{m-q} \\
&= 2mt^m \sum_{q=0}^{m} (-1)^q \frac{m-q}{2m-q} \binom{m}{j} \binom{2m-q}{m-q} = 2mt^m \sum_{q=0}^{m-1} (-1)^q \frac{m-q}{2m-q} \binom{m}{j} \binom{2m-q}{m-q} \\
&= 2mt^m \sum_{q=0}^{m-1} (-1)^q \binom{m}{j} \binom{2m-q-1}{m-q-1} = 2mt^m \sum_{q=0}^{m-1} (-1)^q \binom{m}{j} \binom{2m-q-1}{m-q}.
\end{aligned}$$

Since $\binom{2m-q-1}{m}$ is the coefficient of $z^m$ in $(1+z)^{2m-q-1}$, the last sum is the coefficient of $z^m$ in the polynomial

$$\begin{aligned}
\sum_{q=0}^{m-1} (-1)^q \binom{m}{j} (1+z)^{2m-q-1} &= (1+z)^{2m-1} \left( \left(1-\frac{1}{1+z}\right)^m - \left(\frac{1}{1+z}\right)^m \right) \\
&= (1+z)^{m-1} (z^m - 1)
\end{aligned}$$

so that this coefficient is 1, which gives the result, after multiplying by $\frac{-2mt^m}{2}$. $\square$

## 5.4. Proof of Proposition 3.4.

### 5.4.1. Strategy of proof.
The previous section showed that in order to understand the limiting function, one has to study the convergence of the centered variables

$$(5.13) \qquad V_{k,t}^{(n)} = U_{k,t}^{(n)} - \mathbb{E}[U_{k,t}^{(n)}] = \mathrm{Tr}\left[P_k^{(t)}\left(\frac{A_{n,t}}{\sqrt{n}}\right)\right] - \mathbb{E}\left[\mathrm{Tr}\left[P_k^{(t)}\left(\frac{A_{n,t}}{\sqrt{n}}\right)\right]\right].$$

The study of random variables of the form (5.13) is known as second order asymptotics, which is the analogue of the Central Limit Theorem for random matrices. This study was initiated by the work of Johansson [20] for random matrices sampled from $\beta$-ensembles. In particular, see [25, Section 5.1], for $f$ a polynomial such that
for which of those $\tau$ the limit is not zero. This is the purpose of Lemmas 2.18 and 2.19 that we state and prove now. The end of the proof of Proposition 2.17 will follow afterwards.

**Lemma 2.18.** If $\beta_{\tau} \neq 0$ then each edge of $\overline{G}_{\tau}$ is at least double in $G_{\tau}$ and for each $i \in \{1, \dots, \ell\}$ there is $j \in \{1, \dots, \ell\}$ different from $i$ such that $\overline{G}_{\tau_i}$ and $\overline{G}_{\tau_j}$ share an edge.

*Proof.* Let $\psi : [\mathbf{k}] \to [n]$ which induces $\tau$. If $G_{\psi}$ has a simple edge then there is a unique $i \in \{1, \dots, \ell\}$ such that the edge $(r, r+1) \in E_i$ of $G_{\psi_i}$ for some $r \in [k_i]$ is simple. In that case $\mathbb{E}[a_{\psi_i}] = 0$ since $G_{\psi_i}$ has a simple edge and since $a_{\psi_i(r), \psi_i(r+1)}$ is centered,

$$ \beta_{\tau} = \mathbb{E} \left[ \prod_{j=1}^{\ell} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right] = \mathbb{E} \left[ a_{\psi_i(r), \psi_i(r+1)}^{(s_i)} \right] \mathbb{E} \left[ \prod_{j \neq i} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right] = 0. $$

Let $i \in \{1, \dots, \ell\}$. If there does not exist such index $j$, the random variable $(a_{\psi_i} - \mathbb{E}[a_{\psi_i}])^{(s_i)}$ would be independent of the product $\prod_{j \neq i} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)}$ so that, since $(a_{\psi_i} - \mathbb{E}[a_{\psi_i}])^{(s_i)}$ is centered, $\beta_{\tau}$ would be zero. $\square$

Recall that $k = k_1 + \dots + k_{\ell}$.

**Lemma 2.19.** Suppose that $\beta_{\tau} \neq 0$. Then, $|\tau| \leq k/2$. Moreover, if some connected component of $\overline{G}_{\tau}$ involves three or more $\overline{G}_{\tau_i}$, then the strict inequality $|\tau| < k/2$ holds.

*Proof.* We will actually show the this lemma under the conclusions of Lemma 2.18.

Recall that $|\tau|$ counts the number of vertices in $G_{\tau}$ so that we want to show that $|V_{\tau}| \leq k/2$. A connected component of $\overline{G}_{\tau}$ is formed by some $G_{\tau_{i_1}}, \dots, G_{\tau_{i_s}}$ so that it is enough to prove this inequality for $\tau$ restricted to $[k_{i_1}] \cup \dots \cup [k_{i_s}]$. In other words, we may assume without loss of generality that $\overline{G}_{\tau}$ is connected.

Since each edge is at least double we have that $|\overline{E}_{\tau}| \leq k/2$. The inequality $|V_{\tau}| \leq |\overline{E}_{\tau}| + 1$ tells us that $|V_{\tau}| \leq k/2 + 1$ and we need to understand why $|V_{\tau}|$ cannot be in $(k/2, k/2 + 1]$.

- If $k$ is even and $|V_{\tau}| = k/2 + 1$ then $|\overline{E}_{\tau}| = k/2$ so that $\overline{G}_{\tau}$ is a tree and each edge is double. Since $\tau_i$ traces a closed path in this tree, it must traverse each edge twice, once in each direction by Lemma 2.13. This implies that $\overline{G}_{\tau_i}$ does not share edges with any other $\overline{G}_{\tau_j}$ because each edge of $G_{\tau}$ is double which contradicts $\beta_{\tau} \neq 0$ by Lemma 2.18.

- If $k$ is odd and $|V_{\tau}| = k/2 + 1/2$ then $|\overline{E}_{\tau}| = k/2 - 1/2$ so that again $\overline{G}_{\tau}$ is a tree. But, using Lemma 2.13, a closed path in a tree contains an even number of edges so that each $k_i$ has to be even which contradicts that $k = k_1 + \dots + k_{\ell}$ is odd.

If $|\tau| = k/2$ then $|\overline{E}|$ can be either $k/2$ or $k/2 - 1$. The first case happens when $\overline{G}_{\tau}$ is unicyclic and the second case happens when it is a tree. We need to see why in this case $\ell$ must be 2, i.e., why there can be only two $\overline{G}_{\tau_i}$ forming $\overline{G}_{\tau}$.

- Suppose that $|\tau| = |\overline{E}| = k/2$ so that $\overline{G}_{\tau}$ is unicyclic. Then, each $\tau_i$ has to traverse each edge of the unique cycle at least once because, if not, $\tau_i$ would draw a path on a tree ($\overline{G}_{\tau}$ with that edge removed) so that each edge would be double and, since each edge of $G_{\tau}$ is precisely double, $G_{\tau_i}$ would not share edges with the other $G_{\tau_j}$. But there cannot be three $G_{\tau_i}$ passing through the unique cycle because each edge is double so that there has to be exactly two $G_{\tau_i}$. In particular, $G_{\tau_i}$ is also unicyclic and $G_{\tau}$ is made of $G_{\tau_i}$ and $G_{\tau_j}$ by gluing them along the unique cycle because, if an edge is traversed at least once by $\tau_i$ then it is traversed exactly twice, once in each direction (if not, we would be able to form a cycle passing through this edge).

- Suppose that $|\tau| = |\overline{E}| + 1 = k/2$ so that $\overline{G}_{\tau}$ is a tree. In this case, this implies in particular, that $\overline{G}_{\tau_i}$ is a tree with each of its edges traversed twice by $\tau_i$, once in each direction. Take $\tau_i$ and $\tau_j$ for $i \neq j$ such that $G_{\tau_i}$ and $G_{\tau_j}$ share an edge. This implies that this shared edge is at least “quadruple”. Since $|\overline{E}| = k/2 - 1$, $|\overline{E}| = k/2$ and each edge is at least double we must have that every edge of $G_{\tau}$ is precisely double except for the “quadruple edge. This implies that any other $\overline{G}_{\tau_p}$ cannot share an edge with any
$$ \frac{1}{2\pi} \int_{-2}^{2} f(x) \sqrt{4 - x^2} dx = 0 $$

and {$A_n$} be scaled GUE matrices of size $n$, then {$\text{Tr}[C_k(A_n)]$}$_{k \ge 0}$ converges to independent Gaussians with $\lim_{n \to \infty} \text{Tr}[C_k(A_n)]$ having zero mean and variance $k$, where {$C_k$}$_{k \ge 0}$ are the Chebyshev polynomials scaled to [-2, 2]. Those scaled Chebyshev polynomials {$C_n$}$_{n \ge 0}$ are defined by the recurrence relation $C_{n+1} = xC_n - C_{n-1}$ with $C_0 = 2$ and $C_1 = X$. Comparing with (3.1), one sees that the polynomials which diagonalize the covariance for GUE matrices are exactly the modified Chebyshev polynomials {$P_n^{(1)}$}$_{n \ge 0}$. We give the following definition in [25, Definition 5.2] of having a second order distribution for a family of random matrices.

**Definition 5.14.** Let {$X_N$}$_N$ be a sequence of random matrices. We say that {$X_N$}$_N$ has a second order limiting distribution if there are sequences {$\alpha_k$}$_k$ and {$\alpha_{p,q}$}$_{p,q}$ such that

* $\forall k, \lim_{N \to \infty} \mathbb{E}[\frac{1}{N} \text{Tr}[X_N^k]] = \alpha_k$.

* $\forall p, q, \lim_{N \to \infty} \text{cov}(\text{Tr}[X_N^p], \text{Tr}[X_N^q]) = \alpha_{p,q}$.

* $\forall r > 2, \forall p_1, \dots, p_r \ge 1 : \lim_{N \to \infty} k_r(\text{Tr}[X_N^{p_1}], \dots, \text{Tr}[X_N^{p_r}]) = 0$.

where $k_r$ denotes the r-th cumulant.

