Wegner estimate and level repulsion for Wigner random matrices

TL;DR

This paper proves the semicircle law on optimal scales, establishes a Wegner estimate, and confirms eigenvalue repulsion for Wigner matrices, advancing understanding of their spectral properties.

Contribution

It provides the first proof of the semicircle law on the optimal scale, removes a previous logarithmic factor, and confirms eigenvalue repulsion for Wigner matrices.

Findings

Semicircle law holds on the optimal scale for Wigner matrices.

The averaged density of states is bounded (Wegner estimate).

Eigenvalues exhibit repulsion, supporting universality.

Abstract

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{ESY2}. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.