The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct aymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erdös

Abstract. This work is concerned with finite range bounds on the variance of individual eigenvalues of Wigner random matrices, in the bulkTwo different models of random Hermitian matrices were introduced by Wishart in the twenties and by Wigner in the fifties. Wishart was interested in modeling tables of random data in multivariate analysis and worked on random covariance matrices. In this paper, the results for covariance matrices are very close to those for Wigner matrices. Therefore, it deals mainly with Wigner matrices. Definitions and results regarding covariance matrices are available in the last section.Random Wigner matrices were first introduced by Wigner to study eigenvalues of infinite-dimensional operators in statistical physics (see [17]) and then propagated to various fields of mathematics involved in the study of spectra of random matrices. Under suitable symmetry assumptions, the asymptotic properties of the eigenvalues of a random matrix were soon conjectured to be universal, in the sense they do not depend on the individual distribution of the matrix entries. This opened the way to numerous developments on the asymptotics of various statistics of the eigenvalues of random matrices, such as for example the global behavior of the spectrum, the spacings between the eigenvalues in the bulk of the spectrum or the behavior of the extreme eigenvalues. Two main models have been considered, invariant matrices and Wigner matrices. In the invariant matrix models, the matrix law is unitary invariant and the eigenvalue joint distribution can be written explicitly in terms of a given potential. In the Wigner models, the matrix entries are independent (up to symmetry conditions). The case where the entries are Gaussian is the only model belonging to both types. In the latter case, the joint distribution of the eigenvalues is thus explicitly known and the previous statistics have been completely studied.

We investigate the fluctuations of linear spectral statistics of a Wigner matrix W N deformed by a deterministic diagonal perturbation D N , around a deterministic equivalent which can be expressed in terms of the free convolution between a semicircular distribution and the empirical spectral measure of D N . We obtain Gaussian fluctuations for test functions in C 7 c pRq (C 2 c pRq for fluctuations around the mean). Furthermore, we provide as a tool a general method inspired from Shcherbina and Johansson to extend the convergence of the bias if there is a bound on the bias of the trace of the resolvent of a random matrix. Finally, we state and prove an asymptotic infinitesimal freeness result for independent GUE matrices together with a family of deterministic matrices, generalizing the main result from [Shl18].