From Random Matrices to Stochastic Operators

Details From Random Matrices to Stochastic Operators From Random Matrices to Stochastic Operators

2006-07-19

arxiv.org/abs/math-ph/0607038v2

Physics

Mathematical Physics

41 pages, 5 figures. Submitted to Journal of Statistical Physics. Changes in this revision: recomputed Monte Carlo simulations

Scientific paper

10.1007/s10955-006-9226-4

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.

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