On Rate of Convergence for Universality Limits

Given a probability measure μ on the unit circle 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 ∈ℂ . It is known that under mild assumptions on μ near ζ∈𝕋 , the ratio k_μ ,n(ζ e^u/n, ζ e^v/n)/k_μ ,n(ζ , ζ ) converges to a universal limit S ( u , v ) as n →∞ . We give an estimate for the rate of this convergence for measures μ with finite logarithmic integral.