Level-Spacing Distributions and the Airy Kernel

Abstract

Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\times N$ 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 double scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[\mathrm{Ai}(x){\mathrm{Ai}}^{\prime }(y)-{\mathrm{Ai}}^{\prime }(x)\mathrm{Ai}(y)]/(x-y)$. We announce analogies 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{ô}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{é} 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.

9 pages

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