On the integrable structure of deformed sine kernel determinants

TL;DR

This paper explores deformed sine kernel determinants, linking them to integrable systems and Painlevé equations, enabling explicit solutions for certain PDEs related to free fermion statistics.

Contribution

It establishes a connection between deformed sine kernel determinants and a generalized Painlevé system, providing explicit solutions for related integrable PDEs.

Findings

Connection between deformed sine kernel determinants and generalized Painlevé equations

Explicit solutions for integrable PDEs with specific initial data

Application to bulk statistics of finite temperature free fermions

Abstract

We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function w. For a specific choice of w, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlev\'e equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data.