# Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

Charlie Dworaczek Guera, Ronan Memin

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Charlie Dworaczek Guera, Ronan Memin. Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature. 2025. hal-05314896

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Distributed under a Creative Commons Attribution 4.0 International License
Large deviations at the edge for 1D gases and tridiagonal random matrices at high temperature

Charlie Dworaczek Guera,\* Ronan Memin†

July 21 2025

Abstract

We consider a model for a gas of $N$ confined particles subject to a two-body repulsive interaction, namely the one-dimensional log or Riesz gas. We are interested in the so-called *high temperature* regime, ie where the inverse temperature $\beta_N$ scales as $N\beta_N \to 2P > 0$. We establish, in the log case, a large deviation (LD) principle and moderate deviations estimates for the largest particle $x_{\max}$ when appropriately rescaled. Our result is an extension of [BADG01, Pak20] where such estimates were shown for the largest particle of the $\beta$-ensemble respectively at fixed $\beta_N = \beta > 0$ and $\beta_N \gg N^{-1}$. We show that the corresponding rate function is the same as in the case of iid particles. We also provide LD estimates in the Riesz case. Additionally, we consider related models of symmetric tridiagonal random matrices with independent entries having Gaussian tails; for which we establish the LD principle for the top eigenvalue. In a certain specialization of the entries, we recover the result for the largest particle of the log-gas. We show that LD are created by a few entries taking abnormally large values.

Contents

<table><tr><td><b>1 Introduction</b></td><td><b>2</b></td></tr><tr><td>1.1 Setting of the problem</td><td>2</td></tr><tr><td>1.2 Main results</td><td>4</td></tr><tr><td>1.3 Comparison with the literature</td><td>7</td></tr><tr><td>1.4 Outline of the proofs</td><td>8</td></tr><tr><td>1.4.1 Outline of the proof of Theorem 1.2</td><td>8</td></tr><tr><td>1.4.2 Outline of the proof of Theorem 1.8</td><td>9</td></tr><tr><td><b>2 Proof of Theorems 1.2, 1.3, 1.5, 1.6</b></td><td><b>9</b></td></tr><tr><td>2.1 Large deviation upper bounds, exponential tightness</td><td>10</td></tr><tr><td>2.1.1 Upper bound: right-deviation</td><td>10</td></tr><tr><td>2.1.2 Exponential tightness</td><td>11</td></tr><tr><td>2.1.3 Upper bound: left deviation.</td><td>11</td></tr><tr><td>2.2 Lower bound</td><td>14</td></tr><tr><td><b>3 Matrix interpretation: proof of Theorem 1.8</b></td><td><b>16</b></td></tr></table>

<footnote>\*KTH Stockholm,</footnote>

<footnote>email: chadg@kth.se</footnote>

<footnote>†Institut de mathématiques de Toulouse, France</footnote>

<footnote>email: ronan.memin@math.univ-toulouse.fr</footnote>
# 1 Introduction

## 1.1 Setting of the problem

Let $N \ge 1$ and $\beta_N > 0$, $V: \mathbb{R} \to \mathbb{R}$ growing fast enough at infinity, we are interested in the Gibbs probability measure given by

$$d\mathbb{P}_N(\underline{x}) = p_N(\underline{x})dx_1 \dots dx_N \quad \text{with} \quad p_N(\underline{x}) \stackrel{(\text{def})}{=} \frac{1}{\mathcal{Z}_N} \prod_{i<j}^{N} e^{-\beta_N g(x_i - x_j)} \prod_{i=1}^{N} e^{-V(x_i)}, \quad (1)$$

where the two-body interaction $g$ is either given by the logarithmic repulsion $g(x) = -\log|x|$ or by the Riesz repulsion $g(x) = |x|^{-s}$ with $0 < s < 1$. The partition function $\mathcal{Z}_N$ is the normalization constant which makes $\mathbb{P}_N$ a probability measure on $\mathbb{R}^N$, namely

$$\mathcal{Z}_N = \int_{\mathbb{R}^N} \prod_{i<j}^{N} e^{-\beta_N g(x_i - x_j)} \prod_{i=1}^{N} e^{-V(x_i)} dx_i. \quad (2)$$

The measure (1) is called the (one dimensional) log gas – or $\beta$-ensemble – in the case $g = -\log$, and the Riesz gas in the case $g = |\cdot|^{-s}$. We are interested in this paper in the case where $N\beta_N \xrightarrow{N\to\infty} 2P > 0$, which is called the high temperature regime. In this regime, it was proven in [GZ19], [Lam21, Proposition 1.3] that the sequence of random probability measures $(\hat{\mu}_N)_N$ defined by

$$\hat{\mu}_N \stackrel{(\text{def})}{=} \frac{1}{N} \sum_{i=1}^{N} \delta_{x_i}$$

satisfies a LDP at speed $N$ with a strictly convex, good rate function $I$. The latter is defined by $I \stackrel{(\text{def})}{=} \mathcal{E} - \inf_{\mu \in \mathcal{M}_1(\mathbb{R})} \mathcal{E}(\mu)$ where, for all probability measure $\mu$ absolutely continuous with respect to the Lebesgue measure,

$$\mathcal{E}(\mu) \stackrel{(\text{def})}{=} \int_{\mathbb{R}} V(x) d\mu(x) + P \iint_{\mathbb{R}^2} g(x-y) d\mu(x) d\mu(y) + \int_{\mathbb{R}} \log \frac{d\mu(x)}{dx} d\mu(x), \quad (3)$$

and $\mathcal{E}(\mu) = +\infty$ for non absolutely continuous probability measures with respect to the Lebesgue measure on $\mathbb{R}$. In this energy functional, the last term is called the entropy of the measure $\mu$ and represents the negative physical entropy. Minimizing $\mathcal{E}$ can then be seen as finding a compromise between a minimization of the energy (composed of the two first terms) and a maximization of the physical entropy. The latter is negligible when the inverse temperature of the Gibbs measure $\beta_N$ satisfies $\beta_N \gg N^{-1}$, but has the same order of magnitude as the energy when the correlation between particles is small/the temperature is high (see [DM08, (34)] for a discussion).

We call the unique minimizer of $\mathcal{E}$ the equilibrium measure and denote it $\mu_{\text{eq}}$ (we omit the $V$ and $P$-dependence since these parameters remain fixed throughout the entire article). It is Lebesgue continuous with p.d.f. $\rho_{\text{eq}}$ that satisfies:

$$V(x) + 2P \int_{\mathbb{R}} g(x - y) \rho_{\text{eq}}(y) dy + \log \rho_{\text{eq}}(x) = \lambda_{\text{eq}} \quad x - ae \quad (4)$$

where $\lambda_{\text{eq}}$ is a constant (see [Lam21, Proposition A.2]). In this context, $\rho_{\text{eq}}$ is supported on $\mathbb{R}$ because of the presence of the entropy in the minimizing equation. This is a major difference
with the constant $\beta$ case and the case of a $N$-scaled potential, where the equilibrium measure is compactly supported. This equation can be rewritten as

$$ \forall x \in \mathbb{R}, \rho_{\text{eq}}(x) = \exp\left(-V(x) - 2PU^{\rho_{\text{eq}}}(x) + \lambda_{\text{eq}}\right), \quad U^{\rho_{\text{eq}}}(x) \stackrel{\text{(def)}}{=} \int_{\mathbb{R}} g(x-y)\rho_{\text{eq}}(y)dy. \quad (5) $$

This potential $U^{\rho_{\text{eq}}}$ has recently been identified as the Markov-Krein transform of simple distributions for some specific examples of $V$ and $g = -\log|\cdot|$ in [NTT24]. The density $\rho_{\text{eq}}$ also admits explicit expressions, such as in the case $V_G(x) = x^2/2$ and $g = -\log|\cdot|$ where it is known as the Askey-Wimp-Kerov distribution [AW84]:

$$ \rho_{V_G}(x) = \frac{e^{-\frac{x^2}{2}}}{\sqrt{2\pi} |\hat{f}_{\alpha}(x)|^2}, \quad \hat{f}_{\alpha}(x) \stackrel{\text{(def)}}{=} \sqrt{\frac{P}{\Gamma(P)}} \int_0^{+\infty} t^{P-1} e^{-\frac{t^2}{2} + ixt} dt. $$

Because the limiting empirical measure $\mu_{\text{eq}}$ is supported on the whole real line, the top particle $x_{\text{max}} \stackrel{\text{(def)}}{=} \max_{1 \le i \le N} x_i$ goes to $+\infty$ almost surely. More precisely, [Lam21] establishes (in the general context of log and Riesz interactions in the high temperature regime) that for $V = |\cdot|^{\alpha}$, $x_{\text{max}}$ is almost surely of order $(\log N)^{1/\alpha}$ and that the local statistics at the edge are given by a Poisson point process. Considering the particular case $\beta_N = 0$ of independent particles, these results are well known, see for example [Viv15]. This shows that the correlations introduced by the interaction $g$ in (1) do not affect the nature of the statistics of the top particle in this regime.

In this paper, we show a LDP for the top particle $x_{\text{max}}$, namely a quantification of the probability of a deviation of $x_{\text{max}}$ from its typical behavior in the large $N$ limit.

