  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.
