Fredholm Determinants, Differential Equations and Matrix Models

TL;DR

This paper explores Fredholm determinants associated with integral operators from random matrix models, expressing them via PDE solutions and extending to circular ensembles, revealing new analytical connections.

Contribution

It introduces a method to represent Fredholm determinants of a broad class of kernels in terms of PDE solutions, including an exponential variant for circular ensembles.

Findings

Fredholm determinants can be expressed through PDE systems when kernels satisfy certain differentiation formulas.

The analysis extends to circular ensembles of unitary matrices using an exponential variant.

The approach unifies the treatment of various matrix models via integral operator determinants.

Abstract

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.