In [25], the authors proved that GUE matrices have a second-order limiting distribution with explicit coefficients {$\alpha_k$}$_k$ and {$\alpha_{p,q}$}$_{p,q}$ that can be expressed via non-crossing partitions. In the recent paper [23] on second order fluctuations, the authors computed the limiting covariance of Wigner matrices with some additional hypotheses, see the introduction therein. The limiting second-order covariance depends on the parameters $\theta = \mathbb{E}[a_{1,2}^2], \eta = \mathbb{E}[a_{1,1}^2]$ and $k_4 = \mathbb{E}[a_{1,2}^2 \bar{a}_{1,2}^2] - 2 - |\mathbb{E}[a_{1,2}^2]|^2$ of the Wigner matrix and can be expressed using non-crossing partitions of annulus just as in the case of GUE matrices.

On the other hand, in [12, Lemma 3.4 and 3.5], the authors proved the convergence of the variables {$\text{Tr} \left[ \left( \frac{A_n}{\sqrt{n}} \right)^k \right]$}$_k$ for Girko matrices to some independent Gaussian random variables whose parameters depend on $\eta = \mathbb{E}[a_{1,1}^2]$. Thus, the polynomials {$X^k$}$_{k \ge 0}$ diagonalize the limiting covariance for Girko matrices. Remark that those polynomials correspond to the modified Chebyshev polynomials {$P_n^{(0)}$}$_{n \ge 0}$ at the other endpoint of our interpolation. The statement of Proposition 3.4 can be seen as an extension of the two previous diagonalizations, namely by ordinary Chebyshev polynomials for $t = 1$ and monomials for $t = 0$, of a limit covariance structure that we will now compute. Recall that for $k \ge 1$:

$$ \text{Tr}[A^k] = \sum_{i: [k] \to [n]} a_i. $$

Let $\mathcal{P}(k)$ be the set of partitions of $[k]$. To a function $i : [k] \to [n]$, we associate the partition $\ker i \in \mathcal{P}(k)$ by the relation $u \stackrel{\ker i}{\sim} v \iff i(u) = i(v)$, which regroups the index set $[k]$ in blocks having the same image by $i$. For two functions such that $\ker i = \ker h$, we have $\mathbb{E}[a_i] = \mathbb{E}[a_h]$.

Let us fix integers $k_1, \dots, k_m$, conjugating exponents $s_1, \dots, s_m \in \{0, 1\}$ and $\mathbf{k} = (k_1, \dots, k_m)$. Denote $k = k_1 + \dots + k_m$. With the convention that $x^{(0)} = x$ and $x^{(1)} = \bar{x}$, let us consider

$$ (5.14) \qquad M_{\mathbf{k},t}^{(n)} := n^{-\frac{k}{2}} \mathbb{E} \left[ \prod_{j=1}^{m} \left( \text{Tr}(A_{n,t}^{k_j}) - \mathbb{E}[\text{Tr}(A_{n,t}^{k_j})] \right)^{(s_j)} \right]. $$

The proof of Proposition 5.4 consists in two part. We first prove the convergence of $M_{\mathbf{k},t}^{(n)}$ to $\mathbb{E}[Z_{k_1,t}^{(s_1)} \dots Z_{k_m,t}^{(s_m)}]$ where {$Z_{k,t}$}$_k$ are Gaussian random variables, see Proposition 5.15 below. Then, we prove that the covariance of the family {$Z_{k,t}$}$_k$ is diagonalized by the modified Chebyshev polynomials of Definition 3.1, which is Proposition 5.16 below. Propositions 5.15 and 5.16 are proved in Sections 5.4.3 and 5.4.4 respectively.

**Proposition 5.15** (Convergence to a Gaussian family). Fix $t \in [0, 1]$.

The family {$n^{-\frac{k}{2}}(\text{Tr}(A_{n,t}^k) - \mathbb{E}[\text{Tr}(A_{n,t}^k)])$}$_{k \ge 0}$ converges to a centered Gaussian family {$Z_{k,t}$}$_{k \ge 0}$.
other $G_{\tau_q}$ which cannot happen by Lemma 2.18. So, $G_{\tau}$ is made of two trees $G_{\tau_i}$ and $G_{\tau_j}$ by gluing them along an edge (which may be thought of as a degenerate cycle). $\square$

We turn back to the proof of Proposition 2.17. Lemmas 2.18 and 2.19 already show the Gaussian behavior. Indeed, let us denote by $\mathcal{P}_2(\pi_1, \dots, \pi_\ell)$ the set of partitions $\tau \in \mathcal{P}(\mathbf{k})$ such that $\tau_i = \pi_i$ for every $i \in [\ell]$ and such that $\beta_\tau \neq 0$, i.e., that satisfy the conditions of Lemmas 2.18 and 2.19. The proof of Lemma 2.19 showed that connected components of $G_\tau$ come from two $\tau_i$'s which are paired according to a common cycle or via an edge which will be quadruple if both are rooted plane trees. Then, by Lemma 2.19, there exists a pair partition that we denote $\Pi_\tau$ of $[\ell]$ where a block is $\{i, j\}$ if some connected component of $G_\tau$ is formed by $G_{\tau_i}$ and $G_{\tau_j}$. In particular, $\ell$ has to be even for $\mathcal{P}_2(\pi_1, \dots, \pi_\ell)$ not to be empty. In all cases,

$$ \lim_{n \to \infty} n^{-k/2} \mathbb{E} \left[ \prod_{i=1}^{\ell} (\mathcal{A}_{\pi_i} - \mathbb{E}[\mathcal{A}_{\pi_i}])^{(s_i)} \right] = \sum_{\tau \in \mathcal{P}_2(\pi_1, \dots, \pi_\ell)} \beta_\tau $$

which would be zero if $\ell$ is odd. Denote by $\mathcal{P}_2(\ell)$ the set of pair partitions of $[\ell]$. For each partition $\tau \in \mathcal{P}_2(\pi_1, \dots, \pi_\ell)$ consider the family of partitions $(\tau_a)_{a \in \Pi_\tau}$, where if $a = \{i, j\}$ is a block of $\Pi_\tau$, the partition $\tau_a \in \mathcal{P}_2(\pi_i, \pi_j)$ is the one induced on $[k_i] \sqcup [k_j]$ by $\tau$. We identify $\mathcal{P}_2(\pi_i, \pi_j) \simeq \mathcal{P}_2(\pi_j, \pi_i)$ so that the order is not important. The assignment $\tau \mapsto (\tau_a)_{a \in \Pi_\tau}$ defines a bijection

$$ \mathcal{P}_2(\pi_1, \dots, \pi_\ell) \to \bigsqcup_{\Pi \in \mathcal{P}_2(\ell)} \prod_{\{i,j\} \in \Pi} \mathcal{P}_2(\pi_i, \pi_j). $$

Notice that, since the components of $G_\tau$ are independent, $\beta_\tau = \prod_{a \in \Pi_\tau} \beta_{\tau_a}$ where we defined $\beta_{\tau_a} = \mathbb{E}[(\mathcal{A}_{\tau_i} - \mathbb{E}[\mathcal{A}_{\tau_i}])^{(s_i)}(\mathcal{A}_{\tau_j} - \mathbb{E}[\mathcal{A}_{\tau_j}])^{(s_j)}]$ whenever $a = \{i, j\}$. Then, we may write

$$ \begin{aligned} \sum_{\tau \in \mathcal{P}_2(\pi_1, \dots, \pi_\ell)} \beta_\tau &= \sum_{\Pi \in \mathcal{P}_2(\ell)} \sum_{\tau: \Pi_\tau = \Pi} \beta_\tau \\ &= \sum_{\Pi \in \mathcal{P}_2(\ell)} \sum_{\tau: \Pi_\tau = \Pi} \prod_{a \in \Pi} \beta_{\tau_a} \\ &= \sum_{\Pi \in \mathcal{P}_2(\ell)} \prod_{\{i,j\} \in \Pi} \left( \sum_{\tau \in \mathcal{P}_2(\pi_i, \pi_j)} \beta_\tau \right). \end{aligned} $$

where we have used the bijection described above together with the distributive property. By Isserlis–Wick's theorem, we know that

$$ (2.13) \qquad \mathbb{E} \left[ \prod_{i=1}^{\ell} Y_i^{(s_i)} \right] = \sum_{\Pi \in \mathcal{P}_2(\ell)} \prod_{\{i,j\} \in \Pi} \sum_{\tau \in \mathcal{P}_2(\pi_i, \pi_j)} \beta_\tau $$

if $(Y_1, \dots, Y_\ell)$ is a centered Gaussian vector with covariances

$$ \mathbb{E}[Y_i^{(s_i)} Y_j^{(s_j)}] = \sum_{\tau \in \mathcal{P}(\pi_i, \pi_j)} \beta_\tau. $$

Since the right-hand side of (2.13) is the limit of covariances, we have shown that the family $(n^{-k/2} \mathcal{A}_\pi^{(n)} - n^{-k/2} \mathbb{E}[\mathcal{A}_\pi^{(n)}])_{\pi \in \mathcal{P}(k), k \ge 1}$ converges to a Gaussian family as $n \to \infty$. In fact, in the proof of Lemma 2.19 we already found what graphs will contribute and how.

**Lemma 2.20 (Contributing graphs).** If $\mathcal{P}_2(\pi_i, \pi_j)$ is non-empty then both $\pi_i$ and $\pi_j$ are either plane rooted trees or they are plane rooted unicyclic graphs whose cycles have the same length.

*Proof.* The proof is contained in the proof of Lemma 2.19. $\square$

This concludes the proof of Proposition 2.17. Figure 4 gives an example of contributions described by Lemma 2.20. $\square$

<figure><img src="image_7.png" /></figure>
Let $\varphi^{(t)}$ and $\varphi_c^{(t)}$ be defined by $\varphi^{(t)}(X^p, X^q) = \mathbb{E}[Z_{p,t}Z_{q,t}]$ and $\varphi_c^{(t)}(X^p, X^q) = \mathbb{E}[Z_{p,t}\overline{Z_{q,t}}]$. This notation extends by linearity of $\varphi^{(t)}$, $\varphi_c^{(t)}$ in both arguments to $\varphi^{(t)}(P, Q)$ and $\varphi_c^{(t)}(P, Q)$ for polynomials $P, Q \in \mathbb{C}[X]$.

