Planar equilibrium measure problem in the quadratic fields with a point charge

TL;DR

This paper analyzes a 2D equilibrium measure problem with quadratic potentials and a point charge, revealing phase transitions in droplet topology and connecting to spectral distributions in random matrix theory.

Contribution

It explicitly characterizes the shape and topology of droplets in a quadratic field with a point charge, highlighting phase transitions and their spectral implications.

Findings

Droplets are doubly connected in the post-critical case.

Droplets contain merging boundary points at criticality.

Droplets are disconnected in the pre-critical case.

Abstract

We consider a two-dimensional equilibrium measure problem under the presence of quadratic potentials with a point charge and derive the explicit shape of the associated droplets. This particularly shows that the topology of the droplets reveals a phase transition: (i) in the post-critical case, the droplets are doubly connected domain; (ii) in the critical case, they contain two merging type singular boundary points; (iii) in the pre-critical case, they consist of two disconnected components. From the random matrix theory point of view, our results provide the limiting spectral distribution of the complex and symplectic elliptic Ginibre ensembles conditioned to have zero eigenvalues, which can also be interpreted as a non-Hermitian extension of the Marchenko-Pastur law.