  Using the results on the $1/n$-expansion of the Verblunsky coefficients for a
class of polynomials orthogonal on the unit circle with $n$ varying weight, we
prove that the local eigenvalue statistic for unitary matrix models is
independent of the form of the potential, determining the matrix model. Our
proof is applicable to the case of four times differentiable potentials and of
supports, consisting of one interval.