**Proposition 5.16** (Diagonal covariance for modified Chebyshev polynomials). For all $k, l \ge 1$,

$$ (5.15) \qquad \varphi^{(t)}(P_k^{(t)}, P_l^{(t)}) = kt^k \delta_{k=l}. $$

and

$$ (5.16) \qquad \varphi_c^{(t)}(P_k^{(t)}, P_l^{(t)}) = k\delta_{k=l}. $$

which means that the modified Chebyshev polynomials diagonalize the limiting covariance for the El- liptic Ginibre Ensemble.

Note that Proposition 5.15 together with Proposition 5.16 prove Proposition 3.4. Before turning to the proof of Proposition 5.15, we introduce some definitions and notations to study products of variables $\{\mathrm{Tr}(A_{n,t}^k) - \mathbb{E}[\mathrm{Tr}(A_{n,t}^k)]\}_k$.

5.4.2. Rearrangement of contributions. The main result of this section is Proposition 5.19 below, inspired by [23, Proposition 22], which gives another expression of (5.14) in order to prove the con- vergence to a Gaussian family. We introduce the necessary material here. For each $1 \le j \le m$,

$$ \mathrm{Tr}[A_{n,t}^{k_j}] - \mathbb{E}[\mathrm{Tr}(A_{n,t}^{k_j})] = \sum_{\psi_j: [k_j] \to [n]} (a_{\psi} - \mathbb{E}[a_{\psi}]) $$

where for notation convenience, we dropped the dependence in $t$ in the products $a_\psi$. Thus,

$$ M_{\mathbf{k},t}^{(n)} = n^{-\frac{k}{2}} \sum_{\psi_1, \dots, \psi_m} \mathbb{E} \left[ \prod_{j=1}^{m} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right]. $$

Consider the directed graphs $G_1, \dots, G_m$ associated to each of the functions $\psi_1, \dots, \psi_m$ (i.e., for all $j$, $G_j = (V_j, E_j)$ with $V_j = \mathrm{Im}(\psi_j)$) and $E_j = \{(\psi_j(1), \psi_j(2)), \dots, (\psi_j(k_j-1), \psi_j(k_j)), (\psi_j(k_j), \psi_j(1))\}$.

For notation convenience, it will be useful to group the functions $\psi_1, \dots, \psi_m$ by concatenating their images and view the result as a single function from $[k]$ to $[n]$. We therefore define $\psi: [k] \to [n]$ by $\psi(l + \sum_{p=0}^{j-1} k_p) = \psi_j(l)$ for $1 \le j \le m$ and $1 \le l \le k_j$, with the convention that $k_0 = 0$. For given integers $k_1, \dots, k_m$, the knowledge of $\psi_j: [k_j] \to [n]$, $1 \le j \le m$ is equivalent to the one of $\psi: [k] \to [n]$.

**Example 5.17** (Concatenation of maps). Set $m = 3$, $k_1 = 5$, $k_2 = 5$ and $k_3 = 4$. Let $\psi_1, \psi_2, \psi_3$ be given by $\psi_1 = [1, 3, 1, 2, 4]$, $\psi_2 = [5, 7, 6, 5, 8]$ and $\psi_3 = [4, 2, 1, 3]$ where we represent $\varphi: [r] \to [n]$ by the tuple $[\varphi(1), \dots, \varphi(r)]$. Then, $k = k_1 + k_2 + k_3 = 14$ and $\psi: [14] \to [n]$ is given by $\psi = [1, 3, 1, 2, 4, 5, 7, 6, 5, 8, 4, 2, 1, 3]$. Furthermore, one has $\ker \psi = \{\{1, 3, 14\}, \{2, 12\}, \{4, 13\}, \{5, 11\}, \{6, 9\}, \{7\}, \{8\}, \{10\}\} \in \mathcal{P}(14)$.

The next lemma shows that terms can be grouped by the induced partition $\ker \psi \in \mathcal{P}(k)$.

**Lemma 5.18** (Grouping contributions by their partitions). Let $\psi_1, \dots, \psi_m$ and $\phi_1, \dots, \phi_m$ functions such that the associated functions $\psi$ and $\phi$ from $[k]$ to $[n]$ verify $\ker \psi = \ker \varphi$. Then,

$$ (5.17) \qquad \mathbb{E} \left[ \prod_{j=1}^{m} (a_{\psi_j} - \mathbb{E}[a_{\psi_j}])^{(s_j)} \right] = \mathbb{E} \left[ \prod_{j=1}^{m} (a_{\phi_j} - \mathbb{E}[a_{\phi_j}])^{(s_j)} \right]. $$

*Proof.* For $\psi, \phi: [k] \to [n]$ such that $\ker \psi = \ker \phi$, there exists $\sigma \in S_n$ such that $\psi = \sigma \circ \phi$. By the dependence structure of coefficients $(a_{u,v})_{u,v \ge 1}$,

$$ \mathbb{E} \left[ \prod_{j=1}^{k} a_{\psi(j), \psi(j+1)} \right] = \prod_{(u,v) \in [n]^2} \mathbb{E} \left[ \prod_{j \in I(u,v,\psi)} a_{\psi(j), \psi(j+1)} \right], $$
<figure><img src="image_8.png" /><figcaption>FIGURE 4. Two elements of $\mathcal{P}_2(\pi_1, \pi_2)$ where $\pi_1, \pi_2$ are both rooted unicyclic graphs (left) or both plane rooted trees (right).</figcaption></figure>

**Remark 2.21.** A possible way to prove Proposition 2.4 would be to compute explicitly the limiting Gaussian family corresponding to traces of powers. These Gaussians variables would not be independent. One would have to show that their covariance is diagonalised by the Chebyshev polynomials thereby implying that the variables $(U_k^{(n)})_{k \ge 1}$ are independent. However, in Section 2.2.3, we chose a different approach which provides more intuition as to why the Chebyshev polynomials are the right family for diagonalizing the covariance.

### 2.2.3. Variance identification.

*Proof of Proposition 2.4.* Let us denote by $C_k^{(n)}$ the set of $\psi: [k] \to [n]$ such that the associated graph $\pi_\psi$ is a rooted plane unicyclic graph or a rooted plane tree. For such $\psi$, let denote $c(\psi)$ the set of edges in $E_\psi$ that are single, i.e, which form the cycle of $\bar{G}_\psi$. Note that this set is empty if $\pi_\psi$ is a rooted plane tree. Likewise, denote by $\ell(\psi)$ the set of edges in $E_\psi$ incident to a vertex of degree one in $\bar{G}_\psi$, that is, edges which are leaves in trees anchored on the cycle of $\bar{G}_\psi$. Recall that

$$ \left( n^{-k/2} \mathrm{Tr}(A_n^k) - n^{-k/2} \mathbb{E}[\mathrm{Tr}(A_n^k)] \right) - n^{-k/2} \sum_{\psi \in C_k^{(n)}} (a_\psi - \mathbb{E}[a_\psi]) \xrightarrow{n \to \infty} 0. $$

The idea is to “regularize” these terms and to consider instead, for any $\psi \in C_k^{(n)}$,

$$ \hat{a}_{\psi} := \prod_{e \in c(\psi)} a_e \prod_{e \in \ell(\psi)} (a_e a_{e^*} - t) $$

where for an edge $e = (i, i+1)$, $a_e = a_{\psi(i), \psi(i+1)}$ and $e^* = (i+1, i)$. A nice property of this term is that it has zero expected value even if $\pi_\psi$ is a rooted plane tree. Consider the sum

$$ :\mathrm{Tr}(A_n^k): := \sum_{\psi \in C_k^{(n)}} \hat{a}_{\psi}. $$

By the same arguments as in Section 2.2.2, we know that $(:\mathrm{Tr}(A_n^k):):_{k \ge 1}$ converges to a Gaussian process as $n \to \infty$. For a rooted plane unicyclic graph or a rooted plane tree $\pi$, we define

$$ \hat{A}_{\pi}^{(n)} := \sum_{\substack{\psi: [k] \to [n] \\ \pi_{\psi} = \pi}} \hat{a}_{\psi}. $$

In the same way, $(\hat{A}_{\pi}^{(n)})_\pi$ converges to a Gaussian process indexed by rooted plane unicyclic graphs and rooted plane trees. Let us check that for any pair of rooted plane unicyclic graphs
where $I(u, v, \psi) = \{j \in [k] : (\psi(j), \psi(j+1)) \in \{(u, v), (v, u)\}\}$ with the convention that the expectation is one if $I(u, v) = \emptyset$. Thus, $I(u, v, \phi) = I(\sigma(u), \sigma(v), \psi)$ and since the pairs $\{(a_{u,v}, a_{v,u})\}_{u,v}$ are identically distributed,

$$ \mathbb{E} \left[ \prod_{j=1}^{k} a_{\phi(j), \phi(j+1)} \right] = \prod_{(u,v) \in [n]^2} \mathbb{E} \left[ \prod_{j \in I(u,v,\phi)} a_{\phi(j),\phi(j+1)} \right] \\ = \prod_{(u,v) \in [n]^2} \mathbb{E} \left[ \prod_{j \in I(u,v,\psi)} a_{\psi(j),\psi(j+1)} \right] = \mathbb{E} \left[ \prod_{j=1}^{k} a_{\psi(j),\psi(j+1)} \right]. \quad \square $$

For $\pi \in \mathcal{P}(k)$, denote $a_{\mathbf{k},t}(\pi)$ the value of (5.17) for any functions $\{\psi_j : [k_j] \to [n], 1 \le j \le m\}$ such that $\ker \psi = \pi$. We now have,

$$ M_{\mathbf{k},t}^{(n)} = \sum_{\pi \in \mathcal{P}(k)} n^{-\frac{k}{2}} c_{\mathbf{k}}^{(n)}(\pi) a_{\mathbf{k},t}(\pi) $$

where, for $\pi \in \mathcal{P}(k)$, $c_{\mathbf{k}}^{(n)}(\pi)$ is the number of maps $(\psi_1, \dots, \psi_m), \psi_j : [k_j] \to [n], 1 \le j \le m$ such that $\ker \psi = \pi$. For $\pi \in \mathcal{P}(k)$, denote $\#\pi$ its number of blocks. By choosing an image for each block, one has $c_{\mathbf{k}}^{(n)}(\pi) = (n)_{\#\pi} = n(n-1) \dots (n - \#\pi + 1)$. Note that this number is well-defined if $\#\pi \le n$, which holds for $n$ large enough as $\pi \in \mathcal{P}(k)$ implies that $\#\pi \le k$.