In the log gas case, $ie$ when $g = -\log$, another approach to analyze the measure (1) is given by its link with random matrices. Indeed, specifying $V(x) = x^2/2$, it was shown in [DE02] that with the choice $N\beta_N \xrightarrow{N\to\infty} 2P$, $\mathbb{P}_N$ is the joint law of (unordered) eigenvalues of the tridiagonal matrix

$$ T = \begin{pmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2 & \ddots & \ddots & \\ & & \ddots & \ddots & b_{N-1} \\ & & & b_{N-1} & a_N \end{pmatrix}, $$

where $(a_i)_{1 \le i \le N}$, $(b_i)_{1 \le i \le N-1}$ are independent families of independent random variables, with $a_i \sim \mathcal{N}(0, 1)$ and $b_i \sim \chi_{\beta_N(N-i)}$, where for $\theta > 0$ the law $\chi_{\theta}$ has density

$$ f_{\theta}(x) = \frac{2^{1-\theta/2}}{\Gamma(\theta/2)} x^{\theta-1} e^{-x^2/2} \mathbf{1}_{x>0}. \quad (6) $$

With this representation, it is natural to ask which kind of events on the entries of $T$ produce a large deviation of the top eigenvalue. We establish in the sequel that such large deviations are created by a small number of neighbouring entries taking large values.

**The present paper is composed of two mainly independent parts.**

**The first part** establishes in Theorem 1.2 a large deviation principle for the top particle $x_{\text{max}}$, namely a quantification of the probability of a deviation of $x_{\text{max}}$ from its typical behavior in the
large N limit, for the log case; as well as LD estimates for the Riesz-case. We also provide the moderate deviations estimates.

**The second part** establishes a LD principle for the top eigenvalue of symmetric tridiagonal matrices having independent entries, and uniform (with respect to N) Gaussian tails. Because of the aforementioned tridiagonal matrix representation of the log gas, we recover in particular the result of Theorem 1.2 in the case of a quadratic potential, and give an interpretation of the events producing a LD of the top eigenvalue.

## 1.2 Main results

**Assumptions 1.1** $V(x) = \kappa|x|^{\alpha} + \varphi(x)$ where $\alpha \ge 1$, $\kappa > 0$ and $\varphi(x) = o_{|x|\to\infty}(x^{\alpha})$, where $\varphi$ is $C^2$ outside of a compact set, with $\varphi'(x) = o_{|x|\to\infty}(x^{\alpha-1})$ and for any $a < b$,

$$ \sup_{y \in [a,b]} \varphi''(x+y) = o_{|x|\to\infty}(x^{\alpha-2}). $$

The assumptions above are quite general. The assumption on $\varphi''$ allows us to use the results from [Lam21] about the convergence of the edge towards a Poisson point process when $N \to \infty$, see Lemma 2.7. These assumptions also imply in particular that there exists $c \in \mathbb{R}$ such that for all $a \ge 0$, for all $x \in \mathbb{R}$, $V(x) - a\log(1+|x|) \ge c$, ensuring that the probability measure (1) is well defined when $g = -\log$.

We define the function $I_{\alpha}: \mathbb{R} \mapsto [0, +\infty]$

$$ I_{\alpha}(x) \stackrel{(\text{def})}{=} \begin{cases} x^{\alpha} - 1 & \text{if } x \ge 1 \\ +\infty & \text{if } x < 1. \end{cases} \qquad (7) $$

This function is clearly lower semi-continuous and its level sets are compact, making $I_{\alpha}$ a good rate function. Our main result establishes a full LDP for the top particle of the log gas.

**Theorem 1.2 (LDP for $\beta$-ensembles)** Under Assumptions 1.1, let $g = -\log$. The distribution of $\frac{x_{\max}}{(\kappa^{-1}\log N)^{1/\alpha}}$ under $\mathbb{P}_N$ satisfies a LDP at speed $\log N$ with good rate function $I_{\alpha}$, ie the following estimates hold:

* For any open set $O \subset \mathbb{R}$,

$$ -\inf_{x \in O} I_{\alpha}(x) \le \liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in O \right) \qquad (8) $$

* For any closed set $F \subset \mathbb{R}$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in F \right) \le -\inf_{x \in F} I_{\alpha}(x). \qquad (9) $$

In the Riesz case, we prove the following LD estimates.

**Theorem 1.3 (LD estimates, Riesz case.)** Under Assumptions 1.1, let $g = |\cdot|^{-s}$. Then, for all $x > 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} > x \right) \le -I_{\alpha}(x), $$
and

$$ \lim_{\delta \to 0} \liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in (x-\delta, x+\delta) \right) \geq -I_{\alpha}(x). $$

For all $x < \left(\frac{1-s}{1+s}\right)^{-1/\alpha}$,

$$ \lim_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} < x \right) = I_{\alpha}(x) = -\infty. \quad (10) $$

**Remark 1.4** • Although we expect the full LDP for the top particle in the Riesz case to hold, Theorem 1.3 says nothing about the rate of decay of $\mathbb{P}_N\left(\frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} < x\right)$ in the region $\left(\frac{1-s}{1+s}\right)^{-1/\alpha} \le x < 1$. To do so, extending (10) to all $x < 1$ would be enough.

• We only state our theorem in the case potentials with power growth. It can be seen that for potentials which grow faster to infinity, the LDP looks different. Indeed, in the case of iid particles ($P = 0$) and potential $V = \cosh$, the largest particle satisfies a LDP at speed $\log N$ with rate function given by

$$ I_{\cosh}(x) \stackrel{(\text{def})}{=} \begin{cases} +\infty & \text{if } x < 1 \\ 0 & \text{if } x = 1 \\ +\infty & \text{if } x > 1 \end{cases} $$

• The analogue of this result for $\beta$-ensembles for $\beta > 0$ are often used in truncation argument such as in [BG13, Proposition 2.2] and could be applied to extend the asymptotic expansion of $\log \mathcal{Z}_N$ proven in [DG24] to more general potentials.

• The previous result can be easily extended to the LD for the law of $\frac{x_{\min}}{(\kappa^{-1} \log N)^{1/\alpha}}$.

Establishing Theorems 1.2 and 1.3, we derive a refined upper bound in the case of a left deviation for $x_{\max}$. Such a deviation is much more unlikely than a right one, as it imposes a displacement of a mesoscopic number of particles.

**Theorem 1.5 (Refined upper bound)** Under Assumptions 1.1, set $x_0 = 0$ for $g = -\log$, and $x_0 = s/(1+s)$ for $g = |\cdot|^{-s}$. Then for all $0 < x < 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \log \frac{1}{\mathbb{P}_N\left(\frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} < 1-x\right)} \geq 1 - (1-x)^{\alpha}. $$

It was proven in [Lam21] that the size of the fluctuations of $x_{\max}$ around $(\kappa \log N)^{1/\alpha}$ is of size $\frac{1}{(\log N)^{1-1/\alpha}}$. Hence the probability of a deviation of size $\frac{x}{(\log N)^{\gamma}}$ for $x \neq 0$ and $\gamma < 1-\alpha^{-1}$ should go to 0. This regime is called the *moderate deviations* and the quantification of these probabilities is the object of the following proposition.
**Theorem 1.6 (Moderate deviations)** Under Assumptions 1.1, set $-\alpha^{-1} < \gamma < 1 - \alpha^{-1}$, we have for all $x > 0$ the following estimates:

$$ \limsup_{N \to \infty} \frac{1}{(\log N)^{1-\gamma-\alpha^{-1}}} \log \mathbb{P}_N \left( x_{\max} \ge (\kappa^{-1} \log N)^{1/\alpha} + \frac{x}{(\log N)^\gamma} \right) \le \frac{x}{\alpha} $$

and for $g = -\log$, for all $x > 0$:

$$ \limsup_{N \to \infty} \frac{1}{(\log N)^{1-\gamma-\alpha^{-1}}} \log \log \frac{1}{\mathbb{P}_N \left( x_{\max} \le (\kappa^{-1} \log N)^{1/\alpha} - \frac{x}{(\log N)^\gamma} \right)} \ge \frac{x}{\alpha} $$

In Theorem 1.8, we establish the following LDP for (periodic or not) tridiagonal matrices with independent entries having Gaussian tail. Let $(a_1, \dots, a_N)$, $(b_1, \dots, b_N) \in \mathbb{R}^N$, we consider the symmetric tridiagonal matrix

$$ T = \begin{pmatrix} a_1 & b_1 & & & \\ b_1 & a_2 & b_2 & & \\ & b_2 & \ddots & \ddots & \\ & & \ddots & \ddots & b_{N-1} \\ & & & b_{N-1} & a_N \end{pmatrix} \quad (11) $$

and the corresponding symmetric periodic tridiagonal matrix

$$ T_{\text{per}} = \begin{pmatrix} a_1 & b_1 & & & b_N \\ b_1 & a_2 & b_2 & & \\ & b_2 & \ddots & \ddots & \\ & & \ddots & \ddots & b_{N-1} \\ & & & b_{N-1} & a_N \end{pmatrix}. \quad (12) $$

We assume the entries of these matrices to have uniform Gaussian tail:

**Definition 1.7 (Uniform Gaussian tail)** We say that the family of real random variables $(X_i)_{i \in I}$, where $I$ is any index set, has uniform Gaussian tail if for all $i \in I$,

$$ -\log \mathbb{P}(|X_i| > x) = x^2/2 + o_{x \to +\infty}(x^2), \quad \text{and} \quad -\log \mathbb{P}(X_i > x) = x^2/2 + o_{x \to +\infty}(x^2), $$

where the reminder term $o(x^2)$ is independent of $i \in I$.

Then, we show, recalling the notation $I_\alpha$ of equation (7),

**Theorem 1.8** Let $(a_i)_{i \ge 1}$, $(b_i)_{i \ge 1}$ be independent sequences of independent variables. Assume that $(a_i)_{i \ge 1}$ and $(\sqrt{2b_i})_{i \ge 1}$ have uniform Gaussian tail.

Consider the tridiagonal or periodic tridiagonal model

$$ J = T(a_1, \dots, a_N, b_1, \dots, b_{N-1}) \quad \text{or} \quad J = T_{\text{per}}(a_1, \dots, a_N, b_1, \dots, b_N). $$

Then, $\lambda_{\max}(J)/\sqrt{2\log N}$ satisfies a LDP with rate function $I_2$.

