We apply universality limits to asymptotics of spacing of zeros {x kn } of orthogonal polynomials, for weights with compact support and for exponential weights. A typical result is lim n→∞ x kn − x k+1,n K n (x kn , x kn ) = 1 under minimal hypotheses on the weight, withK n denoting a normalized reproducing kernel. Moreover, for exponential weights, we derive asymptotics for the differentiated kernels:
Abstract. We establish universality in the bulk for …xed exponential weights on the whole real line. Our methods involve …rst order asymptotics for orthogonal polynomials and localization techniques. In particular we allow exponential weights such as jxj 2 g 2 (x) exp ( 2Q (x)), where > 1=2, Q is convex and Q 00 satis…es some regularity conditions, while g is positive, and has uniformly continuous and slowly growing or decaying logarithm.
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