LEVEL-SPACING DISTRIBUTIONS AND THE AIRY KERNEL

Scaling level-spacing distribution functions in the ''bulk of the spectrum'' in random matrix models of N x N hermitian matrices and then going to the limit N --> infinity leads to the Fredholm determinant of the sine kernel sin pi(x - y)/pi(x - y). Similarly a scaling limit at the ''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 kernel of the following properties of the sine kernel: the completely integrable 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 determinant 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.