Edge Statistics for a Class of Repulsive Particle Systems

TL;DR

This paper investigates a class of interacting particle systems that generalize eigenvalue ensembles of Hermitian random matrices, analyzing their edge statistics and transition between universal and non-universal behaviors.

Contribution

It introduces a stochastic linearization method to study non-determinantal ensembles and describes the transition in edge behavior, including moderate deviations, compared to determinantal cases.

Findings

Describes transition from Tracy-Widom to non-universal behavior

Analyzes moderate deviation regimes in detail

Compares effects of averaging in large deviations

Abstract

We study a class of interacting particle systems on $\mathbb{R}$ which was recently investigated by F. G\"otze and the second author [GV14]. These ensembles generalize eigenvalue ensembles of Hermitian random matrices by allowing different interactions between particles. Although these ensembles are not known to be determinantal one can use the stochastic linearization method of [GV14] to represent them as averages of determinantal ones. Our results describe the transition between universal behavior in the regime of the Tracy-Widom law and non-universal behavior for large deviations of the rightmost particle. Moreover, a detailed analysis of the transition that occurs in the regime of moderate deviations, is provided. We also compare our results with the corresponding ones obtained recently for determinantal ensembles [Sch15, EKS15]. In particular, we discuss how the averaging effects the leading order behavior in the regime of large deviations. In the analyis of the averaging procedure we use detailed asymptotic information on the behavior of Christoffel-Darboux kernels that is uniform for perturbative families of weights. Such results have been provided by K. Schubert, K. Sch\"uler and the authors in [KSSV14].