For $\pi \in \mathcal{P}(k)$, denote $G_{\pi,\mathbf{k}} = (V_\pi, E_\pi)$ the union graph $\cup_{j=1}^m G_j$ associated to any function $\psi : [k] \to [n]$ such that $\ker \psi = \pi$. This means that $G_{\pi,\mathbf{k}}$ is the union of $m$ directed graphs which can be constructed from restricted maps $\psi_j$ as above. Thus, one has

$$ V_{\pi} = \mathrm{Im}(\psi) = \psi([k]). $$

$$ E_{\pi} = E_{\pi}^{(1)} \cup \dots \cup E_{\pi}^{(m)} \text{ where for each } j \in [m]. $$

$$ \begin{aligned} E_{\pi}^{(j)} &= \{(\psi(k_{j-1} + 1), \psi(k_{j-1} + 2)), \dots, (\psi(k_{j-1} + k_j - 1), \psi(k_{j-1} + k_j)), (\psi(k_{j-1} + k_j), \psi(k_{j-1} + 1))\} \\ &= \{(\psi_j(1), \psi_j(2)), \dots, (\psi_j(k_j - 1), \psi_j(k_j)), (\psi_j(k_j), \psi_j(1))\}. \end{aligned} $$

By definition, $|E_\pi| = k$ and $|V_\pi| = \#\pi$. The dominating power of $n$ in $M$ is thus

$$ q(\pi) = |V_{\pi}| - \frac{|E_{\pi}|}{2} = \underbrace{\left(\overline{|E_{\pi}|} - \frac{|E_{\pi}|}{2}\right)}_{q_1} + \underbrace{\left(|V_{\pi}| - \overline{|E_{\pi}|}\right)}_{q_2}, $$

where we used the notation $\overline{G} = (V, \overline{E})$ where edge multiplicities are forgotten introduced in Definition 5.3 above. Suppose that $G_{\pi,\mathbf{k}}$ has connected components $(\Gamma_i = (V_i, E_i))_{i \in I}$. Then, one can define $q_i(\pi) = q_{i,1} + q_{i,2}$ for each connected component by the same formula as in Definition 5.3 above restricted to $\Gamma_i = (V_i, E_i)$. There are now two different families of graphs:

* The graphs $(G_j)_{j \in [m]}$ which are directed graphs with $k_j$ edges whose union is $G_{\pi,\mathbf{k}}$.

* The connected components $(\Gamma_i)_{i \in I}$ of the graph $G_{\pi,\mathbf{k}}$.

Denote $\mathcal{D}$ the set of partitions such that the connected components are either double unicyclic or 2-4 tree type. We now state Proposition 5.19, adapted from [23, Proposition 22].

**Proposition 5.19** (Principal contributions are double unicyclic graphs and 2-4 trees). As $n \to \infty$, one has

$$ (5.18) \qquad M_{\mathbf{k},t}^{(n)} = \sum_{\pi \in \mathcal{D}} a_{\mathbf{k},t}(\pi) + o(1). $$

The proof of Proposition 5.19 is based on the same arguments as in [23] and relies on the series of lemmas below which are adaptations of the results in [23].

An example of configuration giving a term in the sum on the right hand side of (5.18) is given in Figure 2 below. Here, we have $m = 4$ directed cycles which give $|I| = 2$ connected components for $G_{\pi,\mathbf{k}}$.
or trees $\pi_1$ and $\pi_2$,

$$ \lim_{n \to \infty} \mathbb{E}[\hat{\mathcal{A}}_{\pi_1}^{(n)} \hat{\mathcal{A}}_{\pi_2}^{(n)}] = 0 \quad \text{and} \quad \lim_{n \to \infty} \mathbb{E}[\hat{\mathcal{A}}_{\pi_1}^{(n)} \overline{\hat{\mathcal{A}}_{\pi_2}^{(n)}}] = 0 $$

whenever one of them is not a cycle, that is, such that either $c(\psi_i) \neq E_{\psi_1}$ or $c(\psi_2) \neq E_{\psi_2}$ for any $\psi_1, \psi_2$ inducing $\pi_1$ and $\pi_2$ respectively. Since the leaves are centered in $\hat{\mathcal{A}}_{\pi_1}^{(n)}$ and $\hat{\mathcal{A}}_{\pi_2}^{(n)}$, the only possible non-vanishing contributions to the variance arise when $\pi_1$ and $\pi_2$ that either both have no leaves or share the same leaves. The case where both have no leaves corresponds exactly to the desired condition mentioned above. Recall that the connected component must consist of double edges so that $\overline{G}_{\pi}$ has either a single cycle, or a quadruple edge in which case $\overline{G}_{\pi}$ is a tree. If $\pi_1$ and $\pi_2$ have a common leaf, this would result in a quadruple edge in $G_{\pi}$. Therefore, both $G_{\pi_1}$ and $G_{\pi_2}$ are plane rooted trees and the condition of having the same leaves implies that both $\overline{G}_{\pi_1}$ and $\overline{G}_{\pi_2}$ consist of only one edge. The latter case corresponds to two cycles of length two. Therefore, the variances that will not have a zero limit are those between two cycles of the same length so that, if $c_k$ denotes the partition $\{\{1\}, \dots, \{k\}\}$, we have in law,

$$ \lim_{n \to \infty} \left( n^{-k/2} \operatorname{Tr}(A_n^k) : -n^{-k/2} \hat{A}_{c_k} \right) = 0. $$

If $Y_k$ denotes the limit in law of $n^{-k/2} \hat{A}_{c_k}$, we would have, for $k \ge 3$,

$$ \mathbb{E}[(Y_k)^2] = k \mathbb{E}[a_{12}a_{21}]^k = kt^k \quad \text{and} \quad \mathbb{E}[|Y_k|^2] = k \mathbb{E}[|a_{12}|^2]^k = k. $$

For $k=2$, we have $\mathbb{E}[(Y_2)^2] = 2\mathbb{E}[(a_{12}a_{21} - t)^2] = 2t^2$ and $\mathbb{E}[(Y_2)^2] = 2\mathbb{E}[|a_{12}a_{21} - t|^2] = 2$ and, for $k=1$, $\mathbb{E}[(Y_1)^2] = \mathbb{E}[a_{11}^2] = t^2$ and $\mathbb{E}[|Y_1|^2] = \mathbb{E}[|a_{11}|^2] = 1$.

It now remains to express the normalised traces : $\operatorname{Tr}(A_n^k)$ : in terms of the traces of Chebyshev polynomials, that is, in terms of the variables $U_k^{(n)}$. For simplicity, we forget the initial instant of $\psi$, i.e., we consider the equivalence relation generated by $\psi \sim \psi \circ \sigma$ with $\sigma : [k] \to [k]$ given by $\sigma(i) = i+1$ and denote by $\mathcal{D}_k^{(n)}$ the quotient of $\mathcal{C}_k^{(n)}$ by this equivalence relation. Notice that $a_\eta$ and $\hat{a}_\eta$ are well-defined for $\eta \in \mathcal{D}_k^{(n)}$ and that, since each equivalence class contains $k$ elements,

$$ \operatorname{Tr}(A_n^k) := \sum_{\psi \in \mathcal{C}_k^{(n)}} \hat{a}_{\psi} = k \sum_{\eta \in \mathcal{D}_k^{(n)}} \hat{a}_{\eta}. $$

We may write the sum of $\hat{a}_\eta$ over $\eta \in \mathcal{D}_k^{(n)}$ as a weighted sum of $a_\theta - \mathbb{E}[a_\theta]$ with $\theta \in \mathcal{D}_{k-2j}^{(n)}$ for $j \ge 0$ since each time we erase a leave we are erasing two edges. Fix $j \ge 0$ and let us find the weight of $\theta \in \mathcal{D}_{k-2j}^{(n)}$. To construct an element from $\mathcal{D}_k^{(n)}$ by adding leaves to $\theta$, we would have to add $j$ of those leaves. We should choose $j$ instants (with possible repetitions) among the $k-2j$ instants in the path $\theta$ to introduce these leaves. This would give us $\binom{k-2j+j-1}{j} = \binom{k-j-1}{j}$ choices. Then, we would have $(n - (k-2j))^j \sim n^j$ choices for the vertices associated to the leaves. This means that the weight of $a_\theta - \mathbb{E}[a_\theta]$ is

$$ \binom{k-j-1}{j} (n - (k - 2j))^j (-t)^j. $$

Note that the case where $k-2j=0$ is not relevant because our variables $\hat{a}_\psi$ are centered. We can write

$$ \begin{aligned} n^{-k/2} \sum_{\psi \in \mathcal{C}_k^{(n)}} \hat{a}_{\psi} &= n^{-k/2} k \sum_{0 \le j \le k/2} \binom{k-j-1}{j} (-t)^j (n - (k-2j))^j \sum_{\theta \in \mathcal{D}_{k-2j}^{(n)}} (a_{\theta} - \mathbb{E}[a_{\theta}]) \\ &= \sum_{0 \le j \le k/2} \frac{k}{k-2j} \binom{k-j-1}{j} (-t)^j \frac{(n - (k-2j))^j}{n^j} \left( n^{-(k-2j)/2} \sum_{\psi \in \mathcal{C}_{k-2j}^{(n)}} (a_{\psi} - \mathbb{E}[a_{\psi}]) \right). \end{aligned} $$

Notice then that whenever $k-2j \ne 0$,

$$ \frac{k}{k-2j} \binom{k-j-1}{j} (-t)^j = \frac{k}{k-j} \binom{k-j}{j} (-t)^j = \alpha_{k-2j}^{(k)} $$
<figure><img src="image_5.png" /><figcaption>FIGURE 2. Four directed cycles giving two connected components: one double unicyclic (left) and one 2-4 tree (right).</figcaption></figure>

**Lemma 5.20** (Contributing graphs have multiple edges). If $a_{\mathbf{k},t}(\pi) \neq 0$, edges of $G_{\pi,\mathbf{k}}$ are multiple.

