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

Abstract

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.

Universality at the Edge for Unitary Matrix Models

“…First, using the properties of CMV matrices, we present F n (z, w) in terms of the "resolvent" of C (n) . After that we use the asymptotic behaviour of the Verblunsky coefficients, obtained in [9], to get an approximation of the "resolvent". The approximation will be given in terms of the Airy functions.…”

“…The following simple representation of ρ plays an important role in our asymptotic analysis (see [9]) Proposition 1.4 Under conditions C1-C3 the density ρ has the form…”