A Riemann Hilbert approach to the study of the generating function associated to the Pearcey process

TL;DR

This paper develops a Riemann-Hilbert approach to derive a Tracy-Widom-like formula and PDE for the generating function of the Pearcey process, linking it to differential equations and occupancy numbers.

Contribution

It introduces a novel Riemann-Hilbert method to analyze the Pearcey process and derives new coupled differential equations and PDEs for its generating function.

Findings

Established a Tracy-Widom-like formula for the Pearcey process

Derived a coupled vector differential equation of order three

Obtained a nonlinear coupled heat equation and a PDE for the generating function

Abstract

Using Riemann-Hilbert methods, we establish a Tracy-Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.