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

TL;DR

This paper demonstrates that gap probabilities in orthogonal and symplectic matrix ensembles can be expressed as $ au$-functions of Painlevé systems, extending known results from unitary ensembles and revealing new relationships among these probabilities.

Contribution

It establishes that all gap probabilities for orthogonal and symplectic ensembles are $ au$-functions of Painlevé systems, providing a unified framework and new insights into their structure.

Findings

Orthogonal ensemble gap probabilities are $ au$-functions of Painlevé systems.

Symplectic ensemble gap probabilities are products or averages of two $ au$-functions.

The product of two $ au$-functions in symplectic case relates to unitary ensemble probabilities.

Abstract

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.