Riemann-Hilbert approach to gap probabilities for the Bessel process

TL;DR

This paper connects the gap probabilities of the Bessel process to Riemann-Hilbert problems and Painlevé equations, providing a new analytical framework for understanding these probabilities.

Contribution

It introduces a Riemann-Hilbert approach to express Bessel process gap probabilities via integrable kernels and derives a Painlevé III equation for the single-time case.

Findings

Expressed Fredholm determinants in terms of integrable kernels

Established a Riemann-Hilbert problem formulation

Derived a Painlevé III equation related to the process

Abstract

We consider the gap probability for the Bessel process in the single-time and multi-time case. We prove that the scalar and matrix Fredholm determinants of such process can be expressed in terms of determinants of integrable kernels \`a la Its-Izergin-Korepin-Slavnov and thus related to suitable Riemann-Hilbert problems. In the single-time case, we construct a Lax pair formalism and we derive a Painlev\'e III equation related to the Fredholm determinant.