On orthogonal and symplectic matrix ensembles
Abstract
The focus of this paper is on the probability, E β(O; J), that a set J consisting of a finite union of intervals contains no eigenvalues for the finite N Gaussian Orthogonal (β=1) and Gaussian Symplectic (β=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 (β=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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- April 1996
- DOI:
- 10.1007/BF02099545
- arXiv:
- arXiv:solv-int/9509007
- Bibcode:
- 1996CMaPh.177..727T
- Keywords:
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- Nonlinear Sciences - Exactly Solvable and Integrable Systems;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- 34 pages. LaTeX file with one figure. To appear in Commun. Math. Physics