Bulk universality for unitary matrix models

A proof of universality in the bulk of spectrum of unitary matrix models, assuming that the potential is globally $C^2$ and locally $C^3$ function (see Theorem 1.2), is given. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. The sin-kernel is obtained as a unique solution of a certain nonlinear integrodifferential equation without using asymptotics of orthogonal polynomials.