Integrable equations associated with the finite-temperature deformation of the discrete Bessel point process

TL;DR

This paper investigates the finite-temperature deformation of the discrete Bessel point process, revealing its connection to integrable equations like the 2D Toda and Painlevé II, and demonstrating continuum limits leading to classical integrable PDEs.

Contribution

It establishes the integrable structure of the finite-temperature discrete Bessel process and links it to well-known integrable equations, extending previous results to a deformed setting.

Findings

Largest particle distribution satisfies 2D Toda and Painlevé II equations.

In continuum limit, the process converges to the finite-temperature Airy point process.

Reductions lead to classical integrable PDEs like KdV.

Abstract

We study the finite-temperature deformation of the discrete Bessel point process. We show that its largest particle distribution satisfies a reduction of the 2D Toda equation, as well as a discrete version of the integro-differential Painlev\'e II equation of Amir-Corwin-Quastel, and we compute initial conditions for the Poissonization parameter equal to 0. As proved by Betea and Bouttier, in a suitable continuum limit the last particle distribution converges to that of the finite-temperature Airy point process. We show that the reduction of the 2D Toda equation reduces to the Korteweg-de Vries equation, as well as the discrete integro-differential Painlev\'e II equation reduces to its continuous version. Our approach is based on the discrete analogue of Its-Izergin-Korepin-Slavnov theory of integrable operators developed by Borodin and Deift.