  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.
