Interacting particle systems at the edge of multilevel Dyson Brownian motions

Abstract

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.

Eigenvalues for the Minors of Wigner Matrices

“…In this paper, we study the microscopic scaling limit of the Wigner corner process both near the spectral edge and in the bulk, and prove they are universal. We show: (i) Near the spectral edge, the corner process exhibit a decoupling phenomenon, as first observed in [24]. Individual extreme particles have Tracy-Widom β distribution; the spacings between the extremal particles on adjacent levels converge to independent Gamma distributions in a much smaller scale.…”

“…More general random matrix corner processes with determinantal structure were studied in [3]. The Hermite β corner process as the β analogue of GUE corner process was introduced by V. Gorin and M. Shkolnikov [23], and they showed in [24] the spacings between the extremal particles on adjacent levels converges to independent Gamma distributions. The bulk scaling limit of Hermite β corner process was understood recently.…”

A construction of infinite Brownian particle systems

“…Hereby, Γ(•) is the Gamma function. We refer to [13] for more details and note that in (4.1) time is slowed down by a factor of two compared to the setting there to concur with the normalizations of Assumption 1.1.…”

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

“…In a different direction, the works of Gorin and Shkolnikov have introduced and studied multilevel systems of Dyson Brownian motion [27,28]. That is, they consider for general β, time dependent triangular arrays whose fixed-time distributions are a generalization (in the parameter β) of the GUE corners process (i.e., the eigenvalues of a GUE matrix and its the top left k ˆk corners), and each level of which is a Dyson Brownian motion.…”