Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight

We present an asymptotic analysis of the Verblunsky coefficients for the polynomials orthogonal on the unit circle with the varying weight \(e^{-nV(\cos x)}\), assuming that the potential \(V\) has four bounded derivatives on \([-1,1]\) and the equilibrium measure has a one interval support. We obtain the asymptotics as a solution of the system of "string" equations.