Gap probability for the hard edge Pearcey process

TL;DR

This paper derives integral representations and asymptotic formulas for the gap probability in the hard edge Pearcey process, a universal model in random matrix theory, using Hamiltonian systems and differential identities.

Contribution

It introduces a novel integral representation of the gap probability and establishes large gap asymptotics including the constant term, advancing understanding of the process's statistical properties.

Findings

Derived integral representation of gap probability.

Established large gap asymptotics including the constant term.

Analyzed asymptotic statistical properties of the counting function.

Abstract

The hard edge Pearcey process is universal in random matrix theory and many other stochastic models. This paper deals with the gap probability for the thinned/unthinned hard edge Pearcey process over the interval $(0,s)$ by working on the relevant Fredholm determinants. We establish an integral representation of the gap probability via a Hamiltonian related a system of coupled differential equations. Together with some remarkable differential identities for the Hamiltonian, we derive the large gap asymptotics for the thinned case, up to and including the constant term. As an application, we also obtain the asymptotic statistical properties of the counting function for the hard edge Pearcey process.