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 mu be a measure on a compact subset of the real line. Assume that mu is absolutely continuous in a neighborhood of some point x in the support and that mu' is bounded above and below near x, which is assumed to be a Lebesgue point of mu'. Then universality holds at x if and only if it holds "along the diagonal," that is, lim(n ->infinity) K-n(x + a/n, x + a/n)/K-n(x, x) = 1 for all real a. The method does not require regularity of the measure mu as did earlier methods. Moreover, the assumption on the diagonal is certainly satisfied in the case of regular measures, so that we obtain another proof of some recent results of Simon and Totik.