Universality in the bulk holds close to given points

Abstract

Let μ be a measure with compact support. Assume that ξ is a Lebesgue point of μ and that μ ′ is positive and continuous at ξ . Let { A n } be a sequence of positive numbers with limit ∞ . We show that one can choose ξ n ∈ [ ξ − A n n , ξ + A n n ] such that lim n → ∞ K n ( ξ n , ξ n + a K ̃ n ( ξ n , ξ n ) ) K n ( ξ n , ξ n ) = sin π a π a , uniformly for a in compact subsets of the plane. Here K n is the n th reproducing kernel for μ , and K ̃ n is its normalized cousin. Thus universality in the bulk holds on a sequence close to ξ , without having to assume that μ is a regular measure. Similar results are established for sequences of measures.