ON ORTHOGONAL AND SYMPLECTIC MATRIX-ENSEMBLES

The focus of this paper is on the probability, E(beta)(0;J), that a se t J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (beta = 1) and Gaussian Symplecti c (beta = 4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (beta = 2) ensembles. Our most explicit new results concern th e distribution of the largest eigenvalue in each of these ensembles. I n the edge scaling limit we show that these largest eigenvalue distrib utions are given in terms of a particular Painleve II function.