Asymptotic expansion of the expected spectral measure of Wigner matrices

TL;DR

This paper derives a detailed asymptotic expansion for the expected spectral measure of Wigner matrices, revealing how it approximates the semicircle law plus correction terms depending on entry moments.

Contribution

It provides the first explicit asymptotic expansion of the expected spectral measure of Wigner matrices with precision 1/n, linking it to moments of the semicircle law and explicit correction measures.

Findings

Asymptotic expansion of spectral measure moments with 1/n precision

Explicit signed measure correction depending on second and fourth moments

Connection between spectral measure and semicircle law plus correction

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.