*Proof.* If one edge $(u,v)$ is single in $G_{\pi,\mathbf{k}}$, the random variable $a_{u,v}$ in centered and independent of the others so that $a_{\mathbf{k},t}(\pi) = 0$. □

**Lemma 5.21** (Graph characterisation of $q_{i,1}, q_{i,2}$). Suppose that $a_{\mathbf{k},t}(\pi) \neq 0$. For each connected component $\Gamma_i$:

(a) $q_{i,1} \le 0$ with equality if and only if each edge in $\Gamma_i$ is double.

(b) $q_{i,2} \le 1$ with equality if and only if $\bar{\Gamma}_i$ is a tree.

*Proof.* Since each edge should be multiple by Lemma 5.20, to each edge in $\bar{\Gamma}_i$ corresponds at least two edges in $\Gamma_i$, so that $|E_i| \ge 2|\bar{E}_i|$. The equality case follows when every edge in $\bar{\Gamma}_i$ gives exactly two edges in $\Gamma_i$ which proves (a). The result (b) is a direct consequence of Proposition 5.4 (i). □

**Lemma 5.22** (Even multiplicity of disconnecting edges). Let $\bar{e} \in \overline{G_{\pi,\mathbf{k}}}$ corresponding to a set of twin edges in some connected component $\Gamma_i$. If the removal of the edges of $\bar{e}$ disconnects $\Gamma_i$, then the multiplicity of edges in $\bar{e}$ coming from each graph $G_j$ is an even number, with an equal number of edges in each direction.

*Proof.* The proof uses the same arguments as in the proof of [23, Lemma 20]. Assume that a graph $G_j$ has $p \ge 1$ twin edges in the group $\bar{e}$ that disconnects $\Gamma_i$, for some $j \in [m]$ and $i \in I$.

Let us prove that $p > 1$. Assume that $e_0 \in G_j$ is the only one edge of $\bar{e}$ coming from $G_j$. The graph $G_j \setminus e_0$ is connected by construction of $G_j$. Then, in $\Gamma_i \setminus \bar{e}$ the source and target of $e_0$ are connected by the path in $G_j \setminus e_0$ which contradicts the assumption that $\bar{e}$ is disconnecting. Thus, $p > 1$.

Assume $p \ge 2$. Start a walk in $G_j$ from the source of $e_0$ and consider $e_1$ its first twin edge in $G_j$ met after $e_0$. Then, $e_0$ and $e_1$ are opposite. Indeed, if they were parallel, removing $e_0$ and $e_1$ would not disconnect $G_j$. This would imply that removing $\bar{e}$ in $\overline{G_{\pi,\mathbf{k}}}$ would not disconnect $\Gamma_i$, leading to the same contradiction as in (i). Thus, $e_0$ and $e_1$ are opposite. Remove the loop from the source of $e_0$ to the target of $e_1$ in $G_j$. The remaining graph has now $p-2$ directed edges in $\bar{e}$. Using induction, one derives that the number of edges in $\bar{e}$ coming from $G_j$ is even with an equal number in each direction. □

**Lemma 5.23** (Characterisation of non-vanishing connected components). If $a_{\mathbf{k},t}(\pi) \neq 0$, then for every connected component $\Gamma_i$ of $G_{\pi,\mathbf{k}}$, one has $q_{i,1} + q_{i,2} \le 0$ with equality if and only if the connected component is either double unicyclic of 2-4 tree type.

Before turning to the proof of Lemma 5.23, we will need the following lemmas.

**Lemma 5.24** (Connectivity condition). If there exists $j \in [m]$ and $i \in I$ such that $G_j = \Gamma_i$, then $a_{\mathbf{k},t}(\pi) = 0$.

*Proof.* For such $j \in [m]$ and $i \in I$ such that $G_j = \Gamma_i$, the random variable $(a_{\psi_j} - \mathbb{E}[a_{\psi_j}])$ is independent of the other random variables $\{(a_{\psi_l} - \mathbb{E}[a_{\psi_l}]) : l \in [m], l \neq j\}$ and thus $a_{\mathbf{k},t}(\pi) = 0$. □

**Lemma 5.25** (Double tree type components have vanishing contribution). If some connected component $\Gamma_i$ is of double tree type, then $a_{\mathbf{k},t}(\pi) = 0$.
where $(\alpha_{k-2j}^{(k)})_j$ are the coefficients of $P_k$ given in (2.2). Since $\frac{(n-(k-2j))^j}{n^j}$ converges to 1, the limit of $n^{-k/2} \sum_{\psi \in C_k^{(n)}} \hat{a}_\psi$ coincides with the limit of

$$ \mathrm{Tr}\left(P_k\left(\frac{A_n}{\sqrt{n}}\right)\right) - \mathbb{E}\left[\mathrm{Tr}\left(P_k\left(\frac{A_n}{\sqrt{n}}\right)\right)\right] = U_k^{(n)} - \mathbb{E}[U_k^{(n)}]. $$

More precisely, their difference goes to zero in law. Since $\mathbb{E}[U_k^{(n)}]$ goes to zero by Proposition 2.14, we obtain that

$$ \lim_{n \to \infty} (U_k^{(n)} - n^{-k/2} \hat{A}_{c_k})_{k \ge 1} = 0 $$

which completes the proof of Proposition 2.4. □

### 3. PROOF OF THEOREM 1.3

For $I \subset [n]$, we denote by $A_I$ the submatrix of $A_{n,t}$ obtained by taking rows and columns with index in $I$. Since one can write

$$ (3.1) \qquad \det \left( (1 + tz^2) - z \frac{A_{n,t}}{\sqrt{n}} \right) = \sum_{k=0}^{n} (1 + tz^2)^{n-k} (-z/\sqrt{n})^k S_k^{(n)} $$

where $S_k^{(n)} = \sum_{I \subset [n]: |I|=k} \det(A_I)$ is coefficient of $w^k$ in the polynomial $\det(1 + wA)$, the expectation $\mathbb{E}[f_n(z)]$ depends only on $t$. It can be written in terms of Hermite polynomials, see Proposition 3.1 below. We will not need this since we can study its asymptotic behavior by considering $A_{n,t}$ sampled from the Elliptic Ginibre ensemble of parameter $t$.

Recall that if $(X_n)_{n \ge 1}$ is a sequence of uniformly integrable random variables which converge in law to $X$, then $\lim_{n \to \infty} \mathbb{E}[X_n] = \mathbb{E}[X]$. Since the second order moment of $\{f_n\}_{n \ge 1}$ is uniformly bounded on compact subsets by Proposition 2.10, the family is uniformly integrable and therefore for $z \in \mathbb{D}$,

$$ (3.2) \qquad \lim_{n \to \infty} \mathbb{E}[f_n(z)] = \mathbb{E}[e^{-F_t(z)}] $$

with

$$ F_t(z) = \sum_{k \ge 1} X_k \frac{z^k}{\sqrt{k}} $$

for a family $(X_k)_{k \ge 1}$ of independent Gaussian random variables satisfying $\mathbb{E}[X_k] = 0$, $\mathbb{E}[X_k^2] = t^k$ and $\mathbb{E}[|X_k|^2] = 1$. Since $|\mathbb{E}[f_n(z)]| \le \sqrt{\mathbb{E}[|f_n(z)|^2]}$ and since $\mathbb{E}[|f_n(z)|^2]$ is uniformly bounded on compact sets by Proposition 2.10, $\mathbb{E}[f_n(z)]$ is a precompact sequence of holomorphic functions by Montel's theorem. This implies that the convergence in (3.2) is uniform on compact sets. It is enough to calculate

$$ \mathbb{E}[e^{-F_t(z)}] = \mathbb{E}\left[e^{-\sum_{k \ge 1} X_k \frac{z^k}{\sqrt{k}}}\right] = e^{\frac{1}{2} \sum_{k \ge 1} \mathbb{E}[X_k^2] \frac{z^{2k}}{k}} = e^{\frac{1}{2} \sum_{k \ge 1} t^k \frac{z^{2k}}{k}} = \sqrt{1 - tz^2} $$

which completes the proof of Theorem 1.3.

For the sake of completeness, we provide an explicit expression of the expectation $\mathbb{E}[f_n(z)]$ in Proposition 3.1.

**Proposition 3.1 (Mean characteristic polynomial).** Let $A_{n,t}$ be a random matrix as in Theorem 1.3. Then, for every $z \in \mathbb{D}$,

$$ (3.3) \qquad \mathbb{E}[f_{n,t}(z)] = e^{-ntz^2/2} \left(\sqrt{\frac{t}{n}}z\right)^n H_n\left(\sqrt{\frac{n}{t}}\left(\frac{1}{z} + tz\right)\right). $$

**Remark 3.2 (Universality of the expectation).** Notice that the expectations involved in $\mathbb{E}[f_n(z)]$ only depend on $\mathbb{E}[a_{1,2}a_{2,1}] = t$. Therefore, $\mathbb{E}[f_{n,t}(z)]$ can be obtained by considering EGE
**Proof.** Assume that $\Gamma_i$ is a connected component of double tree type. Let $E(\overline{\Gamma_i})$ be the edges of $\overline{\Gamma_i}$, each one corresponding to double opposite edges in $G_{\pi, \mathbf{k}}$. By Lemma 5.22, to each edge $\bar{e} \in E(\overline{\Gamma_i})$, one can associate some $j \in [m]$ such that the pair of opposite edges composing $\bar{e}$ comes from $G_j$ only. Denote this map $h: E(\overline{\Gamma_i}) \to [m]$. If $h$ has more than one image point, say $j \neq l$ then $(a_{\psi_j} - E[a_{\psi_j}])$ and $(a_{\psi_l} - E[a_{\psi_l}])$ are independent of all other random variables and thus $a_{\mathbf{k},t}(\pi) = 0$. If the image of $h$ contains only one integer $j \in [m]$, we would have $\Gamma_i = G_j$ so that $a_{\mathbf{k},t}(\pi) = 0$ by Lemma 5.24. $\square$

