LEVEL-SPACING DISTRIBUTIONS AND THE AIRY KERNEL

Scaling level-spacing distribution functions in the ''bulk of the spec trum'' in random matrix models of N x N hermitian matrices and then go ing to the limit N --> infinity leads to the Fredholm determinant of t he sine kernel sin pi(x - y)/pi(x - y). Similarly a scaling limit at t he ''edge of the spectrum'' leads to the Airy kernel [Ai(x) Ai(y) - Ai '(x) Ai(y)]/(x - y). In this paper we derive analogues for this Airy k ernel of the following properties of the sine kernel: the completely i ntegrable system of P.D.E.'s found by Jimbo, Miwa, Mori, and Sato; the expression, in the case of a single interval, of the Fredholm determi nant in terms of a Painleve transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues.