Level-spacing distributions and the Airy kernel

Abstract

Scaling level-spacing distribution functions in the ``bulk of the spectrum''\nin random matrix models of $N\\times N$ hermitian matrices and then going to the\nlimit $N\\to\\infty$, leads to the Fredholm determinant of the sine kernel\n$\\sin\\pi(x-y)/\\pi (x-y)$. Similarly a double scaling limit at the ``edge of the\nspectrum'' leads to the Airy kernel $[{\\rm Ai}(x) {\\rm Ai}'(y) -{\\rm Ai}'(x)\n{\\rm Ai}(y)]/(x-y)$. We announce analogies for this Airy kernel of the\nfollowing properties of the sine kernel: the completely integrable system of\nP.D.E.'s found by Jimbo, Miwa, M{\\^o}ri and Sato; the expression, in the case\nof a single interval, of the Fredholm determinant in terms of a Painlev{\\'e}\ntranscendent; the existence of a commuting differential operator; and the fact\nthat this operator can be used in the derivation of asymptotics, for general\n$n$, of the probability that an interval contains precisely $n$ eigenvalues.\n