Universality for generalized Wigner matrices with Bernoulli distribution

TL;DR

This paper extends the universality of eigenvalue spacing statistics for generalized Wigner matrices to include Bernoulli distributions, using a stronger local semicircle law for improved accuracy.

Contribution

It proves universality for Bernoulli measures in generalized Wigner matrices by developing a new local semicircle law with sharper error estimates.

Findings

Universality now includes Bernoulli distributions.

Improved error estimate on the Stieltjes transform.

Extended previous results to a broader class of distributions.

Abstract

The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.