Interacting particle systems at the edge of multilevel Dyson Brownian motions

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 β random matrix models. Keywords: Beta random matrix models; Dyson Brownian motions; Interacting particle systems with local interactions; Minors of random matrices; Singular stochastic differential equations