**Proof of Lemma 5.23.** By the Lemmas 5.21 and 5.25 excluding double tree types, any connected component of a non-vanishing contribution satisfies $\Gamma_i$ satisfies $q_{i,1} + q_{i,2} < 1$. The next possible value would be $q_{i,1} + q_{i,2} = \frac{1}{2}$ if $(q_{i,1}, q_{i,2}) = (-\frac{1}{2}, 1)$. Should the previous hold, the graph $\overline{\Gamma}_i$ would thus be a tree and $\Gamma_i$ would only have double edges except for an edge with multiplicity three, contradicting Lemma 5.22 as this group of edges would disconnect $G_{\pi, \mathbf{k}}$ and have an odd cardinal. Thus, we have $q_{i,1} + q_{i,2} \le 0$, with equality cases corresponding to $\pi \in \mathcal{D}$ by the characterisations of Lemma 5.23. $\square$

**Proof of Proposition 5.19.** The leading order of $n$ in $M_{\mathbf{k},t}^{(n)}$ is $q(\pi) = \sum_{i \in I} q_i(\pi) = \sum_{i \in I} q_{i,1} + q_{i,2} \le 0$. By Lemma 5.23, the contribution of $\pi$ is non-vanishing if and only if each of its connected components are double unicyclic or 2-4 type which proves Proposition 5.19. $\square$

### 5.4.3. Convergence to a Gaussian family.
This section is devoted to the proof of Proposition 5.15. To prove that the limiting family is Gaussian, let us write $M_{\mathbf{k},t}^{(n)}$ as a Wick product. This is the statement of Proposition 5.27 below which is the analogue of [23, Proposition 33]. Such an expression implies that the family is Gaussian and thus proves Proposition 5.15. Our proof structure follows again the lines of [23].

**Definition 5.26.** For any pair of indexes $\{j, l\}$, denote $\mathcal{P}(j, l)$ the set of partitions $\pi$ of $\{k_{j-1} + 1, \dots, k_j\} \cup \{k_{l-1} + 1, \dots, k_l\}$ such that

(1) Either the graph $G_j \cup G_l$ is of double unicyclic type and both graphs $\overline{G_j}, \overline{G_l}$ are unicyclic. This happens when both graphs $G_j$ and $G_l$ are double unicyclic trees with the same cycle, by pairing the edges in the common cycle.

(2) Either $G_j \cup G_l$ is of 2-4 tree type and both graphs $G_j, G_l$ are double trees. A pair of twin edges in $G_j$ and a pair of twin edges $G_l$ are thus paired to form the group of edges of multiplicity four in $G_j \cup G_l$.

The following proposition shows that graphs contributing to the limit in $M_{\mathbf{k},t}^{(n)}$ are obtained by pairing the graphs $\{G_j\}_{1 \le j \le m}$.

**Proposition 5.27 (Wick product expression of $M_{\mathbf{k},t}^{(n)}$).** We have

$$ (5.19) \qquad M_{\mathbf{k},t}^{(n)} = \sum_{\sigma \in \mathcal{P}_2(m)} \prod_{(j,l) \in \sigma} M_t(j,l) + o(1) $$

as $n \to +\infty$, with

$$ (5.20) \qquad M_t(j,l) = \sum_{\pi \in \mathcal{P}(j,l)} a_{\pi}(\pi) $$

where $a_t(\pi)$ is the common value for $E[(a_{\psi_j} - E[a_{\psi_j}])^{(s_j)}(a_{\psi_l} - E[a_{\psi_l}])^{(s_l)}]$ for $\psi_j, \psi_l$ such that the associated partition of $\{k_{j-1} + 1, \dots, k_j\} \cup \{k_{l-1} + 1, \dots, k_l\}$ is $\pi$.

The proof of Proposition 5.27 relies on Lemma 5.28 that we state below. Recall that by Proposition 5.19, the only graphs $G_1, \dots, G_m$ that can contribute are those for which their union graph has connected components of either double unicyclic or 2-4 type. As products are centered, each such connected component should come from at least two of the $G_i$'s. The next lemma adapted from [23, Lemma 35] shows that each connected component comes from exactly two of the $G_i$'s, which is where the pairings appear.

**Lemma 5.28 (Graph pairings).** Consider $\pi \in \mathcal{P}(k)$ such that $a_{\mathbf{k},t}(\pi) \neq 0$ and assume that $\Gamma_i$ is of double cycle type. Then, there are two different cycles $G_j, G_l$ such that each group of twin edges in the cycle of $\Gamma_i$ consists in an edge from $G_j$ and an edge from $G_l$.
matrices for the same $t$ and is related to the kernel when we see the eigenvalues as a determinantal process. Namely, if $(Z_1, \dots, Z_n) \sim \frac{1}{\sqrt{2}} \prod_{i<j}^n |z_j - z_i|^2 d\mu^{\otimes n}(z_1, \dots, z_n)$, then

$$ \mathbb{E}[\prod_{i=1}^{n} (z - Z_i)] = p_n(z), $$

where $p_n$ is the monic orthogonal polynomial of degree $n$ with respect to $\mu$.

For convenience of the reader, we prefer to give a more direct proof of Proposition 3.1. Recall that $S_k^{(n)} = \sum_{I \subset [n]: |I| = k} \det(A_I)$ are the coefficients in (3.1).

**Lemma 3.3 (Mean coefficient).** For $1 \le k \le n$,

$$ (3.4) \qquad \mathbb{E}[S_k^{(n)}] = \begin{cases} 0 & \text{if } k \in 2\mathbb{N} + 1 \\ \binom{n}{k} (k-1)!! (-t)^{k/2} & \text{if } k \in 2\mathbb{N}. \end{cases} $$

where $(l)!! = l \cdot (l-2) \dots 3 \cdot 1$ for $l \in 2\mathbb{N} + 1$.

*Proof.* Let $I \subset [n]$ such that $|I| = k$. Then,

$$ \mathbb{E}[\det(A_I)] = \sum_{\sigma} (-1)^{\sigma} \mathbb{E} \prod_{i \in I} a_{i, \sigma(i)}. $$

where the sum is over permutations of $I$. The expectation is non-zero if and only if $\sigma$ is a product of transpositions since each term is centered and $(a_{i,j}, a_{j,i})$ is independent of the family $\{(a_{l,k}, a_{k,l}), \{k,l\} \ne \{i,j\}\}$. Thus, $I$ has to be of even cardinal so that $k = 2l$ for some $l \ge 1$. There are $(2l-1)!! = (2l-1)(2l-3)\dots 1$ permutations that are product of transpositions since they are in bijection with pairings of $2l$ elements. Each such permutation gives a contribution of $(\mathbb{E}[a_{1,2}a_{2,1}])^l = t^l$ and has a signature of $(-1)^l$. Therefore,

$$ \mathbb{E}[\det(A_I)] = (2l-1)!!(-t)^l, $$

which only depends on the cardinal of $I$. Thus,

$$ \mathbb{E}[S_{2l}^{(n)}] = \binom{n}{2l} (2l-1)!! (-t)^l $$

□

*Proof of Proposition 3.1.* Let $z \in \mathbb{D}$. Using (3.1), Lemma 3.3 and $\frac{(2k-1)!!}{(2k)!} = \frac{1}{2^k k!}$,

$$ \begin{align*} \mathbb{E}[f_{n,t}(z)] &= e^{-ntz^2/2} \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} (1+tz^2)^{n-2k} \frac{z^{2k}}{n^k} (2k-1)!! (-t)^k \\ &= e^{-ntz^2/2} \left(\sqrt{\frac{t}{n}}z\right)^n \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{(-1)^k n!}{2^k (n-2k)! k!} \left(\sqrt{\frac{n}{t}}\left(\frac{1}{z} + tz\right)\right)^{n-2k} \\ &= e^{-ntz^2/2} \left(\sqrt{\frac{t}{n}}z\right)^n H_n\left(\sqrt{\frac{n}{t}}\left(\frac{1}{z} + tz\right)\right). \end{align*} $$

□

Note that from Proposition 3.1, one can derive the result of Theorem 1.3 by using uniform asymptotics of Hermite polynomials. In particular, this approach does not require the use of Proposition 2.10. Since this alternative method involves more intricate computations, we opted to prove Theorem 1.3 using the approach presented in the beginning of this section.

REFERENCES

[AC23] Y. Ameur and J. Cronvall. “Szegő type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials”. In: *Comm. Math. Phys.* 398.3 (2023), pp. 1291–1348. doi: 10.1007/s00220-022-04539-y.

[ADM23] G. Akemann, M. Duits, and L. D. Molag. “The elliptic Ginibre ensemble: a unifying approach to local and global statistics for higher dimensions”. In: *J. Math. Phys.* 64.2 (2023), pp. 023503, 39. doi: 10.1063/5.0089789.
**Proof.** Denote $E_0$ the set of groups of twin edges in the cycle of $\Gamma_i$. Assume that some $G_j$ has exactly one edge in a group $\bar{e} \in E_0$. Suppose for the sake of a contradiction that there is another group of edge $\bar{e}' \in E_0$ coming from two other cycles $G_{j'}$, $G_{j''}$ with $j' \neq j \neq j''$. The removal of $\bar{e}$ disconnects $G_{\pi,\mathbf{k}} \setminus \bar{e}'$ (which is connected). The fact that only one edge of $\bar{e}$ comes from $G_j$ would contradict 5.22. Thus, $G_j$ has one edge in every element of $E_0$ and since $\Gamma_i$ is a double cycle, there is some other cycle $G_l$ with the same property. Let us show that this other cycle cannot be $G_j$. Assume now that each group in $E_0$ comes from only a single cycle $G_j$. Then, the component $\Gamma_i$ would come from this cycle $G_j$ and thus $G_j$ would be X disconnected from $G_{\pi,\mathbf{k}}$ giving $a_{\mathbf{k},t}(\pi) = 0$. $\square$

**Proof of Proposition 5.27.** Assume that some component $\Gamma_i$ of $G_{\pi,\mathbf{k}}$ is made from at least three cycles among $G_1, \dots, G_m$. We will show that $a_{\mathbf{k},t}(\pi) = 0$.

* If $\Gamma_i$ is a 2-4 type, by Lemma 5.22, the edge of multiplicity four can only come from two graphs $G_j, G_l$ and one of the (at least) three graphs would be disconnected leading to a vanishing contribution.

