On the gap probability of the tacnode process

TL;DR

This paper investigates the mathematical properties of the tacnode process, focusing on its gap probability, integrable structure, and large gap asymptotics, with applications to statistical measures like expectation and variance.

Contribution

It provides an integral representation of the gap probability and computes large gap asymptotics, including the constant term, for the tacnode process.

Findings

Derived an integral representation of the gap probability.

Computed large gap asymptotics including the constant term.

Established a central limit theorem for the counting function.

Abstract

The tacnode process is a universal determinantal point process arising from non-intersecting particle systems and tiling problems. It is the aim of this work to explore the integrable structure and large gap asymptotics for the gap probability of the thinned/unthinned tacnode process over $(-s,s)$. We establish an integral representation of the gap probability in terms of the Hamiltonian associated with a system of differential equations. With the aids of some remarkable differential identities for the Hamiltonian, we also compute large gap asymptotics, up to and including the constant term in the thinned case. As direct applications, we obtain expectation, variance and a central limit theorem for the associated counting function.