Asymptotics of the hard edge Pearcey determinant

TL;DR

This paper derives the asymptotic behavior of the Fredholm determinant linked to the hard edge Pearcey kernel, which is significant in random matrix theory and non-intersecting paths models, using Riemann-Hilbert problem techniques.

Contribution

It introduces a method to analyze the large gap asymptotics of the Pearcey determinant via a Riemann-Hilbert problem approach.

Findings

Asymptotic formulas for the Pearcey determinant at the hard edge.

Connection between the determinant's derivatives and a 3x3 Riemann-Hilbert problem.

Application to large gap probability calculations in related models.

Abstract

We study the Fredholm determinant of an integral operator associated to the hard edge Pearcey kernel. This determinant appears in a variety of random matrix and non-intersecting paths models. By relating the logarithmic derivatives of the Fredholm determinant to a $3 \times 3$ Riemann-Hilbert problem, we obtain asymptotics of the determinant, which is also known as the large gap asymptotics for the corresponding point process.