On finite-temperature Fredholm determinants

We consider finite-temperature deformation of the sine kernel Fredholm determinants acting on the closed contours. These types of expressions usually appear as static two-point correlation functions in the models of free fermions and can be equivalently presented in terms of Toeplitz determinants. The corresponding symbol, or the phase shift, is related to the temperature weight. We present an elementary way to obtain large-distance asymptotic behavior even when the phase shift has a non-zero winding number. It is done by deforming the original kernel to the so-called effective form factors kernel that has a completely solvable matrix Riemann-Hilbert problem. This allows us to find explicitly the resolvent and address the subleading corrections. We recover Szego, Hartwig and Fisher, and Borodin-Okounkov asymptotic formulas.