Communications in Mathematical Physics

, Volume 177, Issue 3, pp 727–754 | Cite as

On orthogonal and symplectic matrix ensembles

  • Craig A. Tracy
  • Harold Widom


The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN 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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Craig A. Tracy
    • 1
  • Harold Widom
    • 2
  1. 1.Department of Mathematics, and Institute of Theoretical DynamicsUniversity of CaliforniaDavisUSA
  2. 2.Department of MathematicsUniversity of CaliforniaSanta CruzUSA

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