Level-Spacing Distributions and the Airy Kernel

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ hermitian matrices and then going to the limit $N\to\infty$, 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 $[{\rm Ai}(x) {\rm Ai}'(y) -{\rm Ai}'(x) {\rm Ai}(y)]/(x-y)$. 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, M{\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\'e} 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.