The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach

TL;DR

This paper develops a Riemann-Hilbert approach to analyze the transition between Pearcey and Airy gap probabilities, revealing their asymptotic relationship and deriving new nonlinear PDEs for the Pearcey process.

Contribution

It introduces a novel Riemann-Hilbert framework for the Pearcey process, connecting Fredholm determinants with isomonodromic tau functions and deriving new PDEs.

Findings

Asymptotic factorization into two Airy processes

Construction of a Lax pair for Pearcey gap probability

Discovery of a new nonlinear PDE for Pearcey process

Abstract

We consider the gap probability for the Pearcey and Airy processes; we set up a Riemann--Hilbert approach (different from the standard one) whereby the asymptotic analysis for large gap/large time of the Pearcey process is shown to factorize into two independent Airy processes using the Deift-Zhou steepest descent analysis. Additionally we relate the theory of Fredholm determinants of integrable kernels and the theory of isomonodromic tau function. Using the Riemann-Hilbert problem mentioned above we construct a suitable Lax pair formalism for the Pearcey gap probability and re-derive the two nonlinear PDEs recently found and additionally find a third one not reducible to those.