Asymptotic expansion of the expected spectral measure of Wigner matrices

Abstract

We compute an asymptotic expansion with precision 1/n of the moments of the expected empirical spectral measure of Wigner matrices of size n with independent centered entries. We interpret this expansion as the moments of the addition of the semicircle law and 1/n times an explicit signed measured with null total mass. This signed measure depends only on the second and fourth moments of the entries.