The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths

TL;DR

This paper introduces new limiting processes called the hard edge tacnode and Pearcey processes for non-intersecting squared Bessel paths, deriving explicit correlation kernels involving Airy functions and Toeplitz determinants.

Contribution

It provides explicit formulas for the correlation kernels of these processes, extending existing results to multi-time settings and involving novel operator representations.

Findings

Derived explicit correlation kernel for the hard edge tacnode process.

Extended the Pearcey process to multi-time correlation kernels.

Rewrote Toeplitz determinants using Borodin-Okounkov type formulas.

Abstract

A system of non-intersecting squared Bessel processes is considered which all start from one point and they all return to another point. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches the hard edge, a limiting critical process is described in the neighbourhood of the touching point which we call the hard edge tacnode process. We derive its correlation kernel in an explicit new form which involves Airy type functions and operators that act on the direct sum of $L^2(\mathbb R_+)$ and a finite dimensional space. As the starting points of the squared Bessel paths are set to 0, a cusp in the boundary appears. The limiting process is described near the cusp and it is called the hard edge Pearcey process. We compute its multi-time correlation kernel which extends the existing formulas for the single-time kernel. Our pre-asymptotic correlation kernel involves the ratio of two Toeplitz determinants which are rewritten using a Borodin-Okounkov type formula.