Edge Rigidity of Dyson Brownian Motion with General Initial Data

TL;DR

This paper investigates the edge behavior of Dyson Brownian motion with general initial data, establishing improved rigidity bounds and demonstrating convergence to Tracy-Widom distribution in short time.

Contribution

It provides new edge rigidity results under weaker initial density assumptions and connects these results to Tracy-Widom distribution convergence.

Findings

Edge fluctuations bounded by $( ext{log } n)^{O(1)} n^{-2/3}$ after a short time

Improved rigidity results under minimal initial density assumptions

Convergence of extreme particle distribution to Tracy-Widom $eta$ distribution

Abstract

In this paper, we study the edge behavior of Dyson Brownian motion with general $\beta$. Specifically, we consider the scenario where the averaged initial density near the edge, on the scale $\eta_*$, is lower bounded by a square root profile. Under this assumption, we establish that the fluctuations of extreme particles are bounded by $(\log n)^{{\rm O}(1)}n^{-2/3}$ after time $C\sqrt{\eta_*}$. Our result improves previous edge rigidity results from [1,24] which require both lower and upper bounds of the averaged initial density. Additionally, combining with [24], our rigidity estimates are used to prove that the distribution of extreme particles converges to the Tracy-Widom $\beta$ distribution in short time.