Asymptotics of the finite-temperature sine kernel determinant

In the present paper, we study the asymptotics of the Fredholm determinant $D(x,s)$ of the finite-temperature deformation of the sine kernel, which represents the probability that there is no particles on the interval $(-x/\pi,x/\pi)$ in the bulk scaling limit of the finite-temperature fermion system. The variable $s$ in $D(x,s)$ is related to the temperature. The determinant also corresponds to the finite-temperature correlation function of one dimensional Bose gas. We derive the asymptotics of $D(x,s)$ in several different regimes in the $(x,s)$-plane. A third-order phase transition is observed in the asymptotic expansions as both $x$ and $s$ tend to positive infinity at certain related speed. The phase transition is then shown to be described by an integral involving the Hastings-McLeod solution of the second Painlev\'e equation.