**Remark 1.9** • The proof of Theorem 1.8 is based on an exponential approximation argument. As we will see, we will approximate $J$ by block diagonal matrices, whose blocks can be assumed to be of size $O(1)$. From this fact, we deduce that the LD of (exponential approximations of) $\lambda_{\max}(J)$ are produced by the LD of a bounded number of entries, which is enough to prove the full LDP for the Gaussian $\beta$-ensembles.
* Taking for $i \ge 1$, $X_i \sim \chi_{u_i}$, with $0 < u_i < C$ for some $C$ independent of $i$, the sequence $(X_i)$ satisfies the previous conditions.

* As mentioned earlier, considering the model $T$ with $(a_i)_{1\le i\le N}$ iid standard Gaussians, and $\sqrt{2b_i}$ is distributed with respect to $\chi_{\beta_N(N-i)}$, by [DE02], the law of the spectrum of $T$ is distributed with respect to the high temperature Gaussian $\beta$-ensemble, and one recovers Theorem 1.2 in this special case.

## 1.3 Comparison with the literature

### Links with extremal value of random matrices

The first LD result in random matrix theory was established by Ben-Arous and Guionnet [BAG97], where the authors showed that the empirical measure of the fixed temperature $\beta$-ensemble — which includes the classical ensembles GOE, GUE and GSE — satisfies a LDP. They provided an expression for the rate function, relying on the explicit joint law of eigenvalues.

A LDP for the largest eigenvalue was first shown for the GOE in [BADG01] in the context of the study of the aging phenomenon in spin glass models. This result was then extended in [AGZ10, Theorem 2.6.6] for general $\beta$-ensembles (at fixed $\beta$) whose partition function satisfies a certain rate condition. This type of LDP was later shown, proving a continuity result for the spherical integrals, for rank-one perturbations of GOE/GUE in [Mai07] and then extended for the joint law of the extremal eigenvalues in [BGGM12]. The case of critical $\beta$-ensembles was also studied in [FGSW15]. The LDP for the largest eigenvalue was recently further generalized in [GKL24], where the authors consider $\beta$-ensembles on a complex contour homeomorphic to the real line. LDP for the law of the maximal eigenvalue of Wigner matrices with variances profiles were shown in [Hus22, DGH24]. In [BC14], the authors establish a LDP for the empirical measure for a Hermitian matrix $X_N$ with sub-Gaussian tails, relying on the fact that LD are caused by a small number of entries being abnormally large. In [Aug16], Augeri built on this idea to deduce a LDP for the largest eigenvalue for the same matrix model. In [GH20], the authors proved a LDP for the largest eigenvalue in the case of general sharp sub-Gaussian entries. The sharpness assumption was then relaxed in [AGH21].

### The $\beta$-ensembles at high temperature

When the interaction is given by $-\log$, a lot is known in the regime $\beta_N \to 0$. A pioneer work at high temperature was [ABG12] where the authors considered the Gaussian $\beta$-ensembles with $N\beta = 2P$ and the so-called Gauss-Wigner crossover when $P \to 0$ and $P \to \infty$. It was also argued in this case that when $P \to -1$, the empirical measure should converge to a Dirac [ABG12, AB19]. Interpolations between the Tracy-Widom and Gumbel distributions which govern respectively the fluctuations at the edge for strongly correlated and weakly correlated systems have been studied in [Joh07, AD14b]. The local statistics in the bulk were then studied in [AD14a], where the authors show that their limiting behavior, described by the Sine$\beta$ point process, converges as $\beta \to 0$ to a Poisson point process. It was later shown in [BGP15] that the local statistics converge, in the case of a quadratic potential, towards those of a Poisson point process similarly to the case of iid particles ($P = 0$). This result was then extended to the edge, general potential, general interaction and general configuration spaces in [Pak18, NT20, Lam21, PGPT24, PGPT25]. The LDP for the empirical measure of the eigenvalues for different matrix models at high temperature were shown in [GM22, MM23] as well as connection with different integrable systems. The same results were shown for various $\beta$-ensembles in [Ma23a] for $\beta \gg \log N/N$; and for the extremal eigenvalues in the same temperature regime in the Laguerre and Jacobi $\beta$-ensembles in [Ma23b, LM23]. Central limit theorems were established in [HL21, NT18, DGM24, MM24] and the asymptotics of the partition function was established in [DG24].
## Sparse and tridiagonal random matrices

Many random graph models can be studied through their adjacency matrices, for example regular graphs or Erdős–Rényi graphs. A lot is known about the extremal eigenvalues of these matrices [BGBK19, BGBK20, ADK21, ADK23, AB23]. LDP were shown for the two largest eigenvalues of sparse Erdős–Rényi ($p \to 0$) in [BG20] in the regime $p \gg n^{-1/2}$. The sparsity regime was then extended to $p \sim n^{-1}$ in [BBG21]. In [GN22, GHN24], the authors were able to show the LDP for a sparsity parameter $p = O(n^{-1})$ in the case of respectively Gaussian weights and general weights on edges. It is very similar to the LDP obtained in this paper. The situation described in Remark 1.9, ie that the LD of $\lambda_{max}$ is produced by a bounded number of entries, is reminiscent of the conclusion of [Aug16], where the matrix of interest is a Wigner matrix whose entries have tails heavier than Gaussian. An interesting case of application of Theorem 3.1 is taking $(a_i)_{i \ge 1}$ iid standard Gaussians, and $(\sqrt{2}b_i)_{i \ge 1}$ iid with law $\chi_{2P}$ for some $P > 0$. Then, the periodic matrix $J_{per}$ is distributed as the Lax matrix of the periodic Toda chain with pressure $P$ at thermal equilibirum, [Spo20]. This model has attracted attention lately, both as a useful model towards the understanding of Generalized Hydrodynamics (GHD) for integrable systems, and via its links with the high temperature $\beta$-ensemble ; see for example [Spo23, Maz22, GM22, MM24, Agg25b, Agg25a] and references therein.

## 1.4 Outline of the proofs

### 1.4.1 Outline of the proof of Theorem 1.2

We establish the LDP by proving a weak LDP and exponential tightness.

* The weak LDP for $(\frac{x_{max}}{(\kappa^{-1} \log N)^{1/\alpha}})_{N \ge 1}$ with rate function $I_{\alpha}$ holds if, by [DZ09, Theorem 4.1.11] if the following holds:

$$ \lim_{\delta \downarrow 0} \liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in (x-\delta, x+\delta) \right) = \\ \lim_{\delta \downarrow 0} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in (x-\delta, x+\delta) \right) = -I_{\alpha}(x). $$

* Exponential tightness holds if there is a sequence of compact sets $(K_M)_{M \ge 1}$ such that

$$ \lim_{M \to \infty} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\kappa^{-1} \log N)^{1/\alpha}} \in K_M^c \right) = -\infty. $$

Together these results are enough to show the full LDP see [AGZ10, Theorem 2.6.6]. The outline of the proof of the LDP for the log case, as well as LD estimates for the Riesz gas and the moderate deviations estimates are established in

1. Propositions 2.1 and 2.5 show respectively the large and moderate deviations upper bound for closed sets of the form $[x, +\infty)$ for $x \ge 1$, and $(-\infty, x]$ for $x < 1$. For right deviations, we use a Wegner type estimate and for left deviations, we obtain a pointwise bound on $p_N$ defined in (1) to show that up to a negligible remainder $\mathfrak{R}_N$, it holds that

$$ p_N(\underline{x}) \le e^{\mathfrak{R}_N} \prod_{i=1}^{N} \rho_{\text{eq}}(x_i). $$
2. As a direct consequence, we deduce the exponential tightness in Corollary 2.2 by taking $K_M \stackrel{(\text{def})}{=} [0, M]$. The event $\{x_{\max} < 0\}$ leads to a LD of the empirical measure which is of speed $N$ [GZ19], while $\log \mathbb{P}_N(x_{\max} > M(\kappa^{-1} \log N)^{1/\alpha}) \le -f(M) \log N$, with $f(M) \xrightarrow{M \to \infty} +\infty$.

3. We establish in Proposition 2.8 the weak LD lower bound. We restrict the configuration to a deviation of only one particle while the others remain in a very likely set. Since by [Lam21], we know the size of the fluctuations, we are able to bound by below $\log p_{N-1}$ and conclude by Jensen inequality.

### 1.4.2 Outline of the proof of Theorem 1.8

The proof of Theorem 1.8 relies on a truncation procedure on the entries of the matrix, similar to the procedure of [Aug16], showing that only the big entries of the matrix at stake matter for the LD. Let us describe here the case of the (nonperiodic) tridiagonal model $T_N$.

For $\varepsilon > 0$, we let $T_N^\varepsilon = (T_{i,j} \mathbf{1}_{\{|J_{i,j}| \ge \varepsilon\sqrt{2\log N}\}})_{1 \le i,j \le N}$ be the truncated matrix with entries larger than $\varepsilon\sqrt{2\log N}$. Then Lemma 3.6 shows that the normalized top eigenvalue of $T_N^\varepsilon$ is exponentially equivalent to the one of $T_N$: more precisely, let $\mu_N = \lambda_{\max}(T_N)/\sqrt{2\log N}$ and define $\mu_N^\varepsilon$ similarly. Then, for any $\varepsilon > 0$, we have

$$ \lim_{\delta \to 0} \lim_{N} \frac{1}{\log N} \log \mathbb{P}(|\mu_N^\varepsilon - \mu_N| > \varepsilon) = -\infty. \quad (13) $$

By [DZ09, Theorem 4.2.16], to establish the existence of a LDP for $(\mu_N)_{N \ge 1}$, it suffices to show that for any $\varepsilon > 0$ the sequence $(\mu_N^\varepsilon)_{N \ge 1}$ satisfies a LDP with rate function $I^{(\varepsilon)}$. The sequence $(\mu_N)_{N \ge 1}$ then satisfies a LDP with rate function $I(x) \stackrel{(\text{def})}{=} \sup_{\delta>0} \liminf_{\varepsilon \to 0} \inf_{z:|z-x|<\delta} I^{(\varepsilon)}(z)$. Now, because we assume the entries of $T_N$ to have Gaussian tails, a lot of entries in $T_N^\varepsilon$ vanish. In particular, $T_N^\varepsilon$ can be assumed to be a block diagonal matrix, $T_N^\varepsilon = \text{diag}(B_1, \dots, B_{k_N})$ where the blocks $(B_i)_{1 \le i \le k_N}$ are independent and can be assumed – at the scale of LD – to be of finite size (ie independent from $N$). The top eigenvalue of $T_N^\varepsilon$ is then

$$ \lambda_{\max}^{\varepsilon} = \max_{1 \le i \le k_N} \lambda_{\max}(B_i). \quad (14) $$

By Lemma 3.5 we then see that $\lambda_{\max}(B_i)$ has Gaussian tail. From this fact, decomposition (14), and the exponential approximation (13), we will deduce that for any $\varepsilon > 0$, $I^{(\varepsilon)} = I_2$, thus concluding for the LD of $\lambda_{\max}(T_N)$ with rate function $I_2$.

