Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes

Abstract

We study Fredholm determinants of a class of integral operators, whose kernels can be expressed as double contour integrals of a special type. Such Fredholm determinants appear in various random matrix and statistical physics models. We show that the logarithmic derivatives of the Fredholm determinants are directly related to solutions of the Painlevé II hierarchy. This confirms and generalizes a recent conjecture by Le Doussal, Majumdar, and Schehr. In addition, we obtain asymptotics at $\pm \infty$ for the Painlevé transcendents and large gap asymptotics for the corresponding point processes.

arXiv comment34 pages, 5 figures

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• Unitary matrix models and random partitions: Universality and multi-criticality
• Gap probability for the hard edge Pearcey process
• Multicritical random partitions
• Interface fluctuations associated with split Fermi seas
• Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution
• Classical group matrix models and universal criticality
• A fully noncommutative Painlevé II hierarchy: Lax pair and solutions related to Fredholm determinants
• Universal Cusp Scaling in Random Partitions
• Asymptotics of the deformed higher order Airy-kernel determinants and applications
• A $\dbar$-steepest descent method for oscillatory Riemann-Hilbert problems