Fredholm determinant solutions of the Painlev\'e II hierarchy and gap probabilities of determinantal point processes

Mattia Cafasso (1) , Tom Claeys , Manuela Girotti

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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 Painlev\'e 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 Painlev\'e transcendents and large gap asymptotics for the corresponding point processes.

Domaines: Physique mathématique [math-ph]

Mattia Cafasso, Tom Claeys, Manuela Girotti. Fredholm determinant solutions of the Painlev\'e II hierarchy and gap probabilities of determinantal point processes. 2019. ⟨hal-02049070⟩