## Acknowledgements

The authors would like to thank the organizers of the "Random Matrices and Free probability" summer school organized in Toulouse in June 2024 where this project was initiated. The authors also thank Gaultier Lambert, Michel Pain and Jonathan Husson for fruitful discussions and suggestions.

# 2 Proof of Theorems 1.2, 1.3, 1.5, 1.6

We first notice that in both log gas and Riesz cases, we can assume $\kappa = 1$, ie $V(x) = |x|^\alpha + \varphi(x)$, via considering the system $y := \kappa^{1/\alpha} \underline{x}$, which amounts to replacing $\varphi$ by $\varphi(\cdot/\kappa^{1/\alpha})$, and $g$ by $g(\cdot/\kappa^{1/\alpha})$, for which the proofs are unchanged. In this section, we therefore assume $\kappa = 1$.
## 2.1 Large deviation upper bounds, exponential tightness

### 2.1.1 Upper bound: right-deviation

We start the proof by showing the upper bound in the case $x \ge 1$, ie when the rate function $I_\alpha$ is finite.

**Proposition 2.1 (Upper bound: right-deviation)** For all $x \ge 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \ge x \right) \le -I_{\alpha}(x). \quad (15) $$

Furthermore, let $-\alpha^{-1} < \gamma < 1 - \alpha^{-1}$ and $x > 0$, the following estimate holds:

$$ \limsup_{N \to \infty} \frac{1}{(\log N)^{1-\gamma-\alpha^{-1}}} \log \mathbb{P}_N \left( x_{\max} \ge (\log N)^{1/\alpha} + \frac{x}{(\log N)^\gamma} \right) \le \frac{-x}{\alpha} \quad (16) $$

**Proof** By [Lam21, Proposition 2.1], there exists $C > 0$, such that we have the following estimate for the density of the first marginal $\rho_N(u)$ for all $u \in \mathbb{R}$:

$$ \rho_N(u) \le C e^{-V(x)+2P \log(1+|u|)1_{g=-\log}}. $$

Thus let $x \ge 1$, for (15), by a union bound we have:

$$ \begin{aligned} \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \ge x \right) &\le N \mathbb{P}_N \left( \frac{x_1}{(\log N)^{1/\alpha}} \ge x \right) \\ &\le N C \int_{x(\log N)^{1/\alpha}}^{+\infty} e^{-|u|^\alpha - \varphi(u) + 2P \log(1+u) 1_{g=-\log}} du \\ &\le N (\log N)^{1/\alpha} C \int_{x}^{+\infty} e^{-(\log N)s^\alpha - \varphi((\log N)^{1/\alpha}s) + 2P \log(1+(\log N)^{1/\alpha}s) 1_{g=-\log}} ds. \end{aligned} $$

Setting $\phi(x) \stackrel{(\text{def})}{=} -\varphi(x) + 2P \log(1+x) 1_{g=-\log}$ and since $\partial_s (e^{-(\log N)s^\alpha + \phi((\log N)^{1/\alpha}s)})$ is negative, for $N$ large enough, by Assumption 1.1, there exists $C > 0$, such that for all $s \ge x$

$$ \begin{aligned} e^{-(\log N)s^\alpha + \phi((\log N)^{1/\alpha}s)} &= \frac{-1}{\alpha s^{\alpha-1} \log N - (\log N)^{1/\alpha} \phi'(s(\log N)^{1/\alpha})} \partial_s \left( e^{-(\log N)s^\alpha + \phi((\log N)^{1/\alpha}s)} \right) \\ &\le \frac{-C}{s^{\alpha-1} \log N} \partial_s \left( e^{-(\log N)s^\alpha + \phi((\log N)^{1/\alpha}s)} \right). \end{aligned} $$

Thus, we obtain the following bound, using that $\alpha \ge 1$:

$$ \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \ge x \right) \le C \frac{N(\log N)^{1/\alpha}}{x^{\alpha-1} \log N} \exp \left( - (\log N)x^{\alpha} + \phi((\log N)^{1/\alpha}x) \right), \quad (17) $$

Taking the logarithm and letting $N \to \infty$, yields the conclusion by Assumption 1.1. The proof of (16) is obtained in the exact same way.
### 2.1.2 Exponential tightness

**Corollary 2.2 (Exponential tightness)** The sequence $(\frac{x_{\max}}{(\log N)^{1/\alpha}})_N$ is exponentially tight, ie there exists a sequence $(K_M)_{M \ge 1}$ of compacts of $\mathbb{R}$ such that

$$ \lim_{M \to \infty} \limsup_{N \to \infty} \frac{1}{\log N} \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \in K_M^c \right) = -\infty. \qquad (18) $$

**Proof** Indeed let $M \ge 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \in [0, M]^c \right) = \max \{a, b_M\} $$

where

$$ a \stackrel{(\text{def})}{=} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} < 0 \right) $$

and

$$ b_M \stackrel{(\text{def})}{=} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} > M \right). $$

But $a = -\infty$ by the LDP for the empirical measure which is at speed $N$, see [GZ19], and

$$ b_M \le -I_{\alpha}(M) \xrightarrow{M \to +\infty} -\infty $$

by Proposition 2.1, proving (18) with $K_M = [0, M]$. $\square$

<figure><img src="image_3.png" /></figure>

### 2.1.3 Upper bound: left deviation.

We now prove an upper bound on the probability of a left-deviation of $x_{\max}$. Our strategy relies on a comparison with the edge of a system of iid particles distributed according to $\otimes^N \mu_{\text{eq}}$ where $\mu_{\text{eq}}$ is the unique minimizer of (3). In the latter case, the precise asymptotic of a left deviation is known. Prior to that, we need to understand how the function $U^{\rho_{\text{eq}}}$ behaves under small perturbations. The following lemma gives few useful properties about its derivative.

**Lemma 2.3** The function $U^{\rho_{\text{eq}}}$ is differentiable on $\mathbb{R}$ and its derivative is a bounded function on $\mathbb{R}$.

**Proof** The proof is done for the log case in [DGM24, Lemma 2.3]. For the Riesz case,

$$ [U^{\rho_{\text{eq}}}]'(x) = -s \int_{\mathbb{R}} \frac{\rho_{\text{eq}}(y)\text{sgn}(x-y)dy}{|x-y|^{s+1}} $$

where $f$ stands for the Cauchy principal value. For $x \in \mathbb{R}$, we have

$$ \left| \int_{|x-y|>1} \frac{\rho_{\text{eq}}(y)\text{sgn}(x-y)dy}{|x-y|^{s+1}} \right| \le 1. $$

And finally, for any $\varepsilon > 0$,

$$ \left| \int_{\varepsilon < |x-y| < 1} \frac{\rho_{\text{eq}}(y) \text{sgn}(x-y) dy}{|x-y|^{s+1}} \right| \le \left| \int_{|x-y| < 1} \frac{[\rho_{\text{eq}}(y) - \rho_{\text{eq}}(x)] \text{sgn}(x-y) dy}{|x-y|^{s+1}} \right| \\ + \rho_{\text{eq}}(x) \left| \int_{\varepsilon < |z| < 1} \frac{\text{sgn}(z) dy}{|z|^{s+1}} \right| \le \frac{2}{1-s} \|\rho'_{\text{eq}}\|_{\infty}. $$

Above, we used that the integral on the second line is equal to zero. Hence taking $\varepsilon \to 0$, we obtain that $[U^{\rho_{\text{eq}}}]'$ is bounded. $\square$

<figure><img src="image_4.png" /></figure>
We are now able to compare the law of the edge of $\mathbb{P}_N$ to the one of $\mu_{\text{eq}}^{\otimes N}$.

**Lemma 2.4** Let $x > 0$,

$$ \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} < (1-x)^{1/\alpha} \right) \le e^{\Re N} \mu_{\text{eq}}^{\otimes N} \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} < (1-x)^{1/\alpha} \right) $$

where $\Re N = O(\log N)$ for the case $g = -\log$ and $O(N^{\frac{2s}{1+s}})$ for the case $g = |\cdot|^{-s}$.

