Aron Wennman: Planar orthogonal polynomials and boundary universality in the random normal matrix model

Abstract: This reports on recent joint work with Håkan Hedenmalm.

Motivated by questions concerning the boundary behavior of the correlation kernel in the random normal matrix model, we study planar orthogonal polynomials with respect to exponentially varying weights. We obtain a complete asymptotic expansion of the orthogonal polynomials, reminiscent of Carleman’s classical theorem on planar orthogonal polynomials on a simply connected domain, which allows us to obtain the universal boundary decay profile of the eigenvalues.

In the talk we will discuss this asymptotic expansion, in particular we focus on a new technique which decomposes planar orthogonality into orthogonality along a curve family which foliates a planar region. 

The talk is a continuation of last week’s analysis seminar given by Håkan, but we aim for it to be self-contained.