Edge Rigidity of Dyson Brownian Motion with General Initial Data

Understand

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.