**Proof** We prove this bound by bounding the density $p_N$ pointwise. To do so, we use energy estimates. Since $\mathcal{E}(\hat{\mu}_N) = +\infty$ for all $\underline{x} \in \mathbb{R}^N$, we need to approximate it by an absolutely continuous measure with respect to Lebesgue. As in [DGM24, Section 3.1], we introduce $\tilde{\mu}_N$, the regularization of $\hat{\mu}_N$ by setting, for any $(x_i)_i \in \mathbb{R}^N$, the following: we first apply a permutation $\sigma$ which orders the particles $x_{\sigma(1)} \le \dots \le x_{\sigma(N)}$ and then we define $(y_i)_i \in \mathbb{R}^N$ as follows:

$$ y_1 \stackrel{\text{(def)}}{=} x_1, \quad \forall k \in [0, N-1], \quad y_{k+1} \stackrel{\text{(def)}}{=} y_k + \max\{x_{\sigma(k+1)} - x_{\sigma(k)}, N^{-b}\}. $$

Finally, we define $\tilde{\mu}_N$ by

$$ \tilde{\mu}_N \stackrel{\text{(def)}}{=} \lambda_{N-a} * \frac{1}{N} \sum_{i=1}^{N} \delta_{y_{\sigma^{-1}(i)}} = \lambda_{N-a} * \frac{1}{N} \sum_{i=1}^{N} \delta_{y_i}, \quad (19) $$

where $\lambda_{N-a}$ is the uniform measure on $[0, N^{-a}]$. When $g = -\log$, we take $a = 5$ and $b = 3$, while for the case $g = |\cdot|^{-s}$, we take $a = b = \frac{2}{1+s}$.

By using [DGM24, Proposition 1.6], $\mathcal{Z}_N \ge e^{N\mathcal{E}(\mu_{\text{eq}})+c}$ for some constant $c$ depending on $V, P$ and $s$ in the Riesz case. Using that for all $i \ne j$ one has $|x_i - x_j| \le |y_i - y_j|$ and that $g$ is non-increasing, we have:

$$ \log p_N(\underline{x}) \le -N\mathcal{E}(\mu_{\text{eq}}) - \frac{\beta_N}{2} \sum_{i \ne j} g(x_i - x_j) - \sum_{i=1}^{N} V(x_i) + O(1). $$

Using that for all $x \ne 0$ and $|h| \le \frac{|x|}{2}$ we have $|g(x+h) - g(x)| \le C|g'(x)|.|h|$ for a uniform constant $C > 0$, let $u, v \in [0, N^{-a}]$, and $i \ne j$, we have:

$$ |g(y_i + u - y_j - v) - g(y_i - y_j)| \le C|u - v| |g'(y_i - y_j)| \le C.N^{-a}|g'(N^{-b})| \stackrel{\text{(def)}}{=} A_N. $$

Above, we used that $|y_i - y_j| > N^{-b}$ for all $i \ne j$. The RHS is a $O(N^{-2})$ in the log case and $O(N^{\frac{2s}{1+s}})$ in the Riesz case. Now summing over $i \ne j$ and integrating against $dudv$, we obtain:

$$ -\frac{\beta_N}{2} \sum_{i \ne j} g(y_i - y_j) \le CN^2 \beta_N A_N - \frac{\beta_N}{2} \sum_{i \ne j} \iint_{[0, N^{-a}]} g(y_i - y_j + u - v) d\lambda_{N-a}(u) d\lambda_{N-a}(v) \\ \le CN^2 \beta_N A_N - \frac{\beta_N N^2}{2} \iint_{\mathbb{R}^2} g(x - y) d\tilde{\mu}_N(x) d\tilde{\mu}_N(y) + B_N \quad (20) $$

where

$$ B_N \stackrel{\text{(def)}}{=} N \frac{\beta_N}{2} \iint_{[0, N^{-a}]} g(u - v) d\lambda_{N-a}(u) d\lambda_{N-a}(v) $$
is a $O(\log N)$ in the log case and $O(N^{\frac{2s}{1+s}})$ in the Riesz case, in particular $NA_N = O(B_N)$. Recalling the definition of $\mathcal{E}$ (3), we obtain:

$$ \log p_N(\underline{x}) \leq -N(\mathcal{E}(\tilde{\mu}_N) - \mathcal{E}(\mu_{\text{eq}})) + N \int_{\mathbb{R}} [V(x) + \log \rho_{\text{eq}}(x)] d\tilde{\mu}_N(x) - \sum_{i=1}^{N} V(x_i) + O(B_N) $$

Finally we use the equation satisfied by $\rho_{\text{eq}}$:

$$ \log \rho_{\text{eq}}(x) = -V(x) - 2PU^{\rho_{\text{eq}}}(x) + C_{\text{eq}} $$

and the fact that $\mathcal{E}(\tilde{\mu}_N) - \mathcal{E}(\mu_{\text{eq}}) > 0$ to deduce that

$$ \begin{aligned} \log p_N(\underline{x}) &\leq -N^2 \beta_N \iint_{\mathbb{R}^2} g(x-y)d(\tilde{\mu}_N(x) - \hat{\mu}_N)(x)d\mu_{\text{eq}}(y) + \sum_{i=1}^N \log \rho_{\text{eq}}(x_i) + O(B_N) \\ &\leq N^2 \beta_N \int_{\mathbb{R}} U^{\rho_{\text{eq}}}(x) d(\tilde{\mu}_N(x) - \hat{\mu}_N)(x) + \sum_{i=1}^N \log \rho_{\text{eq}}(x_i) + O(B_N) \end{aligned} \quad (21) $$

Finally, we use Lemma 2.3 to deduce that

$$ \begin{aligned} \left| N \int_{\mathbb{R}} U^{\rho_{\text{eq}}}(x) d(\tilde{\mu}_N(x) - \hat{\mu}_N)(x) \right| &\leq \left\| \frac{d}{dx} U^{\rho_{\text{eq}}} \right\|_{\infty} \left( \sum_{i=1}^{N} |y_i - x_i| + N \int_{0}^{N^{-a}} u d\lambda_{N^{-a}}(u) \right) \\ &\leq \left\| \frac{d}{dx} U^{\rho_{\text{eq}}} \right\|_{\infty} O \left( \left[ N^{\frac{2s}{1+s}} + N^{\frac{-1+s}{1+s}} \right] \mathbf{1}_{g=|\cdot|^{-s}} + [N^{-1} + N^{-4}] \mathbf{1}_{g=-\log} \right) \end{aligned} \quad (22) $$

where we use that $|x_i - y_i| \geq (i-1)N^{-b}$. These estimations lead to the following bound

$$ p_N(\underline{x}) \leq e^{O(B_N)} \prod_{i=1}^{N} \rho_{\text{eq}}(x_i). $$

This concludes the proof.

<figure><img src="image_5.png" /></figure>

The following proposition deduces from Lemma 2.4 an upper bound on left deviations.

**Proposition 2.5 (Upper bound: left-deviation)** Set $x_0 = 0$ for $g = -\log$, and $x_0 = \frac{2s}{1+s}$ for $g = |\cdot|^{-s}$. For all $x_0 < x < 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \log \frac{1}{\mathbb{P} \left( x_{\max} \le \left( (1-x) \log N \right)^{1/\alpha} \right)} \ge x. \quad (23) $$

Furthermore, for $g = -\log$ and for all $\gamma$ such that $-\alpha^{-1} < \gamma < 1 - \alpha^{-1}$, we have for all $x > 0$:

$$ \limsup_{N \to \infty} \frac{1}{(\log N)^{1-\gamma-\alpha^{-1}}} \log \log \frac{1}{\mathbb{P}_N \left( x_{\max} \le (\log N)^{1/\alpha} - \frac{x}{(\log N)^\gamma} \right)} \ge \frac{x}{\alpha} \quad (24) $$

**Proof** For (23), by direct computation, it is not hard to see that, as $N$ goes to infinity:

$$ \mu_{\text{eq}}^{\otimes N} \left( x_{\max} \le \left( (1-x) \log N \right)^{1/\alpha} \right) \le \exp(-N^x + o(N^x)). $$
Thus, using Lemma 2.4, for the log-case, we obtain:

$$ \log \mathbb{P}_N \left( x_{\max} \le \left( (1-x) \log N \right)^{1/\alpha} \right) \le -N^x + o(N^x) + O(\log N) $$

which allows us to conclude. Similarly, for the Riesz-case, we obtain

$$ \log \mathbb{P}_N \left( x_{\max} \le \left( (1-x) \log N \right)^{1/\alpha} \right) \le -N^x + o(N^x) + O(N^{\frac{2s}{1+s}}). $$

Thus for any $x > \frac{2s}{1+s}$, we can deduce the desired bound. (24) is obtained in the exact same way. □

In particular, we have the immediate following corollary which gives the upper bound on left-deviations.

**Corollary 2.6** For all $0 < x < 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( \frac{x_{\max}}{(\kappa \log N)^{1/\alpha}} \le (1-x)^{1/\alpha} \right) = -I_{\alpha}(x). $$

## 2.2 Lower bound

As shown by [Lam21, Theorems 1.7, 1.8], and in the case of the potential $V(x) = |x|^\alpha$, the maximum of the configuration fluctuates around $E_N \sim (\log N)^{1/\alpha}$, both in the Riesz and the log cases. The edge $E_N$ is identified in the Riesz case by [Lam21, Theorem 3.2] through

$$ \frac{Ne^{-V(E_N)}}{e^{\lambda_{\text{eq}}V'(E_N)}} \xrightarrow{N \to \infty} 1, \qquad (25) $$

where $\lambda_{\text{eq}}$ is the constant of (4). In the log-gas case, the equation for $E_N$ is similar and given by [Lam21, Theorem 3.4]

$$ \frac{Ne^{-V(E_N)+\beta N \log E_N}}{e^{\lambda_{\text{eq}}V'(E_N)}} \xrightarrow{N \to \infty} 1. \qquad (26) $$

More precisely, recalling Assumptions 1.1, as a direct application of [Lam21, Theorems 3.2, 3.4], the following convergence towards a Poisson point process (PPP) at the edge holds. We will deduce that the distribution of the largest particle converges towards a Gumbel law.

