Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases

2011/04/26 by M. Bertola, Marco Bertola, Mattia Cafasso +1 · 2 citations
Mathematics · Physics and Astronomy · #Airy function #Economics #Focus (optics) #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Optics #Order (exchange) #Physics #Point processes and geometric inequalities #Pure mathematics #Random Matrices and Applications #Riemann hypothesis #Statistical Methods and Bayesian Inference #math-ph #math.MP #math.PR #nlin.SI

paper · pdf · doi:10.1016/j.physd.2012.01.003

published as Physica D, 241, 2012, 2237-2245 · 18 pages, 1 figure

arxiv created 2011/04/26 · openalex publication_date 2012/01/19 · arxiv updated 2013/06/06 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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.

Citations

• A Brownian-Motion Model for the Eigenvalues of a Random Matrix
• Differential Equations for Dyson Processes
• Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit
• DIFFERENTIAL EQUATIONS FOR QUANTUM CORRELATION FUNCTIONS
• The Dependence on the Monodromy Data of the Isomonodromic Tau Function
• Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Large n Limit of Gaussian Random Matrices with External Source, Part II

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