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 $e^{-nV(\cos x)}$,assumingthatthepotential$V$hasfourboundedderivativeson$[-1,1]$ and the equilibrium measure has a one interval support. We obtain the asymptotics as a solution of the system of “string” equations.