* If $\Gamma_i$ is double unicyclic, Lemma 5.28 shows that there are two graphs that are paired to constitute its cycle. Since every other edge has multiplicity two outside the cycle, one would also have at least one disconnected graph and a vanishing contribution.

Thus, the only non-vanishing contributions come from a pairing of unicyclic double trees $G_j, G_l$ paired together to form either a double unicyclic graph and a 2-4 tree. $\square$

By Iserlis-Wick's lemma, the proof of Proposition 5.15 is complete.

5.4.4. **Computation of the limiting covariance.** To prove Proposition 5.16, we need to compute the asymptotic covariance of the previous Gaussian family and show that it is diagonal for the modified Chebyshev polynomials. Let us take $m = 2$ and consider $\pi \in \mathcal{P}(1,2)$ a partition associated to any function $\psi$ such that the associated union graph $G_{\pi,\mathbf{k}} = G_{k_1} \cup G_{k_2}$ is a double unicyclic graph or a 2-4 tree. Let us compute the value of

$$a_t(\pi) = \mathbb{E} \left[ (a_{\psi_1} - \mathbb{E}[a_{\psi_1}])^{(s_1)} (a_{\psi_2} - \mathbb{E}[a_{\psi_2}])^{(s_2)} \right].$$

If $G_{\pi,\mathbf{k}}$ is of double unicycle type, then $G_{k_1}$ and $G_{k_2}$ are unicyclic double trees with the same cycle, either in the same direction or opposite. If they are opposite (respectively parallel), we say that $\pi \in DU^{\text{opp}}$ (respectively $\pi \in DU^{\text{par}}$).

**Lemma 5.29 (Covariance $\varphi^{(t)}(X^{k_1}, X^{k_2})$).** Let us fix $s_1 = s_2 = 0$ and suppose that $G_{\pi,\mathbf{k}}$ is a double unicyclic graph with a cycle of length $l \ge 1$.

(i) If $\pi \in DU^{\text{opp}}$, $a_t(\pi) = t^{(k_1+k_2)/2}$.

(ii) If $\pi \in DU^{\text{par}}$, $a_t(\pi) = t^{(k_1+k_2)/2} \delta_{l=1}$.

(iii) If $\pi \in FT$, $a_t(\pi) = t^{(k_1+k_2)/2}$.

**Proof.** Suppose that $G_{\pi,\mathbf{k}} \in DU^{\text{opp}}$. Then, the cycle length $l$ is at least 3 since a cycle of length 2 would only belong to one of the graphs $G_{k_1}, G_{k_2}$ and the two graphs would be independent of each other. Each of $G_{k_1}$ and $G_{k_2}$ both have $l$ simple edges so that $\mathbb{E}[a_{\psi_1}] = \mathbb{E}[a_{\psi_2}] = 0$. Moreover, $a_{\psi_1}a_{\psi_2}$ is the product of $\frac{k_1+k_2}{2}$ independent variables with the same distribution as $a_{1,2}a_{2,1}$ which gives (i). For $G_{\pi,\mathbf{k}} \in DU^{\text{par}}$, if $l \neq 1$, since $\mathbb{E}[a_{1,2}^2] = 0$, one would have $a_t(\pi) = 0$. The only parallel contribution comes from $l = 1$ where the cycle of $G_{\pi,\mathbf{k}}$ is a double loop edge which gives $\mathbb{E}[a_{1,1}^2] = t$ and thus $a_t(\pi) = t \cdot t^{(k_1+k_2)/2-1} = t^{(k_1+k_2)/2}$ proving (ii). If $G_{\pi,\mathbf{k}} \in FT$, the quadruple edge gives a contribution of $\mathbb{E}[a_{1,2}^2 a_{2,1}^2] = 2t^2$ while other opposite edges give a contribution of $t$. Therefore, $\mathbb{E}[a_{\psi_1} a_{\psi_2} - \mathbb{E}[a_{\psi_1} a_{\psi_2}]] = (2t^2 - t^2)t^{(k_1+k_2)/2-2}$ which proves (iii). $\square$

**Lemma 5.30 (Covariance $\varphi_c^{(t)}(X^{k_1}, X^{k_2})$).** Let us fix $s_1 = 0$ and $s_2 = 1$. We have, where $l$ is the common cycle length.

(i) If $\pi \in DU^{\text{opp}}$, $a_t(\pi) = 0$.

(ii) If $\pi \in DU^{\text{par}}$, $a_t(\pi) = t^{(k_1+k_2)/2-l}$.

(iii) If $\pi \in FT$, $a_t(\pi) = t^{(k_1+k_2)/2-2}$.

**Proof.** We have $\mathbb{E}[a_{1,2}\overline{a_{2,1}}] = 0$ so any opposite cycle has zero expectation proving (i). For $G_{\pi,\mathbf{k}} \in DU^{\text{par}}$, since $\mathbb{E}[a_{1,2}\overline{a_{1,2}}] = \mathbb{E}|a_{1,2}|^2 = 1 = \mathbb{E}|a_{1,1}|^2$ and the rest of the double edges outside the
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cycle have expectation $t$, one derives (ii). If $G_{\pi, \mathbf{k}} \in \mathcal{F}T$, the quadruple edge gives a contribution of $\mathbb{E}[|a_{1,2}|^2|a_{2,1}|^2] = 1 + t^2$ so that $a_t(\pi) = t^{(k_1+k_2)/2-2}(\mathbb{E}[|a_{1,2}|^2|a_{2,1}|^2] - t^2) = t^{(k_1+k_2)/2-2}$. $\square$

We are now interested in the number of partitions in $\mathcal{P}(i, j)$. In [23, Lemma 39], the authors introduced a bijection that allows us to count the number of double unicyclic graphs as well as the number of 2-4 trees. Let us first assume that $k_1$ and $k_2$ are both even. Recall that they necessarily have the same parity. Define $\mathcal{G}_{2l}$ the set of all possible graphs obtained which are of double unicyclic type having a cycle length of $2l$ for $l \ge 1$. We state the result of [23, Lemma 39], as well as its extension to $2-4$ tree types as discussed after [23, Definition 40].

**Lemma 5.31** (Non-crossing annular pairings enumeration). For each $l \ge 1$, there is a bijection from $\mathcal{G}_{2l}$ to the set $\mathcal{NC}_2^{(l)}(k_1, k_2)$ of non crossing pairings of the $k_1, k_2$ annulus with $l$ through strings. Furthermore, there is a bijection from $\mathcal{F}\mathcal{T}$ to the set $\mathcal{NC}_2^{(2)}(k_1, k_2)$ of non crossing pairings of the $k_1, k_2$ annulus with 2 through strings.

Denote $\mathcal{nc}^{(l)}(p, q)$ the cardinal of $\mathcal{NC}_2^{(l)}(p, q)$. By [24, Proof of Lemma 22] and reference [26] therein one can have an explicit expression of this quantity given by

$$ (5.21) \qquad \mathcal{nc}^{(l)}(p, q) = l \binom{p}{\frac{p-l}{2}} \binom{q}{\frac{q-l}{2}}. $$

We summarise the results of this section for the value of $M_t(k_1, k_2)$ in the table below.

<table><caption>TABLE 1. Values of $M_t(k_1, k_2)$.</caption><thead><tr><th></th><th>s<sub>0</sub> = s<sub>1</sub> = 0</th><th>s<sub>0</sub> = 0, s<sub>1</sub> = 1</th></tr></thead><tbody><tr><td>k<sub>1</sub>, k<sub>2</sub> even</td><td>&sum;<sub>l=1</sub><sup>k<sub>1</sub>/2</sup> nc<sup>(2l)</sup>(k<sub>1</sub>, k<sub>2</sub>)t<sup>(k<sub>1</sub>+k<sub>2</sub>)/2</sup></td><td>&sum;<sub>l=1</sub><sup>k<sub>1</sub>/2</sup> nc<sup>(2l)</sup>(k<sub>1</sub>, k<sub>2</sub>)t<sup>(k<sub>1</sub>+k<sub>2</sub>)/2-2l</sup></td></tr><tr><td>k<sub>1</sub>, k<sub>2</sub> odd</td><td>&sum;<sub>l=1</sub><sup>(k<sub>1</sub>+1)/2</sup> nc<sup>(2l-1)</sup>(k<sub>1</sub>, k<sub>2</sub>)t<sup>(k<sub>1</sub>+k<sub>2</sub>)/2</sup></td><td>&sum;<sub>l=1</sub><sup>(k<sub>1</sub>+1)/2</sup> nc<sup>(2l-1)</sup>(k<sub>1</sub>, k<sub>2</sub>)t<sup>(k<sub>1</sub>+k<sub>2</sub>)/2-2l+1</sup></td></tr></tbody></table>

To end the proof of Proposition 5.16, we need to compute $\varphi^{(t)}(P_k^{(t)}, P_l^{(t)})$ and $\varphi_c^{(t)}(P_k^{(t)}, P_l^{(t)})$ to show that those quantities vanish when $k \ne l$. For any $t \in [0, 1]$ and $k \ge 0$, we have

$$ P_k^{(t)} = \sum_{j=0}^{k/2} \alpha_{k-2j}^{(k,t)} X^{k-2j}, $$

where the coefficients $\alpha_{k-2j}^{(k,t)}$ are those of (5.9). Recall the following facts:

* $\varphi^{(1)}$ is diagonalized by the usual Chebyshev polynomials $\{P_k^{(1)}\}_{k \ge 1}$, see [25, Theorem 5.1], so that one has:
$$ (5.22) \qquad \varphi^{(1)}(P_k^{(1)}, P_l^{(1)}) = k\delta_{k=l}. $$

* We have the scaling relation: $P_k^{(t)}(x) = \sqrt{t^k} P_k^{(1)}(\frac{x}{\sqrt{t}})$, so that $\alpha_{k-2j}^{(k,t)} = \sqrt{t^{2j}} \alpha_{k-2j}^{(k,1)}$. Since $\varphi^{(t)}(X^p, X^q) = \sqrt{t^{p+q}} \varphi^{(1)}(X^p, X^q) = \mathcal{nc}(p, q)\sqrt{t^{p+q}}$, for every $k, l \ge 1$ one has,

$$ \begin{align*} \varphi^{(t)}(P_k^{(t)}, P_l^{(t)}) &= \sum_{i,j} \alpha_{k-2i}^{(k,t)} \alpha_{l-2j}^{(l,t)} \varphi^{(t)}(X^{k-2i}, X^{l-2j}) \\ &= \sum_{i,j} \alpha_{k-2i}^{(k,t)} \alpha_{l-2j}^{(l,t)} \sqrt{t^{k+l-2i-2j}} \varphi^{(1)}(X^{k-2i}, X^{l-2j}) \\ &= \sum_{i,j} \alpha_{k-2i}^{(k,1)} \alpha_{l-2j}^{(l,1)} \sqrt{t^{k+l}} \varphi^{(1)}(X^{k-2i}, X^{l-2j}) \\ &= \sqrt{t^{k+l}} \varphi^{(1)}(P_k^{(1)}, P_l^{(1)}) \\ &= k\sqrt{t^{k+l}} \delta_{k=l} \end{align*} $$
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QUENTIN FRANÇOIS: CEREMADE, CNRS, UNIVERSITÉ PARIS-DAUPHINE, UNIVERSITÉ PSL, 75016 PARIS, FRANCE & DMA, ÉCOLE NORMALE SUPÉRIEURE, UNIVERSITÉ PSL, CNRS, 75005 PARIS, FRANCE.

