Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \ge 1$

Abstract

For general $\beta \ge 1$, we consider Dyson Brownian motion at equilibrium and prove convergence of the extremal particles to an ensemble of continuous sample paths in the limit $N\to \infty$. For each fixed time, this ensemble is distributed as the Airy${}_{\beta }$ random point field. We prove that the increments of the limiting process are locally Brownian. When $\beta >1$ we prove that after subtracting a Brownian motion, the sample paths are almost surely locally $r$-H{ö}lder for any $r<1-(1+\beta {)}^{-1}$. Furthermore for all $\beta \ge 1$ we show that the limiting process solves an SDE in a weak sense. When $\beta =2$ this limiting process is the Airy line ensemble.

32 Pages. Draft, comments welcome

References in corpus (4)

• From Random Matrices to Stochastic Operators
• Invariant $\beta$-ensembles and the Gauss-Wigner crossover
• Edge statistics of Dyson Brownian motion
• Tridiagonal Models for Dyson Brownian Motion