Bessel kernel determinants and integrable equations

TL;DR

This paper derives differential equations for Bessel determinantal point process statistics, linking them to integrable nonlinear PDEs, Sturm-Liouville equations, and Painlevé equations, using Riemann-Hilbert problem techniques.

Contribution

It establishes a novel connection between Bessel process statistics and integrable PDEs, expanding the understanding of their mathematical structure and relations to classical special functions.

Findings

Statistics satisfy an integrable nonlinear PDE.

Identities relate solutions of PDEs to Sturm-Liouville equations.

Degenerate limits recover known Painlevé V equations.

Abstract

We derive differential equations for multiplicative statistics of the Bessel determinantal point process depending on two parameters. In particular, we prove that such statistics are solutions to an integrable nonlinear partial differential equation describing isospectral deformations of a Sturm-Liouville equation. We also derive identities relating solutions to the integrable partial differential equation and to the Sturm-Liouville equation which imply an analogue for Painlev\'e V of Amir-Corwin-Quastel "integro-differential Painlev\'e II equation". This equation reduces, in a degenerate limit, to the system of coupled Painlev\'e V equations derived by Charlier and Doeraene for the generating function of the Bessel process, and to the Painlev\'e V equation derived by Tracy and Widom for the gap probability of the Bessel process. Finally, we study an initial value problem for the integrable partial differential equation. The approach is based on Its-Izergin-Korepin-Slavnov theory of integrable operators and their associated Riemann-Hilbert problems.