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

Abstract

AbstractWe 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 Painlevé 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 Painlevé 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.