Moderate deviations for the eigenvalue counting function of Wigner matrices

Abstract

We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem and applies localization results by Erd\"os, Yau and Yin. Moreover we investigate families of covariance matrices as well.