On orthogonal and symplectic matrix ensembles

Abstract

The focus of this paper is on the probability, $E_\\beta(0;J)$, that a set $J$\nconsisting of a finite union of intervals contains no eigenvalues for the\nfinite $N$ Gaussian Orthogonal ($\\beta=1$) and Gaussian Symplectic ($\\beta=4$)\nEnsembles and their respective scaling limits both in the bulk and at the edge\nof the spectrum. We show how these probabilities can be expressed in terms of\nquantities arising in the corresponding unitary ($\\beta=2$) ensembles. Our most\nexplicit new results concern the distribution of the largest eigenvalue in each\nof these ensembles. In the edge scaling limit we show that these largest\neigenvalue distributions are given in terms of a particular Painlev\\'e II\nfunction.\n