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

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

2002-03-25

arxiv.org/abs/math-ph/0203049v1

Physics

Mathematical Physics

AMS-Latex

Scientific paper

10.1088/0951-7715/15/3/325

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $\tau$-functions for certain Painlev\'e 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 $\tau$-functions for certain Painlev\'e 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 $\tau$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $\tau$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $\tau$-functions is the gap probability in the corresponding orthogonal symmetry case.

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