**Lemma 2.7 (PPP at the edge)** Assume that $(E_N)_N$ satisfies respectively (25) in the Riesz case ($g = |\cdot|^{-s}$), and (26) in the log gas case ($g = -\log$). Then,

$$ E_N \xrightarrow{N \to \infty} (\log N)^{1/\alpha}, \qquad (27) $$

and the sequence of random measures $(\Xi_N)_N$ defined by

$$ \Xi_N \stackrel{(\text{def})}{=} \sum_{j=1}^{N} \delta_{V'(E_N)(x_j - E_N)} $$

converges towards an inhomogeneous PPP with intensity $\theta(x) = e^{-x}$, meaning that for all Borel set $A \subset \mathbb{R}$ such that $\int_A e^{-x} dx < \infty$, the random variable $\Xi_N(A)$ converges towards a Poisson random variable with mean $\int_A e^{-x} dx$.

**Proof** The only point that needs to be checked is the asymptotic (27) of $E_N$. It is established for the log gas in [DGM24, Theorem 4.1], and the proof for the Riesz gas is identical. □
As a consequence, we have:

$$ \mathbb{P}_N (\forall j, x_j < E_N) = \mathbb{P}_N (\Xi_N(0, \infty) = 0) \underset{N \to \infty}{\longrightarrow} e^{-1} > 0. \quad (28) $$

**Proposition 2.8 (Lower bound)** Let $x \in \mathbb{R}$, the following bound holds:

$$ \lim_{\delta \downarrow 0} \liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \in (x - \delta, x + \delta) \right) \geq -I_{\alpha}(x). $$

**Proof** When $x < 1$, $I_{\alpha}(x) = -\infty$ and there is nothing to prove, while for $x = 1$, since $\frac{1}{(\log N)^{1-1/\alpha}}$ is the size of the fluctuations of $x_{\max}$ around $(\log N)^{1/\alpha}$, we have:

$$ \frac{1}{\log N} \log \mathbb{P}_N \left( \left| \frac{x_{\max}}{(\log N)^{1/\alpha}} - 1 \right| < \delta \right) \geq \frac{1}{\log N} \log \mathbb{P}_N \left( \left| \frac{x_{\max}}{(\log N)^{1/\alpha}} - 1 \right| < \frac{1}{\log N} \right) \underset{N \to \infty}{\longrightarrow} 0 $$

and we conclude by noticing that $-I_{\alpha}(1) = 0$. Let $x > 1$ and $\delta > 0$ small enough. We begin by the case $x > 1$. To show that the above inequality is true, we restrict ourselves to the configurations such that there is exactly one particle abnormally far away from the finite $N$ expected support – namely $[-E_N, E_N]$ – and the $N-1$ remaining particles are located in this expected support. Let

$$ I_N \stackrel{(\text{def})}{=} (-\infty, E_N]. $$

By noticing that

$$ \bigcup_{i=1}^{N} \left\{ \left| \frac{x_i}{(\log N)^{1/\alpha}} - x \right| < \delta, \forall j \neq i, x_j \in I_N \right\} \subset \left\{ \left| \frac{x_{\max}}{(\log N)^{1/\alpha}} - x \right| < \delta \right\}, $$

by Lemma 2.7, we have by exchangeability

$$ \begin{aligned} \mathbb{P}_N \left( \left| \frac{x_{\max}}{(\log N)^{1/\alpha}} - x \right| < \delta \right) &\ge N \mathbb{P}_N \left( \left| \frac{x_1}{(\log N)^{1/\alpha}} - x \right| < \delta, \forall j > 1, x_j \in I_N \right) \\ &= N \frac{\mathcal{Z}_{N-1}^{\beta_N}}{\mathcal{Z}_N^{\beta_N}} \int_{(x-\delta)(\log N)^{1/\alpha}}^{(x+\delta)(\log N)^{1/\alpha}} e^{-V(x_1)} \int_{I_N^{N-1}} e^{-\beta_N \sum_{i=2}^N g(x_1-x_i)} d\mathbb{P}_{N-1}^{\beta_N}(x_2, \dots, x_N) dx_1 \\ &= 2\delta (\log N)^{1/\alpha} \mathbb{P}_{N-1}^{\beta_N} (\forall j \ge 2, x_j \in I_N) N \frac{\mathcal{Z}_{N-1}^{\beta_N}}{\mathcal{Z}_N^{\beta_N}} \\ &\quad \times \int_{x-\delta}^{x+\delta} e^{-V(s(\log N)^{1/\alpha})} \int_{I_N^{N-1}} (\log N)^{(N-1)\beta_N/\alpha} e^{-\beta_N \sum_{i=2}^N g\left(s-\frac{x_i}{(\log N)^{1/\alpha}}\right)} d\mu(s, x_2, \dots, x_N), \end{aligned} $$

where the probability measure $\mu$ is supported on $[x - \delta, x + \delta] \times I_N^{N-1}$ defined by

$$ d\mu(s, \dots, x_N) \stackrel{(\text{def})}{=} \mathbf{1}_{|s-x|<\delta} \frac{ds}{2\delta} \otimes \mathbf{1}_{\cap_j x_j \in I_N} \frac{d\mathbb{P}_{N-1}^{\beta_N}(x_2, \dots, x_N)}{\mathbb{P}_{N-1}^{\beta_N}(\forall j \ge 2, x_j \in I_N)}. $$

By [Lam21, Corollary 2.4], it holds that $\frac{\mathcal{Z}_{N-1}^{\beta_N}}{\mathcal{Z}_N^{\beta_N}} \underset{N \to \infty}{\longrightarrow} C$ for some $C > 0$ and by (28) there exists $p > 0$ such that

$$ \mathbb{P}_{N-1}^{\beta_N} (\forall i \in [2, N], x_i \in I_N) \underset{N \to \infty}{\longrightarrow} p. $$
Taking the logarithm and applying Jensen's inequality, we obtain:

$$
\begin{align*}
\log \mathbb{P}_N \left( \left| \frac{x_{\max}}{(\log N)^{1/\alpha}} - x \right| < \delta \right) &\ge \log N + o(\log N) \\
&\quad + \int_{\mathbb{R}^N} \left[ -s^{\alpha} \log N - \varphi(s(\log N)^{1/\alpha}) - \beta_N \sum_{i=2}^N g \left( s - \frac{x_i}{(\log N)^{1/\alpha}} \right) \right] d\mu(s, x_2, \dots, x_N) \\
&\ge \log N + o(\log N) \\
&\quad - \log N (x + \delta)^{\alpha} - \int_{x-\delta}^{x+\delta} \varphi(s(\log N)^{1/\alpha}) \frac{ds}{2\delta} - \int_{\mathbb{R}^N} \beta_N \sum_{i=2}^N g \left( s - \frac{x_i}{(\log N)^{1/\alpha}} \right) d\mu(s, x_2, \dots, x_N).
\end{align*}
$$

Since by Assumptions 1.1 we have for $\delta > 0$ fixed,

$$
\frac{1}{2\delta} \int_{x-\delta}^{x+\delta} \varphi(s(\log N)^{1/\alpha})ds = o(\log N),
$$

we thus have

$$
\frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \in (x-\delta, x+\delta) \right) \ge -I_{\alpha}(x) + o(1) \\
\frac{-1}{\log N} \int_{[x-\delta, x+\delta] \times I_N^{N-1}} \beta_N \sum_{i=2}^{N} g \left( s - \frac{x_i}{(\log N)^{1/\alpha}} \right) d\mu(s, x_2, \dots, x_N). \quad \square
$$

For $\delta$ small enough, it holds $x - 1 - 2\delta > 0$, and for $N$ big enough $\frac{E_N}{(\log N)^{1/\alpha}} < 1 + \delta$. Hence for all $s \in (x - \delta, x + \delta)$, $y \in I_N$ we have $|s - y/(\log N)^{1/\alpha}| \ge x - 1 - 2\delta$, therefore

$$
\int_{[x, x+\delta] \times I_N^{N-1}} \beta_N \sum_{i=2}^{N} \log \left| s - \frac{x_i}{(\log N)^{1/\alpha}} \right| d\mu(s, x_2, \dots, x_N) \ge -(N-1)\beta_N g(x-1-2\delta).
$$

We conclude that

$$
\liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}_N \left( \frac{x_{\max}}{(\log N)^{1/\alpha}} \in (x-\delta, x+\delta) \right) \geq 1 - (x+\delta)^{\alpha},
$$

and letting $\delta \to 0$ yields the conclusion.

3 Matrix interpretation: proof of Theorem 1.8

In this section, we consider random matrix models of symmetric tridiagonal matrices with indepen-
dent entries having Gaussian tails, and establish the LDP for the top eigenvalue Theorem (1.8).

Recall we consider models

$$
T = \begin{pmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \\
& b_2 & \ddots & \ddots & \\
& & \ddots & b_{N-1} & \\
& & & b_{N-1} & a_N
\end{pmatrix}, \quad
T_{\text{per}} = \begin{pmatrix}
a_1 & b_1 & & & b_N \\
b_1 & a_2 & b_2 & & \\
& b_2 & \ddots & \ddots & \\
& & \ddots & \ddots & b_{N-1} \\
b_N & & & b_{N-1} & a_N
\end{pmatrix}.
$$
We recall here for convenience the rate function

$$I_2(x) = \begin{cases} x^2 - 1 & \text{if } x \ge 1 \\ +\infty & \text{if } x < 1 \end{cases} \qquad (29)$$

**Theorem 3.1** Let $(a_i)_{i \ge 1}$, $(b_i)_{i \ge 1}$ be independent sequences of independent variables. Assume that $(a_i)_{i \ge 1}$ and $(\sqrt{2}b_i)_{i \ge 1}$ have uniform Gaussian tail as in Definition 1.7. Consider the tridiagonal or periodic tridiagonal model

$$J = T(a_1, \dots, a_N, b_1, \dots, b_{N-1}) \quad \text{or} \quad J = T_{\text{per}}(a_1, \dots, a_N, b_1, \dots, b_N).$$

Then, $\mu_N \stackrel{(\text{def})}{=} \lambda_{\max}(J)/\sqrt{2\log N}$ satisfies a LDP with rate function $I_2$ (29).

