A large deviation principle for Wigner matrices without Gaussian tails

TL;DR

This paper establishes a large deviation principle for the spectral measure of Wigner matrices with heavy-tailed entries, revealing the structure of deviations in terms of free convolution with explicit rate functions.

Contribution

It introduces a large deviation principle for Wigner matrices with sub-Gaussian tails, extending understanding to heavy-tailed distributions with explicit rate functions.

Findings

Large deviation principle with speed $n^{1+rac{eta}{2}}$ for spectral measures.

Rate function finite only for measures of the form $ ext{semicircle} oxplus u$.

Explicit expressions for the rate function in terms of the $eta$-th moment of $ u$.

Abstract

We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha}}$ for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\sqrt{n}$ with speed $n^{1+\alpha /2}$ with a good rate function $J(\mu)$ that is finite only if $\mu$ is of the form $\mu=\mu_{\mathrm{sc}}\boxplus\nu$ for some probability measure $\nu$ on $\mathbb {R}$, where $\boxplus$ denotes the free convolution and $\mu_{\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(\mu_{\mathrm{sc}}\boxplus\nu)$ in terms of the $\alpha$th moment of $\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.