Fixed energy universality for generalized Wigner matrices

TL;DR

This paper proves the fixed energy universality conjecture for generalized Wigner matrices, establishing that local spectral statistics are universal at a fixed energy level in the bulk spectrum without averaging.

Contribution

It introduces a homogenization theory of Dyson Brownian motion to demonstrate microscopic universality from mesoscopic statistics for generalized Wigner matrices.

Findings

Universal local spectral statistics at fixed energy in the bulk spectrum.

Extension of universality results to fixed energy without averaging.

Development of a homogenization approach for Dyson Brownian motion.

Abstract

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.