Relationships between $\tau$-function and Fredholm determinant expressions for gap probabilities in random matrix theory

TL;DR

This paper demonstrates that gap probabilities at the edges of orthogonal random matrix ensembles can be expressed as Fredholm determinants, linking $ au$-functions with these determinants and unifying different symmetry classes.

Contribution

It establishes a new connection between $ au$-functions and Fredholm determinants for gap probabilities across multiple random matrix symmetry classes.

Findings

$ au$-functions can be expressed as Fredholm determinants

Fredholm determinants unify gap probability expressions across symmetries

Extension of results to include generating function parameters

Abstract

The gap probabilities at the hard and soft edges of scaled random matrix ensembles with orthogonal symmetry are known in terms of $\tau$-functions. Extending recent work relating to the soft edge, it is shown that these $\tau$-functions, and their generalizations to contain a generating function parameter, can be expressed as Fredholm determinants. These same Fredholm determinants also occur in exact expressions for gap probabilities in scaled random matrix ensembles with unitary and symplectic symmetry.