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

Abstract

It has recently been emphasized that all known exact evaluations of gapprobabilities for classical unitary matrix ensembles are in factτ-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 caseof symplectic matrix ensembles, all exact evaluations, eitherknown or derivable from the existing literature, are identifiedas the mean of two τ-functions, both of which correspond toHamiltonians satisfying the same differential equation, differing only in the boundary condition.Furthermore the product of these two τ-functions gives the gapprobability in the corresponding unitary symmetry case, while one of theseτ-functions is the gap probability in the corresponding orthogonal symmetry case.