From random matrices to stochastic operators
The authors introduce ideas allowing to interpret classical random matrix models as finite difference schemes for stochastic differential operators. In random matrix theory, three major types of large \(N\) eigenvalue behaviors are described by sine ensembles, Airy ensembles and Bessel ensembles. In this paper, three particular stochastic operators are studied, each associated with one of the above 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 suggestion for a stochastic sine operator, which would display bulk behavior, associated with the sine kernel.
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