Edge scaling limit of Dyson Brownian motion at equilibrium for general $β\geq 1$

Abstract

For general β ľ 1, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit N Ñ 8. For each fixed time, this ensemble is distributed as the Airy β random point field. We prove that the increments of the limiting process are locally Brownian. When β ą 1 we prove that after subtracting a Brownian motion, the sample paths are almost surely locally r-Hölder for any r ă 1 ´p1 `βq ´1. Furthermore for all β ľ 1 we show that the limiting process solves an SDE in a weak sense. When β " 2 this limiting process is the Airy line ensemble.