On rate of convergence for universality limits

TL;DR

This paper investigates the rate at which the universality limits of reproducing kernels on the unit circle converge, providing estimates under certain conditions on the measure.

Contribution

It offers a quantitative estimate for the convergence rate of universality limits for measures with finite logarithmic integral.

Findings

Derived explicit convergence rate estimates.

Applicable to measures with finite logarithmic integral.

Enhances understanding of universality limit behavior.

Abstract

Given a probability measure $\mu$ on the unit circle $\mathbb{T}$, consider the reproducing kernel $k_{\mu,n}(z_1, z_2)$ in the space of polynomials of degree at most $n-1$ with the $L^2(\mu)$-inner product. Let $u, v \in \mathbb{C}$. It is known that under mild assumptions on $\mu$ near $\zeta \in \mathbb{T}$, the ratio $k_{\mu,n}(\zeta e^{u/n}, \zeta e^{v/n})/k_{\mu,n}(\zeta, \zeta)$ converges to a universal limit $S(u, v)$ as $n \to \infty$. We give an estimate for the rate of this convergence for measures $\mu$ with finite logarithmic integral.