Universality Limits in the Bulk for Arbitrary Measures on Compact Sets

We present a new method for establishing universality limits in the bulk, based on the theory of entire functions of exponential type. Let be a measure on a compact subset of the real line. Assume that is absolutely continuous in a neighborhood of some point x in the support, and that 0 is bounded above and below near x, which is assumed to be a Lebesgue point of 0. Then universality holds at x i¤ it holds "along the diagonal", that is lim n!1 Kn x+ a n ; x+ a n Kn (x; x) = 1; for all real a. The method does not require regularity of the measure as did earlier methods. Moreover, the assumption on the diagonal is certainly satis…ed in the case of regular measures, so that we obtain another proof of some recent results of Simon and Totik. 1. Introduction and Results Let be a …nite positive Borel measure with compact support supp[ ] and in…nitely many points in the support. Then we may de…ne orthonormal polynomials pn (x) = nx n + :::; n > 0; n = 0; 1; 2; ::: satisfying the orthonormality conditions Z pnpmd = mn: Throughout we use w = d dx to denote the Radon-Nikodym derivative of . We say that is regular (in the sense of Stahl and Totik [23]) if lim n!1 1=n n = 1 cap (supp [ ]) ; where cap denotes logarithmic capacity. When the support is [ 1; 1], the condition reduces to lim n!1 1=n n = 2: Date : December 2, 2007. Research supported by NSF grant DMS0400446 and US-Israel BSF grant 2004353 1