Universality of Wigner random matrices: a Survey of Recent Results

TL;DR

This survey reviews recent advances demonstrating that the local spectral statistics of large Wigner matrices are universal, matching classical Gaussian ensembles, regardless of specific distribution details, under broad conditions.

Contribution

It establishes universality of local eigenvalue statistics for Wigner matrices with minimal assumptions, using a new local relaxation flow approach and extending results to non-identically distributed entries.

Findings

Eigenvalue densities converge to the Wigner semicircle law at small scales

Eigenvectors are fully delocalized in the bulk

Local eigenvalue statistics match those of Gaussian ensembles if moments align

Abstract

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and the Gaussian Symplectic Ensemble (GSE), 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. As a main input, we establish that the density of eigenvalues converges to the Wigner semicircle law and this holds even down to the smallest possible scale, and, moreover, we show that eigenvectors are fully delocalized. These results hold even without the condition that the matrix elements are identically distributed, only independence is used. In fact, we give strong estimates on the matrix elements of the Green function as well that imply that the local statistics of any two ensembles in the bulk are identical if the first four moments of the matrix elements match. Universality at the spectral edges requires matching only two moments. We also prove a Wegner type estimate and that the eigenvalues repel each other on arbitrarily small scales.