Necessary and Sufficient conditions for Universality limits

Abstract

In this talk I present necessary and sufficient conditions for universality limits for orthogonal polynomials on the real line. One of our results is that the Christoffel-Darboux kernel has sine kernel asymptotics at a point ξ, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at ξ and that is the Lebesgue measure. This includes all prior results with absolutely continuous or singular measures. In fact, sine kernel asymptotics is a special case of a more general theory which also includes hard edge universality limits; we show that the Christoffel-Darboux kernel has a regularly varying scaling limit if and only if the orthogonality measure has a unique tangent measure at ξ and that is not the point mass at ξ. In this case the limit kernel is expressible in terms of confluent hypergeometric functions. This talk is based on a joint work in progress with Milivoje Lukic and Harald Woracek.