Universality of local spectral statistics of random matrices

TL;DR

This paper reviews recent advances in proving the universality of local spectral statistics of large random matrices, emphasizing the role of Dyson Brownian motion and extending results beyond classical symmetry classes.

Contribution

It presents a comprehensive solution to the universality conjecture for both invariant and non-invariant ensembles, highlighting the intrinsic mechanism of local ergodicity of Dyson Brownian motion.

Findings

Universality holds for a wide class of random matrices regardless of symmetry.

Dyson Brownian motion's local ergodicity explains spectral universality.

Results include local relaxation times and eigenvector delocalization.

Abstract

The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given by a log-gas with inverse temperature $\beta = 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles. For $\beta \not \in \{1, 2, 4\}$, there is no matrix model behind this model, but the statistical physics interpretation of the log-gas is still valid for all $\beta > 0$. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of $V$. In this article, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian motion is the intrinsic mechanism behind the universality. Furthermore, we review the solution of Dyson's conjecture on the local relaxation time of the Dyson Brownian motion. Related questions such as delocalization of eigenvectors and local version of Wigner's semicircle law will also be discussed.