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

Abstract

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels à 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.

18 pages, 1 figure

References in corpus (4)

• The Pearcey Process
• Differential Equations for Dyson Processes
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation

Cited by in corpus (10)

• Cauchy-Laguerre two-matrix model and the Meijer-G random point field
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• CORRIGENDUM: The dependence on the monodromy data of the isomonodromic tau function
• The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Darboux transformations and random point processes
• Differential equations for the KPZ and periodic KPZ fixed points
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• Gaussian perturbations of hard edge random matrix ensembles
• Riemann-Hilbert approach to gap probabilities for the Bessel process