Ratio asymptotics and zero density for orthogonal polynomials with varying Verblunsky coefficients
Rostyslav Kozhan, František Štampach
TL;DR
The paper analyzes orthogonal polynomials on the unit circle with Verblunsky coefficients that vary with an additional index, establishing ratio asymptotics when n/N → t and the Verblunsky coefficients converge to a constant or are asymptotically periodic. It then characterizes the asymptotic zero density of paraorthogonal polynomials and, under an additional condition, of orthogonal polynomials, in locally constant and locally periodic regimes, using two complementary proofs: a potential-theoretic approach via Widom’s lemma and a moment method. Key contributions include explicit limit objects in terms of m- and Schur-functions and finite-gap data, as well as a robust framework that extends to periodic and almost-periodic settings and to the corresponding OPUC through balayage. The results illuminate the global eigenvalue distribution in varying-parameter OPUC ensembles and connect to equilibrium measures and finite-gap theory, with explicit examples and densities. This advances understanding of how local Verblunsky behavior governs global zero patterns and spectral measures in the varying-parameter regime.
Abstract
We study asymptotic behavior of orthogonal polynomials on the unit circle with varying Verblunsky coefficients $α_{n,N}$. First, we give a streamlined proof of ratio asymptotics for orthogonal and paraorthogonal polynomials in the case when $α_{n,N}$ is asymptotically constant or asymptotically periodic as $n/N\to t>0$. Second, we determine the asymptotic zero density of paraorthogonal polynomials in the locally constant and locally periodic regimes, and obtain analogous results for orthogonal polynomials under an additional condition $|α_{n-1,N}|^{1/n}\to 1$.