On orthogonal and symplectic matrix ensembles |
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Authors: | Craig A Tracy Harold Widom |
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Institution: | (1) Department of Mathematics, and Institute of Theoretical Dynamics, University of California, 95616 Davis, CA, USA;(2) Department of Mathematics, University of California, 95064 Santa Cruz, CA, USA |
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Abstract: | The focus of this paper is on the probability,E
(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal ( =1) and Gaussian Symplectic ( =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 ( =2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painlevé II function. |
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