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

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.

43 pages, 7 figures. Final version with minor changes. Accepted for publication on International Mathematical Research Notices

References in corpus (7)

• The Pearcey Process
• Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• The dependence on the monodromy data of the isomonodromic tau function
• Universality of the Pearcey process
• Universal scaling limits of matrix models, and (p,q) Liouville gravity
• From the Pearcey to the Airy process

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• Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case
• Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
• A Riemann-Hilbert approach to the lower tail of the KPZ equation
• Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
• From the Pearcey to the Airy process
• Asymptotics of Fredholm determinant associated with the Pearcey kernel
• Nonlinear PDEs for gap probabilities in random matrices and KP theory
• Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation
• The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
• 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
• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Darboux transformations and random point processes
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process
• Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel
• On the deformed Pearcey determinant
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
• The Malgrange Form and Fredholm Determinants
• A Riemann Hilbert approach to the study of the generating function associated to the Pearcey process
• Riemann-Hilbert approach to gap probabilities for the Bessel process
• Large gap asymptotics at the hard edge for product random matrices and Muttalib-Borodin ensembles
• Gap probability for products of random matrices in the critical regime