On boundary confinements for the Coulomb gas

TL;DR

This paper introduces a new family of boundary confinements for Coulomb gas ensembles, exploring their effects on particle distributions and limits, including a novel point field emerging when the droplet boundary becomes ill-defined.

Contribution

It proposes a continuum of boundary conditions interpolating between free and hard edges, and analyzes their impact on the Coulomb gas and random matrix ensembles, including a new limiting point field.

Findings

Existence of a new limit point field when the droplet boundary is ill-defined.

Analysis of scaling limits and maximum modulus distribution under various boundary confinements.

Introduction of fuzzy boundary ensembles with particles more likely outside the boundary.

Abstract

We introduce a family of boundary confinements for Coulomb gas ensembles, and study them in the two-dimensional determinantal case of random normal matrices. The family interpolates between the free boundary and hard edge cases, which have been well studied in various random matrix theories. The confinement can also be relaxed beyond the free boundary to produce ensembles with more fuzzy boundaries, i.e., where the particles are more and more likely to be found outside of the boundary. The resulting ensembles are investigated with respect to scaling limits and distribution of the maximum modulus. In particular, we prove existence of a new point field - a limit of scaling limits to the ultraweak point when the droplet ceases to be well defined.