τ-function evaluation of gap probabilities in orthogonal and symplectic matrix ensembles

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 Painleve 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 Painleve 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 these τ-functions is the gap probability in the corresponding orthogonal symmetry case.

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