Transition asymptotics of Toeplitz determinants and emergence of Fisher–Hartwig representations

Abstract

We compute the transition asymptotics (double-scaling limits) of Toeplitz determinants generated by symbols f t possessing Fisher–Hartwig singularities. The symbols f t that we consider depend on a parameter t such that f t has one Fisher–Hartwig singularity when t > 0 and two Fisher–Hartwig singularities when t = 0. Unlike in the other studies of the transition asymptotics of Toeplitz determinants, our setting involves the emergence of Fisher–Hartwig representations as . We use the Riemann–Hilbert problem for orthogonal polynomials and its connection to Painlevé transcendents to obtain the asymptotics. We apply our results to study a special correlator known as the emptiness formation probability (EFP) for the one-dimensional anisotropic XY spin-1/2 chain in a transverse magnetic field, and describe its transition between different regions in the phase diagram across critical lines.