Szegő theorem and tau functions

We show that the constant term (the first non-trivial term) in the Szegő–Widom asymptotic formula for the determinants of certain special truncated block-Toeplitz matrices (loops) is the tau function for the solution (corresponding to this loop) of a soliton equation of parabolic type. We discuss consequences of this fact concerning the analytic properties of local holomorphic solutions of such equations.