Email address: quentin.francois@dauphine.psl.eu

DAVID GARCÍA-ZELADA: LABORATOIRE DE PROBABILITÉS, STATISTIQUE ET MODÉLISATION, UMR CNRS 8001, SORBONNE UNIVERSITÉ, 4 PLACE JUSSIEU, 75005 PARIS, FRANCE.

Email address: david.garcia-zelada@sorbonne-universite.fr
which proves (5.15). It remains to prove (5.16). Suppose that $k_1 \leftarrow 2k_1$ and $k_2 \leftarrow 2k_2$ are both even. Then,

$$
\begin{align*}
\varphi_c^{(t)}(P_{2k_1}^{(t)}, P_{2k_2}^{(t)}) &= \sum_{r,s \ge 0} \alpha_{2k_1-2r}^{(2k_1,t)} \alpha_{2k_2-2s}^{(2k_2,t)} \varphi_c^{(t)}(X^{2k_1-2r}, X^{2k_2-2s}) \\
&= \sum_{r,s \ge 0} \alpha_{2k_1-2r}^{(2k_1,t)} \alpha_{2k_2-2s}^{(2k_2,t)} \sum_{l \ge 1} nc^{(2l)}(2k_1-2r, 2k_2-2s) t^{\frac{2k_1+2k_2}{2}-(r+s+2l)} \\
&= \sum_{l \ge 1} t^{k_1+k_2-2l} \sum_{r,s \ge 0} \alpha_{2k_1-2r}^{(2k_1,1)} \alpha_{2k_2-2s}^{(2k_2,1)} nc^{(2l)}(2k_1-2r, 2k_2-2s).
\end{align*}
$$

Let $l \ge 1$ be fixed. For $r \le k_1 - l$, $s \le k_2 - l$ using (5.21), the innermost sum is

$$
\begin{align*}
& \sum_{r,s \ge 0} \alpha_{2k_1-2r}^{(2k_1,1)} \alpha_{2k_2-2s}^{(2k_2,1)} nc^{(2l)}(2k_1-2r, 2k_2-2s) \\
&= (2k_1 \cdot 2k_2 \cdot 2l) \sum_{r,s \ge 0} \frac{(-1)^{r+s}}{(2k_1-r)(2k_2-s)} \binom{2(k_1-r)}{k_1-r-l} \binom{2(k_2-s)}{k_2-s-l} \binom{2k_1-r}{r} \binom{2k_2-s}{s} \\
(5.23) \quad &= 8lk_1k_2 \left( \sum_{r=0}^{k_1-l} \frac{(-1)^r}{(2k_1-r)} \binom{2(k_1-r)}{k_1-r-l} \binom{2k_1-r}{r} \right) \left( \sum_{s=0}^{k_2-l} \frac{(-1)^s}{(2k_2-s)} \binom{2(k_2-s)}{k_2-s-l} \binom{2k_2-s}{s} \right).
\end{align*}
$$

We claim that if $k_1 \neq k_2$, one of the two sums in (5.23) is zero. Let us take $k \ge l + 1$. One has

$$
\begin{align}
(5.24) \quad & \sum_{r=0}^{k-l} \frac{(-1)^r}{(2k-r)} \binom{2(k-r)}{k-r-l} \binom{2k-r}{r} = \sum_{r=0}^{k-l} \frac{(-1)^r}{(2k-r)} \binom{k-l}{r} \binom{2k-r}{k-l} \nonumber \\
(5.25) \quad & = \frac{1}{k-l} \sum_{r=0}^{k-l} (-1)^r \binom{k-l}{r} \binom{2k-r-1}{k+l-r}. \nonumber
\end{align}
$$

The binomial coefficient $\binom{2k-r-1}{k+l-r}$ is the coefficient of $z^{k+l-r}$ in the development of $(1+z)^{2k-r-1}$. Thus, the sum (5.25) is the coefficient of $z^{k+l}$ in the development of $Q(z) = \sum_{r=0}^{k-l} (-1)^r \binom{k-l}{r} (1+z)^{2k-r-1} z^r$. However,

$$
Q(z) = (1+z)^{2k-1} \sum_{r=0}^{k-l} \binom{k-l}{r} \left(\frac{-z}{1+z}\right)^r = (1+z)^{k+l-1}
$$

so that the coefficient of $z^{k+l}$ in $Q(z)$ is zero. If $k_1 = k_2 = l$, the two sums in (5.23) are equal to $\frac{1}{2l}$ so that

$$
(5.26) \qquad \varphi_c^{(t)}(P_{2k_1}^{(t)}, P_{2k_2}^{(t)}) = (2k_1) \delta_{k_1=k_2}
$$

which is (5.16) for even indexes. The computations are essentially the same for odd values of $k_1 \leftarrow 2k_1 + 1$ and $k_2 \leftarrow 2k_2 + 1$:

$$
\begin{align}
(5.27) \qquad & \varphi_c^{(t)}(P_{2k_1+1}^{(t)}, P_{2k_2+1}^{(t)}) = \sum_{r,s \ge 0} \alpha_{2k_1+1-2r}^{(2k_1+1,t)} \alpha_{2k_2+1-2s}^{(2k_2+1,t)} \varphi_c^{(t)}(X^{2k_1+1-2r}, X^{2k_2+1-2s}) \nonumber \\
(5.28) \qquad & = \sum_{l \ge 1} t^{k_1+k_2-2l+2} \sum_{r,s \ge 0} \alpha_{2k_1+1-2r}^{(2k_1+1,1)} \alpha_{2k_2+1-2s}^{(2k_2+1,1)} nc^{(2l-1)}(2k_1+1-2r, 2k_2+1-2s). \nonumber
\end{align}
$$
For a fixed value of $l \ge 1$, the innermost sum is

$$
\begin{align}
& \sum_{r,s \ge 0} \alpha_{2k_1+1-2r}^{(2k_1+1,1)} \alpha_{2k_2+1-2s}^{(2k_2+1,1)} nc^{(2l-1)}(2k_1+1-2r, 2k_2+1-2s) \nonumber \\
& \qquad \times \sum_{r,s \ge 0} \frac{(-1)^{r+s}}{(2k_1+1-r)(2k_2+1-s)} \binom{2(k_1-r)+1}{k_1+1-r-l} \binom{2(k_2-s)+1}{k_2+1-s-l} \binom{2k_1+1-r}{r} \binom{2k_2+1-s}{s} \tag{5.29} \\
& = \left( \sum_{r=0}^{k_1+1-l} \frac{(-1)^r}{(2k_1+1-r)} \binom{2(k_1-r)+1}{k_1+1-r-l} \binom{2k_1+1-r}{r} \right) \left( \sum_{s=0}^{k_2+1-l} \frac{(-1)^s}{(2k_2-s)} \binom{2(k_2-s)+1}{k_2+1-s-l} \binom{2k_2+1-s}{s} \right) \nonumber \\
& \qquad \times (2k_1+1)(2k_2+1)(2l-1). \nonumber
\end{align}
$$

Using the same argument as above, the two sums in (5.29) vanish, except when $k_1 + 1 = k_2 + 1 = l$
where they are both equal to $\frac{1}{2l+1}$. Therefore,

$$
(5.30) \qquad \varphi_c^{(t)}(P_{2k_1+1}^{(t)}, P_{2k_2+1}^{(t)}) = (2k_1+1)\delta_{k_1=k_2},
$$

which ends the proof of statement (5.16) in Proposition 5.16.

**Remark 5.32.** Let us verify that for $t = 0$, one finds the result of [12]. We have that

$$
\varphi^{(t)}(P_k^{(t)}, P_l^{(t)}) = 0
$$

according to Proposition 5.16 which is indeed equal to $\mathbb{E}[a_{1,2}^2]^k = 0$. On the other hand,

$$
\varphi_c^{(t)}(X^p, X^q) = nc^{(p)}(p, q)\delta_{p,q} = p\delta_{p,q}
$$

where the last equality comes from the fact that a non-crossing pairing of the $(p,p)$ annulus with $p$
through strings is determined by the choice of one through string only. Thus, $(V_k^{(n)})_k$ converges to
a Gaussian family $(V_k)_k$ of independent centered variables with parameters $\mathbb{E}[|V_k|^2] = k$, $\mathbb{E}[V_k^2] = 0$
which coincides with the Gaussian family in [12, Theorem 1.2].

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QUENTIN FRANÇOIS: CEREMADE, UMR CNRS 7534, UNIVERSITÉ PARIS-DAUPHINE, PSL RESEARCH UNIVERSITY, PLACE DU MARÉCHAL DE LATTRE DE TASSIGNY 75016 PARIS, FRANCE.

Email address: quentin.francois@dauphine.psl.eu

DAVID GARCÍA-ZELADA: LABORATOIRE DE PROBABILITÉS, STATISTIQUE ET MODÉLISATION, UMR CNRS 8001, SORBONNE UNIVERSITÉ, 4 PLACE JUSSIEU, 75005 PARIS, FRANCE.

Email address: david.garcia-zelada@sorbonne-universite.fr