LEVEL SPACING DISTRIBUTIONS AND THE BESSEL KERNEL

Scaling models of random N x N hermitian matrices and passing to the l imit N --> infinity leads to integral operators whose Fredholm determi nants describe the statistics of the spacing of the eigenvalues of her mitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others as well, the kernel one obtains by scaling in the ''b ulk'' of the spectrum is the ''sine kernel'' sin pi(x-y)/pi(x-y). Resc aling the GUE at the ''edge'' of the spectrum leads to the kernel Ai(x )Ai'(y)-Ai'(x)Ai(y)/x-y , where Ai is the Airy function. In previous w ork we found several analogies between properties of this ''Airy kerne l'' and known properties of the sine kernel: a system of partial diffe rential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of interval s; a representation of the Fredholm determinant in terms of a Painleve transcendent in the case of a single interval; and, also in this case , asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completel y analogous properties for a class of kernels which arise when one res cales the Laguerre or Jacobi ensembles at the edge of the spectrum, na mely J(alpha)(square-root x) square-root yJ(alpha)'(square-root y) - s quare-root x J(alpha)'(square-root x) J(alpha) (square-root y)/2(x - y ) where J(alpha)(z) is the Bessel function of order alpha. In the case s alpha = -/+ 1/2 these become, after a variable change, the kernels w hich arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an as ymptotic expansion we derive will generalize ones found by Dyson for t he Fredholm determinants of these kernels.