Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

We apply universality limits to asymptotics of spacing of zeros {} fenced(x_kn) of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result isunder(lim, n → ∞) fenced(x_kn - x_{k + 1, n}) over(K, ̃)_n fenced(x_kn, x_kn) = 1under minimal hypotheses on the weight, with over(K, ̃)_n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:K_n^{fenced(r, s)} fenced(x, x) = underover(∑, k = 0, n - 1) p_k^fenced(r) fenced(x) p_k^fenced(s) fenced(x) .