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{\bf $τ$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

P. J. Forrester·N. S. Witte
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Abstract

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $τ$-functions for certain Painlevé systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $τ$-functions for certain Painlevé systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $τ$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $τ$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $τ$-functions is the gap probability in the corresponding orthogonal symmetry case.

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P. J. Forrester, N. S. Witte. 2002-03-25. {\bf $τ$-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}. https://doi.org/10.1088/0951-7715%2F15%2F3%2F325

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