Papers › Level-Spacing Distributions and the Airy Kernel
Level-Spacing Distributions and the Airy Kernel
Craig A. Tracy, Harold Widom
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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, 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.
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