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

TL;DR

This paper studies the edge scaling limit of Dyson Brownian motion at equilibrium for all $eta \, \geq \, 1$, showing convergence to a continuous process with Brownian increments and establishing regularity properties.

Contribution

It proves the convergence of extremal particles to a continuous process for general $eta \, \geq \, 1$, extending known results and characterizing the process via an SDE.

Findings

Convergence of extremal particles to a continuous sample path ensemble.

Limiting process has locally Brownian increments.

Sample paths are almost surely locally Hölder continuous for $eta > 1$.

Abstract

For general $\beta \geq 1$, 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{\"o}lder for any $r<1-(1+\beta)^{-1}$. Furthermore for all $\beta \geq 1$ we show that the limiting process solves an SDE in a weak sense. When $\beta=2$ this limiting process is the Airy line ensemble.