Level-spacing Distributions and the Airy Kernel

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models ofN×N hermitian matrices and then going to the limitN→∞ leads to the Fredholm determinant of thesine kernel sinπ(x−y)/π(x−y). Similarly a scaling limit at the “edge of the spectrum” leads to theAiry 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, Môri, and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlevé transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for generaln, of the probability that an interval contains preciselyn eigenvalues.