Fixed energy universality for Dyson Brownian motion

TL;DR

This paper proves that Dyson Brownian motion exhibits fixed energy universality under certain conditions, extending previous results to a broader class of random matrix ensembles by using homogenization theory and mesoscopic analysis.

Contribution

It establishes fixed energy universality for Dyson Brownian motion with deterministic initial data, generalizing prior results to a wider range of random matrices.

Findings

Eigenvalue statistics match GOE/GUE after short time

Fixed energy universality holds under bounded density conditions

Mesoscopic CLT for linear statistics of Dyson Brownian motion

Abstract

We consider Dyson Brownian motion for classical values of $\beta$ with deterministic initial data $V$. We prove that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time $t \gtrsim 1/N$ if the density of states of $V$ is bounded above and below down to scales $\eta \ll t$ in a window of size $L \gg \sqrt{t}.$ Our results imply that fixed energy universality holds for essentially any random matrix ensemble for which averaged energy universality was previously known. Our methodology builds on the homogenization theory developed in [BEYY] which reduces the microscopic problem to a mesoscopic problem. As an auxiliary result we prove a mesoscopic central limit theorem for linear statistics of various classes of test functions for classical Dyson Brownian motion.