Large deviations for the largest eigenvalue of Rademacher matrices

Abstract

In this article, we consider random Wigner matrices, that is, symmetric matrices such that the subdiagonal entries of $X_{n}$ are independent, centered and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This is satisfied by the Rademacher law and the uniform law on $[-\sqrt{3},\sqrt{3}]$. We extend our result to complex entries Wigner matrices and Wishart matrices.