On rate of convergence for universality limits

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.

arXiv comment17 pages, 2 figures