Tau-function of discrete isomonodromy transformations and probability

TL;DR

This paper introduces a tau-function for rational d-connections, links it to continuous tau-functions, computes it for difference Painleve equations, and connects it to gap probabilities in discrete random matrix models.

Contribution

It defines a new tau-function for rational d-connections, relates it to classical tau-functions, and applies it to analyze gap probabilities in discrete random matrix models.

Findings

Tau-function matches Jimbo-Miwa-Ueno tau-function in the continuous limit.

Explicit tau-function computations for difference Painleve V and VI.

Gap probabilities in discrete models can be expressed as tau-functions.

Abstract

We introduce the tau-function of a rational d-connection and its isomonodromy transformations. We show that in a continuous limit our tau-function agrees with the Jimbo-Miwa-Ueno tau-function, compute the tau-function for the isomonodromy transformations leading to difference Painleve V and difference Painleve VI equations, and prove that the gap probability for a wide class of discrete random matrix type models can be viewed as the tau-function for an associated d-connection.