Edge Rigidity of Dyson Brownian Motion with General Initial Data

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 ${\eta }_{\ast }$, is lower bounded by a square root profile. Under this assumption, we establish that the fluctuations of extreme particles are bounded by $(\log n{)}^{\mathrm{O}(1)}{n}^{-2/3}$ after time $C\sqrt{{\eta }_{\ast }}$. 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.

52 pages, 4 figures