Convergence of local statistics of Dyson Brownian motion

TL;DR

This paper investigates how quickly the local eigenvalue statistics of Dyson Brownian motion converge to those of GOE/GUE ensembles for short times, under certain regularity conditions on the initial data.

Contribution

It establishes convergence of local statistics for Dyson Brownian motion with deterministic initial data under minimal regularity assumptions, extending previous results.

Findings

Local statistics match GOE/GUE after short time t

Convergence holds when initial density is bounded and away from zero

Results rely on coupling, regularity, and eigenvalue rigidity techniques

Abstract

We analyze the rate of convergence of the local statistics of Dyson Brownian motion to the GOE/GUE for short times $t=o(1)$ with deterministic initial data V . Our main result states that if the density of states of $V$ is bounded both above and away from $0$ down to scales $\ell \ll t$ in a small interval of size $G \gg t$ around an energy $E_0$, then the local statistics coincide with the GOE/GUE near the energy $E_0$ after time $t$. Our methods are partly based on the idea of coupling two Dyson Brownian motions from [6], the parabolic regularity result of [15], and the eigenvalue rigidity results of [21].