Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails

TL;DR

This paper establishes a large deviation principle for the largest eigenvalue of Wigner matrices with non-Gaussian tails, revealing how tail behavior influences eigenvalue fluctuations at a specific exponential scale.

Contribution

It extends large deviation results to Wigner matrices with heavy tails characterized by sub-Gaussian decay, providing a new rate function depending solely on tail distributions.

Findings

Large deviation principle with speed N^{α/2}

Rate function depends only on tail distribution

Applicable to matrices with polynomial decay tails

Abstract

We prove a large deviation principle for the largest eigenvalue of Wigner matrices without Gaussian tails, namely such that the distribution tails $\mathbb{P}( |X_{1,1}|>t)$ and $\mathbb{P}(|X_{1,2}|>t)$ behave like $e^{-bt^{\alpha}}$ and $e^{-at^{\alpha}}$ respectively for some $a,b\in (0,+\infty)$ and $\alpha\in (0,2)$. The large deviation principle is of speed $N^{\alpha/2}$ and with a good rate function depending only on the tail distribution of the entries.