Håkan Hedenmalm: Planar orthogonal polynomials and boundary universality in the random normal matrix model

Abstract:
This reports on joint work with A. Wennman.
The study of the asymptotics of orthogonal polynomials has a long history. In the 1920s, a number of new results appeared, with contributions of Szegö and Carleman. In light of recent interest in the random normal matrix model, it has become clear that those results need to be extended beyond the given settings of arc length measure of a closed loop and area measure inside, respectively. The case at hand corresponds to a soft edge, with a rather general planar weight, and complete asymptotics are obtained. As a simple consequence, the blow-up of a boundary point in the ensemble gives erf asymptotics, under rather minimal assumptions. Our methods are based on a flow of loops approach, which we call the orthogonal flow foliation. This foliation is a new device, although it has its roots in the study of Hele-Shaw flow.