Bulk Universality for Unitary Matrix Models

We give 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. The proof is based on the determinant formulas for correlation functions in terms of polynomials orthogonal on the unit circle. We do not use asymptotics of orthogonal polynomials. We obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation.