Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian 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 $e^{-b{t}^{\alpha }}$ and $e^{-a{t}^{\alpha }}$ 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.