**Remark 3.2** The upper bound argument is ultimately based on the fact that if $B$ is a given symmetric (periodic or not) tridiagonal matrix of size $d$, then (with $B_{d,d+1} = B_{1,d}$)

$$\lambda_{\max}(B) \le (\text{tr}(B^2))^{1/2} = \left( \sum_{i=1}^{d} B_{i,i}^2 + 2B_{i,i+1}^2 \right)^{1/2},$$

and if the entries of $B$ are so that:

1. The diagonal entries $(B_{i,i})_i$ have uniform Gaussian tails

2. The off-diagonal entries $B_{i,i+1}$ are so that $(\sqrt{2}B_{i,i+1})_i$ has quadratic, sub-Gaussian (or exactly Gaussian) tails,

then this latter bound has Gaussian tail by Lemma 3.5.

The following lemma will allow us to deduce the LD lower bound, and to control the spectrum of a matrix with small entries. In the following, we will take the indices of the entries modulo $N$. In the nonperiodic model $T$, we consider $b_N = 0$.

**Lemma 3.3** Let $J = T$ or $J = T_{\text{per}}$, for arbitrary $(a_i)_{1 \le i \le N}$, $(b_i)_{1 \le i \le N}$. Let

$$\rho(J) = \max_{\lambda \in \text{Sp}(J)} |\lambda|$$

be the spectral radius of $J$. Then,

$$\max_i (a_i) \le \lambda_{\max}(J) \le \rho(J) \le \max_i (|a_i| + |b_{i-1}| + |b_i|).$$

**Proof** For the lower bound, write $J = PDP^\top$ with $P$ orthonormal and $D = \text{diag}(\lambda_1, \dots, \lambda_N)$.

Then, for any $1 \le i \le N$, it holds that $a_i = J_{ii} = \sum_{j=1}^N p_{i,j}^2 \lambda_j \le \lambda_{\max}$.

For the upper bound on $\rho(J)$, recall that the spectrum of any $N \times N$ matrix is contained in the union of the Gerschgorin disks:

$$\text{Sp}(J) \subset \bigcup_{i=1}^{N} \bar{B} \left( J_{ii}, \sum_{j \ne i} |J_{ij}| \right),$$
where $\bar{B}(x, \delta)$ is the closed ball centered at $x$ with radius $\delta$. Using the tridiagonal structure of $J$, we deduce that for some $1 \le i \le N$,

$$|\lambda_{\max}| \le |\lambda_{\max} - a_i| + |a_i| \le |b_{i-1}| + |b_i| + |a_i|. \qquad (30)$$

The same bound (30) also holds for some $i$ for $|\lambda_{\min}|$, and because $\rho(J) = \max\{|\lambda_{\min}|, |\lambda_{\max}|\}$, we conclude on the second inequality. □

**Remark 3.4** We observe that Lemma 3.3 allows us to conclude on the LD lower bound for $x \ge 1$ in Theorem 3.1. Indeed, by Lemma 3.3, we have for $x \ge 1$

$$\liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( \max_{1 \le i \le N} a_i \ge x \sqrt{2 \log N} \right) \le \liminf_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( \lambda_{\max} \ge x \sqrt{2 \log N} \right), \quad (31)$$

and since the $a_i$'s are independent with uniform Gaussian tail, one establishes that the left hand side is $-I_2(x)$. Furthermore, note that for $x < 1$,

$$\limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( \lambda_{\max} \le x \sqrt{2 \log N} \right) \le \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( \max_{1 \le i \le N} a_i \le x \sqrt{2 \log N} \right). \quad (32)$$

As before, the right hand side is $-\infty$, and we conclude on the left LD for the top eigenvalue.

We will use the following result for the $L^2$ norm of independent variables with Gaussian tails.

**Lemma 3.5 (Gaussian tail estimate)** Let $(X_i)_{i \ge 1}$ be a sequence of independent variables with uniform Gaussian tails. Let $d \ge 1$ and $\mathcal{I}_d$ be the set of subsets $I \subset [1, +\infty]$ such that $\text{Card}(I) \le d$. For all $I \in \mathcal{I}_d$, let

$$Z_I = \left( \sum_{i \in I} X_i^2 \right)^{1/2}$$

be the euclidean norm of $(X_i)_{i \in I}$. Then, the family $(Z_I)_{I \in \mathcal{I}_d}$ has uniform Gaussian tail.

**Proof (of Lemma 3.5)** We start with the case $d=2$. Let $i_1, i_2 \in I$ and $X := X_{i_1}, Y := X_{i_2}$. We want to show that the variable $Z := (X^2 + Y^2)^{1/2}$ satisfies as $x \to +\infty$:

$$-\log \mathbb{P}(Z > x) = x^2/2 + o(x^2),$$

where the remainder is independent of $(i_1, i_2) \in I^2$. Let $n \ge 1$ and $x > 0$, we decompose

$$\mathbb{P}(|X| > x) \le \mathbb{P}(Z^2 > x^2) \le \mathbb{P}(|X| > x) + \sum_{k=0}^{n-1} \mathbb{P} \left( X^2 \in \left[ \frac{k}{n}x^2, \frac{k+1}{n}x^2 \right], Y^2 > x^2 \left( 1 - \frac{k+1}{n} \right) \right) \\ \stackrel{(\text{def})}{=} \mathbb{P}(|X| > x) + S_n.$$

By assumption, $\log \mathbb{P}(|X| > x) = -x^2/2 + o(x^2)$, where the remainder is independent of $i_1 \in I$. We now establish that $S_n \le e^{-x^2/2+o(x^2)}$ in the previous decomposition, with $o(x^2)$ independent of $i_1, i_2$. We have

$$S_n \le \sum_{k=0}^{n-1} \mathbb{P}(X^2 \ge kx^2/n, Y^2 \ge x^2(1-(k+1)/n)) = \sum_{k=0}^{n-1} \mathbb{P}(X^2 \ge kx^2/n) \mathbb{P}(Y^2 \ge x^2(1-(k+1)/n)) \\ \le n \max_{0 \le k \le n-1} \mathbb{P}(X^2 \ge kx^2/n) \mathbb{P}(Y^2 \ge x^2(1-(k+1)/n)).$$
Set $n = \lfloor x \rfloor$. We have by assumption for $0 \le k \le n - 1 = \lfloor x \rfloor - 1$, and using that $x/\lfloor x \rfloor \le 1$,

$$ 
\begin{aligned}
-\log \mathbb{P}(X^2 \ge kx^2/n) - \log \mathbb{P}(Y^2 \ge x^2(1-(k+1)/n)) &= \frac{kx}{2} + o(kx) + \frac{x^2 - x(k+1)}{2} + o(x^2 - kx) \\
&= \frac{x^2}{2} + o(x^2),
\end{aligned}
$$

where we used that $k \le \lfloor x \rfloor$, thus both terms $o(kx)$ and $o(x^2 - kx)$ are bounded by some $o(x^2)$ independent of $k$. Thus, $-\log S_n \le -\log \lfloor x \rfloor + x^2/2 + o(x^2) = x^2/2 + o(x^2)$. Therefore, adding up both estimates we have

$$ \mathbb{P}(|X| > x) = e^{-x^2/2+o(x^2)} \le \mathbb{P}(Z > x) \le e^{-x^2/2+o(x^2)} + e^{-x^2/2+o(x^2)} = e^{-x^2/2+o(x^2)}, $$

where the terms $o(x^2)$ are independent of the indices $i_1, i_2$. This establishes the case $d = 2$. In the general case, $Z^2 = (X_{i_1}^2 + \dots + X_{i_{d-1}}^2) + X_{i_d}^2$. By induction, $X \stackrel{(\text{def})}{=} (X_{i_1}^2 + \dots + X_{i_{d-1}}^2)^{1/2}$ has Gaussian tail, uniformly in $(i_1, \dots, i_{d-1}) \in I^{d-1}$, and $Y \stackrel{(\text{def})}{=} X_{i_d}^2$ has Gaussian tail uniformly in $i_d \in I$. We conclude by the case $d = 2$. $\square$

For $x \ge 1$, the upper bound on $\lambda_{\max}$ in Lemma 3.3 is too crude to capture the LD upper bound. Indeed, if $X, Y$ are standard Gaussians, then $|X| + |Y|$ doesn't have Gaussian tail. Instead, we will show that we can assume, at the scale of LD, that only a few entries of $J$ matter. Then we will be able to give a precise LD upper bound.

For $\varepsilon > 0$, and $J = T$ or $J = T_{\text{per}}$, we decompose $J_N = J_N^{\varepsilon} + S_N^{\varepsilon}$, where

$$ J_N^{\varepsilon} \stackrel{(\text{def})}{=} (J_{i,j} \mathbf{1}_{\{|J_{i,j}| \ge \varepsilon \sqrt{2 \log N}\}})_{1 \le i,j \le N} \quad (33) $$

is the matrix where we cut the entries smaller than $\varepsilon\sqrt{2 \log N}$, and

$$ S_N^{\varepsilon} \stackrel{(\text{def})}{=} J_N - J_N^{\varepsilon} = (J_{i,j} \mathbf{1}_{\{|J_{i,j}| < \varepsilon \sqrt{2 \log N}\}})_{1 \le i,j \le N}. $$

We have the following exponential approximation for the top eigenvalue of $J_N$.

