From the Pearcey to the Airy process

TL;DR

This paper explores the connection between the Pearcey and Airy processes in random matrix theory, demonstrating how to approximate Pearcey process statistics using the Airy process through PDEs governing gap probabilities.

Contribution

It establishes a method to approximate Pearcey process multi-time statistics with the Airy process via PDEs, linking two important random matrix models.

Findings

Derived PDE for Pearcey gap probabilities

Established approximation of Pearcey by Airy process

Connected two key random matrix models

Abstract

Putting dynamics into random matrix models leads to finitely many nonintersecting Brownian motions on the real line for the eigenvalues, as was discovered by Dyson. Applying scaling limits to the random matrix models, combined with Dyson's dynamics, then leads to interesting, infinite-dimensional diffusions for the eigenvalues. This paper studies the relationship between two of the models, namely the Airy and Pearcey processes and more precisely shows how to approximate the multi-time statistics for the Pearcey process by the one of the Airy process with the help of a PDE governing the gap probabilities for the Pearcey process.