On rate of convergence for universality limits

Roman Bessonov

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arXiv · 2306.13722

On rate of convergence for universality limits

Roman Bessonov

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Abstract

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

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BibTeX RIS

Roman Bessonov. 2023-06-23. On rate of convergence for universality limits. https://arxiv.org/abs/2306.13722

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