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 are no particles in 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. This determinant also corresponds to the finite-temperature correlation function of the 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 Painlevé equation.