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

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