Level-spacing distributions and the Airy kernel

Abstract

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 sin π ( x − y)/ π( 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, 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.