Universality for outliers in weakly confined Coulomb-type systems

TL;DR

This paper studies the asymptotic behavior of outliers in weakly confined Coulomb systems and random polynomials, revealing dependence on the shape and charge distribution of uncharged regions, with results on convergence to Bergman point processes.

Contribution

It establishes the limiting outlier processes depend only on the uncharged region shape and charge, introducing new universality results for Coulomb gases and polynomial zeros.

Findings

Outliers in simply connected regions converge to Bergman point processes.

Limiting outlier processes depend on the uncharged region shape and charge.

Outliers in different regions are asymptotically independent.

Abstract

This work concerns weakly confined particle systems in the plane, characterized by a large number of outliers away from a droplet where the bulk of the particles accumulate in the many-particle limit. We are interested in the asymptotic behavior of outliers for two classes of point processes: Coulomb gases at determinantal inverse temperature confined by a regular background, and a class of random polynomials. We observe that the limiting outlier process only depends on the shape of the uncharged region containing them, and the global net excess charge. In particular, for a determinantal Coulomb gas confined by a sufficiently regular background measure, the outliers in a simply connected uncharged region converge to the corresponding Bergman point process. For a finitely connected uncharged region $\Omega$, a family of limiting outlier processes arises, indexed by the (Pontryagin) dual of the fundamental group of $\Omega$. Moreover, the outliers in different uncharged regions are asymptotically independent, even if the regions have common boundary points. The latter result is a manifestation of screening properties of the particle system.