Abstract

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--, o¢, leads to the Fredholm determinant of the sine kernel sin n(x-y)/n(x-y). Similarly a double scaling limit at the "edge of the spectrum " leads to the Airy kernel [Ai (x)Ai ' (y)- Ai ' (x)Ai (y) ] / (x- y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE's found by Jimbo, Miwa, M6ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev6 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.

Citations