Level-Spacing Distributions and the Airy Kernel

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sinπ(x− y)/π(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, 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 general n, of the probability that an interval contains precisely n eigenvalues.