Interacting particle systems at the edge of multilevel Dyson Brownian motions

TL;DR

This paper investigates the asymptotic behavior of particle spacings at the edge of multilevel Dyson Brownian motions as levels grow large, revealing a decoupling of global interactions and emergence of local interaction systems.

Contribution

It introduces a novel limiting particle system with local interactions derived from multilevel Dyson Brownian motions in the large-level limit.

Findings

Global interactions become negligible at the edge in the limit

Limiting systems are Brownian versions of asymmetric exclusion processes

First instance of local interaction particle systems in general β random matrix models

Abstract

We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with local interactions in the context of general $\beta$ random matrix models.