Limits of determinantal processes near a tacnode

TL;DR

This paper analyzes a Markov process of interlacing particles near a tacnode, deriving the local correlation kernel and showing it interpolates between the Pearcey process and GUE minor process as the domain degenerates.

Contribution

It derives the local correlation kernel around a tacnode in a particle system and demonstrates its interpolation between two well-known processes.

Findings

Derived the local correlation kernel at the tacnode transition

Showed the process interpolates between Pearcey and GUE minor processes

Analyzed the behavior of the particle system as the domain degenerates

Abstract

We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter $\eps> 0$. The domain has two cusps, one pointing up and one pointing down. In the limit $\varepsilon\downarrow 0$ the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime $\varepsilon \downarrow 0$. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.