Scaling limits of random normal matrix processes at singular boundary points

TL;DR

This paper develops a method for analyzing microscopic limits of normal matrix ensembles at singular boundary points, revealing new determinantal point fields that differ from regular boundary or bulk behaviors.

Contribution

It introduces a novel approach for studying boundary singularities in normal matrix ensembles and identifies new types of determinantal point fields at these points.

Findings

Existence of new determinantal point fields at singular boundary points

Method applicable to ensembles with and without boundary restrictions

Differences between singular boundary and regular boundary or bulk behaviors

Abstract

We give a method for taking microscopic limits of normal matrix ensembles. We apply this method to study the behaviour near certain types of singular points on the boundary of the droplet. Our investigation includes ensembles without restrictions near the boundary, as well as hard edge ensembles, where the eigenvalues are confined to the droplet. We establish in both cases existence of new types of determinantal point fields, which differ from those which can appear at a regular boundary point, or in the bulk.