On Rate of Convergence for Universality Limits

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 \(n \rightarrow \infty \). We give an estimate for the rate of this convergence for measures \(\mu \) with finite logarithmic integral.