On Rate of Convergence for Universality Limits

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