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

Abstract

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