Universality limits in the bulk for varying measures

Universality limits are a central topic in the theory of random matrices. We establish universality limits in the bulk of the spectrum for varying measures, using the theory of entire functions of exponential type. In particular, we consider measures that are of the form W_n^{2 n} (x) d x in the region where universality is desired. W_n does not need to be analytic, nor possess more than one derivative-and then only in the region where universality is desired. We deduce universality in the bulk for a large class of weights of the form W^{2 n} (x) d x, for example, when W = e^{- Q} where Q is convex and Q^′ satisfies a Lipschitz condition of some positive order. We also deduce universality for a class of fixed exponential weights on a real interval.