Local limits for orthogonal polynomials for varying measures via universality

Abstract We consider orthogonal polynomials p n e − 2 n Q n , x for varying measures and use universality limits to prove ”local limits” lim n → ∞ p n e − 2 n Q n , y j n + z K n y j n , y j n p n e − 2 n Q n , y j n e − n Q n ′ y j n K n y j n , y j n z = cos π z . Here y j n is a local maximum point of p n e − n Q n in the ”bulk” of the support, K n y j n , y j n is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local limits at the ”soft edge” of the spectrum, which involve the Airy function.

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