Universality of Wigner Random Matrices

TL;DR

This paper proves that the local eigenvalue statistics of large Wigner matrices with general distributions match those of classical Gaussian ensembles, demonstrating universality in spectral behavior.

Contribution

It establishes universality of local eigenvalue statistics for Wigner matrices with subexponential decay distributions using a novel local relaxation flow approach.

Findings

Eigenvalue statistics in the bulk match GOE/GUE predictions

Wigner semicircle law holds on minimal scales

Eigenvectors are fully delocalized and eigenvalues repel each other

Abstract

We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the local eigenvalue statistics in the bulk of the spectrum for these matrices coincide with those of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. We also show that the Wigner semicircle law holds locally on the smallest possible scales and we prove that eigenvectors are fully delocalized and eigenvalues repel each other on arbitrarily small scales.