Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients

We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients ${\alpha }_{n,N}$ when the ratio $n\mathrm{/}N$ converges as $n,N\to \mathrm{\infty }$. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case of asymptotically constant and asymptotically periodic coefficients ${\alpha }_{n,N}$. Second, we determine the asymptotic zero distribution of paraorthogonal polynomials in the locally constant and locally periodic regimes. Analogous results are obtained for orthogonal polynomials under a mild additional condition on the varying coefficients.