From Random Matrices to Stochastic Operators

TL;DR

This paper reinterprets classical random matrix models as finite difference schemes for stochastic differential operators, linking them to specific local eigenvalue behaviors and kernels.

Contribution

It introduces a novel perspective by viewing random matrix models as finite difference schemes for stochastic operators, connecting them to known eigenvalue behaviors.

Findings

Stochastic Airy operator models soft edge behavior with Airy kernel.

Stochastic Bessel operator models hard edge behavior with Bessel kernel.

Proposes a stochastic sine operator to model bulk eigenvalue behavior.

Abstract

We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonly arise, each associated with a familiar class of local eigenvalue behavior. The stochastic Airy operator displays soft edge behavior, associated with the Airy kernel. The stochastic Bessel operator displays hard edge behavior, associated with the Bessel kernel. The article concludes with suggestions for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.