Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case

TL;DR

This paper extends the Riemann-Hilbert approach to multi-time processes, specifically the Airy and Pearcey cases, expressing matrix Fredholm determinants via integrable kernels and relating them to Riemann-Hilbert problems, and re-derives a PDE for the two-time Airy process.

Contribution

It generalizes the single-time Riemann-Hilbert framework to multi-time processes, connecting Fredholm determinants with integrable kernels and Riemann-Hilbert problems.

Findings

Expressed multi-time matrix Fredholm determinants using integrable kernels.

Related determinants to Riemann-Hilbert problems for Airy and Pearcey processes.

Re-derived a third order PDE for the two-time Airy process.

Abstract

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels \`a la Its-Izergin-Korepin-Slavnov (IIKS) and hence related to suitable Riemann-Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.