BULK UNIVERSALITY HOLDS POINTWISE IN THE MEAN, FOR COMPACTLY SUPPORTED MEASURES DORON S. LUBINSKY Abstract. Let µ be a measure with compact support, with orthonor- mal polynomials {pn}, and associated reproducing kernels {Kn}. We show that without any global assumptions on the measure, a weak local condition leads to the bulk universality limit in the mean. For example, if µ ′ ≥ C> 0 in some open interval J , then at each Lebesgue point ξ of J , and for each r> 0, lim m→∞ 1 m m X n=1 sup |u|,|v|≤r ˛ ˛ ˛ ˛ ˛ ˛ Kn “ ξ + u ˜ Kn(ξ,ξ) ,ξ + v ˜ Kn(ξ,ξ) ” Kn (ξ,ξ) - sin π (u - v) π (u - v) ˛ ˛ ˛ ˛ ˛ ˛ =0. In particular, we don’t assume regularity of the measure µ. 1. Introduction Let µ be a finite positive Borel measure with compact support and infin- itely many points in the support. Define orthonormal polynomials p n (x)= γ n x n + ··· , γ n > 0, n =0, 1, 2,... , satisfying the orthonormality conditions  p j p k dµ = δ jk . Throughout we use µ ′ to denote the Radon-Nikodym derivative of µ. The nth reproducing kernel for µ is K n (x, y)= n−1  k=0 p k (x) p k (y) , (1.1) and the normalized kernel is  K n (x, y)= µ ′ (x) 1/2 µ ′ (y) 1/2 K n (x, y) . (1.2) In the theory of n by n random Hermitian matrices (the so-called uni- tary case), there arise probability distributions on the eigenvalues that are Date : August 3, 2011. 1991 Mathematics Subject Classification. 15A52, 41A55, 65B99, 42C99. Key words and phrases. Universality Limits, Random matrices, Maximal functions, Green’s functions, Christoffel functions. Research supported by NSF grant DMS1001182 and US-Israel BSF grant 2008399. 1