Bulk universality holds in measure for compactly supported measures

摘要

Let µ be a measure with compact support, with orthonormal polynomials {p n } and associated reproducing kernels {K n }. We show that bulk universality holds in measure in {ξ: µ′(ξ) 0}. More precisely, given ɛ, r 0, the linear Lebesgue measure of the set {ξ: µ′(ξ) 0} and for which $$mathop {sup }limits_{left| u right|,left| v right| leqslant r} left| {frac{{K_n (xi + u/tilde K_n (xi ,xi ),xi + v/tilde K_n (xi ,xi ))}} {{K_n (xi ,xi )}}} right. - left. {frac{{sin pi (u - v)}} {{pi (u - v)}}} right| geqslant varepsilon$$ approaches 0 as n → ∞. There are no local or global regularity conditions on the measure µ.