Universality in the bulk holds close to given points

Let mu be a measure with compact support. Assume that xi is a Lebesgue point of mu and that mu' is positive and continuous at xi. Let {A(n)} be a sequence of positive numbers with limit infinity. We show that one can choose xi(n) is an element of [xi - A(n)/n, xi + A(n)/n] such that n ->(lim)infinity K-n(xi(n), xi(n) + a/(K) over tilde (n)(xi(n), xi(n)))/K-n(xi(n), xi(n)) = sin pi a/pi a, uniformly for a in compact subsets of the plane. Here K-n, is the nth reproducing kernel for mu, and (K) over tilde (n) is its normalized cousin. Thus universality in the bulk holds on a sequence close to xi, without having to assume that mu is a regular measure. Similar results are established for sequences of measures. (C) 2009 Elsevier Inc. All rights reserved.