Bulk universality for generalized Wigner matrices

TL;DR

This paper proves that generalized Wigner matrices exhibit universal eigenvalue spacing statistics in the bulk, matching classical ensembles, under broad conditions, and establishes a local semicircle law for band matrices.

Contribution

It demonstrates bulk universality for generalized Wigner matrices with minimal assumptions and extends results to band matrices with a local semicircle law.

Findings

Eigenvalue spacing in the bulk matches GUE/GOE statistics.

Bulk universality holds under broad variance conditions.

Local semicircle law applies to band matrices with bandwidth M.

Abstract

Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$ with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.