**Lemma 3.6** Let $\mu_N \stackrel{(\text{def})}{=} \lambda_{\max}(J_N)/\sqrt{2 \log N}$, and $\mu_N^{(\varepsilon)} \stackrel{(\text{def})}{=} \lambda_{\max}(J_N^{\varepsilon})/\sqrt{2 \log N}$. Then, for all $\delta > 0$, we have

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} \left( |\mu_N - \mu_N^{(\varepsilon)}| > \delta \right) \xrightarrow{\varepsilon \to 0} -\infty. $$

**Proof** Using that for real symmetric matrices one has

$$ \lambda_{\max}(A) = \max_{\|x\|=1} \langle Ax, x \rangle, $$

one sees that $|\lambda_{\max}(J) - \lambda_{\max}(J^{\varepsilon})| \le \rho(S^{\varepsilon})$. Now, by Lemma 3.3, we have

$$ \rho(S_N^{\varepsilon}) \le \max_{1 \le i \le N} \{|S_{i,i}^{\varepsilon}| + |S_{i,i-1}^{\varepsilon}| + |S_{i-1,i}^{\varepsilon}|\} \le 3\varepsilon\sqrt{2 \log N}. $$

Now, for any $\delta > 0$, we deduce that for $\varepsilon < \delta/3$

$$ \mathbb{P} \left( |\mu_N - \mu_N^{(\varepsilon)}| > \delta \right) = 0, $$

and the result follows by taking the log. $\square$
<figure><img src="image_6.png" /><figcaption>Figure 1: Example where $b_2 = b_3 = b_6 = 0$. $a_5 = 0$ but it doesn't change the structure of the block.</figcaption></figure>

Let us now note that for a tridiagonal matrix $J = T(\mathbf{a}, \mathbf{b})$ having vanishing diagonal entries $b_{i_1}, \dots, b_{i_k} = 0$, we can decompose $J$ into sub blocks. As an example, More generally, we define the block decomposition

**Definition 3.7 (Block decomposition)** Let $J = T(\mathbf{a}, \mathbf{b})$ be a symmetric tridiagonal matrix. Assume that $J$ has exactly $k$ vanishing off diagonal entries:

$$b_{i_1}, b_{i_2}, \dots, b_{i_k} = 0 \quad \text{and} \quad b_i \neq 0 \text{ otherwise.}$$

We note the block decomposition $J = \text{diag}(B^{(1)}, \dots, B^{(k+1)})$, where $B^{(\ell)} = (J_{i,j})_{i_\ell-1<i,j\le i_\ell}$, where we set $i_0 = 1$, $i_{k+1} = N$.

Furthermore, we denote $d_\ell = i_\ell - i_{\ell-1}$ the dimension of $B_\ell$, and

$$d_{\max}(J) = \max_{1 \le \ell \le k+1} d_\ell = \max_{1 \le \ell \le k+1} \{i_\ell - i_{\ell-1}\}. \quad (34)$$

**Remark 3.8** Notice that we can make use of a block decomposition also in the case of a periodic tridiagonal matrix, namely $J = T_{\text{per}}(\mathbf{a}, \mathbf{b})$, after a change of basis, upon supposing that at least one off diagonal entry vanishes. Indeed, if say $b_{i^*} = 0$, then denoting $(e_1, \dots, e_N)$ the canonical basis of $\mathbb{R}^N$, representing $J$ in the shifted basis $(e_{i^*}, e_{i^*+1}, \dots, e_N, e_1, \dots, e_{i^*-1})$ we have

$$J = UT(\mathbf{a}', \mathbf{b}')U^T$$

where $U$ is the change of basis matrix and $\mathbf{a}', \mathbf{b}'$ are obtained from $\mathbf{a}$ and $\mathbf{b}$ by shifting the coordinates by $i^*$ modulo $N$. Now, for any $\varepsilon > 0$,

$$\log \mathbb{P}(\forall i, |b_i| > \varepsilon \sqrt{2 \log N}) = \sum_{i=1}^{N} \log \mathbb{P}(|b_i| > \varepsilon \sqrt{2 \log N}) \underset{N \to +\infty}{\sim} -N\varepsilon^2 \log N,$$

where we used the Gaussian tail of the entries. Thus, we see that

$$\frac{1}{\log N} \log \mathbb{P}(\forall i, |b_i| > \varepsilon \sqrt{2 \log N}) \underset{N \to +\infty}{\longrightarrow} -\infty,$$

and we deduce that at the scale of LD, we can assume that at least one off diagonal entry of $T_{\text{per}}^{\varepsilon}(\mathbf{a}, \mathbf{b})$ vanishes. By the previous change of basis argument, we reduce the problem of the LD of $\lambda_{\max}$ to the non periodic case.
In view of Remark 3.8, for the rest of the section, we assume that we are in the non periodic case $J = T(\mathbf{a}, \mathbf{b})$.

We will now show that in the regime of LD, we can assume that $d_{\max}(J_N^\varepsilon)$ is finite.

**Lemma 3.9** Let $\varepsilon > 0$, and let $J_N^\varepsilon$ given by (33). Then,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (d_{\max}(J_N^\varepsilon) \ge d) \underset{d \to +\infty}{\longrightarrow} -\infty. $$

**Proof** Let $d \ge 1$, the size of the blocks in the block decomposition of $J_N^\varepsilon$ is determined by the vanishing off-diagonal entries: on the event $\{d_{\max}(J_N^\varepsilon) \ge d\}$, there is at least some $1 \le i \le N-d+1$ such that $|b_i|, |b_{i+1}|, \dots, |b_{i+d-1}| \ge \varepsilon\sqrt{2\log N}$. Thus by a union bound on the starting position $i$ of such a sequence, independence and uniform Gaussian tail, we can assume without loss of generality that $i=1$, and

$$ \begin{aligned} \log \mathbb{P} (d_{\max}(J_N^\varepsilon) \ge d) &\le \log N + \sum_{k=1}^d \log \mathbb{P} (|b_k| > \varepsilon\sqrt{2\log N}) \\ &\le \log N + \sum_{k=1}^d (-\varepsilon^2 \log N + o(\log N)) = \log N - d\varepsilon^2 \log N + o(\log N). \end{aligned} $$

Dividing by $\log N$ and letting $d \to +\infty$, we deduce the claim. $\square$

We now have everything to prove Theorem 3.1.

**Proof (Proof of Theorem 3.1)** Recall that by the discussion preceding Lemma 3.9, we can reduce the proof to the nonperiodic tridiagonal case $J = T(\mathbf{a}, \mathbf{b})$. Recall also that the LD lower bound for $x \ge 1$ was already established in (31).

We now establish the LD upper bound for $x \ge 1$. By the exponential approximation of Lemma 3.6, recalling the notation $\mu_N = \lambda_{\max}/\sqrt{2\log N}$, we have

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N > x) \le \lim_{\delta \to 0} \liminf_{\varepsilon \to 0} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x - \delta). \quad (35) $$

Now for fixed $\varepsilon > 0$, we have for any $d \ge 1$, recalling the notation $d_{\max}$ (34),

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x) = \max \left\{ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x, d_{\max}(J_N^\varepsilon) \le d) \ ; \ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x, d_{\max}(J_N^\varepsilon) > d) \right\}. $$

Using Lemma 3.9, taking $d \to +\infty$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x) = \limsup_{d \to +\infty} \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P} (\mu_N^\varepsilon > x, d_{\max}(J_N^\varepsilon) \le d). \quad (36) $$

Now, if $\lambda \in \text{sp}(J_N^\varepsilon)$, there is some $1 \le \ell \le k_N^\varepsilon$ such that $\lambda \in \text{sp}(B_\ell)$. Then, on $\{d_{\max}(J_N^\varepsilon) \le d\}$,

$$ \begin{aligned} |\lambda| \le \text{tr}(B_\ell^2)^{1/2} &= \left( \sum_{i=1}^{d_\ell} B_\ell(i, i)^2 + 2B_\ell(i, i+1)^2 \right)^{1/2} \\ &= \left( \sum_{i=1}^{d_\ell} J_N^\varepsilon(i_{\ell-1} + i, i_{\ell-1} + i)^2 + 2J_N^\varepsilon(i_{\ell-1} + i, i_{\ell-1} + i+1)^2 \right)^{1/2} \\ &\le \left( \sum_{i=1}^d J_N(i_{\ell-1} + i, i_{\ell-1} + i)^2 + 2J_N(i_{\ell-1} + i, i_{\ell-1} + i+1)^2 \right)^{1/2} \stackrel{\text{(def)}}{=} Z_\ell^{(d)}, \end{aligned} $$
where we used that $d_{\ell} \le d$, and that $|J_{N}^{\epsilon}(i, j)| \le |J_{N}(i, j)|$. Thus

$$ \mathbb{P}(\mu_{N}^{\epsilon} > x, d_{\max}(J_{N}^{\epsilon}) \le d) \le \mathbb{P}\left(\max_{1 \le \ell \le k_{N}^{\epsilon}} Z_{\ell}^{(d)} > x\sqrt{2\log N}\right) \le \mathbb{P}\left(\max_{1 \le \ell \le N} Z_{\ell}^{(d)} > x\sqrt{2\log N}\right), $$

where we used $k_{N}^{\epsilon} \le N$ and for $k_{N}^{\epsilon} + 1 \le \ell \le N$ we took $Z_{\ell}^{(d)}$ standard Gaussian.

Using that $Z_{\ell}^{(d)}$ has Gaussian tail independent of $N$ by the fact that each $J(i, i)$ and $\sqrt{2}J(i, i + 1)$ have Gaussian tail and Lemma 3.5, we conclude by an union bound that for any fixed $d \ge 1$, and for $x \ge 1$,

$$ \limsup_{N \to \infty} \frac{1}{\log N} \log \mathbb{P}(\mu_{N}^{\epsilon} > x, d_{\max} \le d) \le -I_{2}(x). $$

By equations (36) and (35), we conclude for the LD upper bound in the case $x \ge 1$. Finally, the left deviations (i.e. the case $x < 1$) were established by (32), which allows us to conclude on the full LDP. $\square$

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