Large Deviations and Linear Statistics for Potential Theoretic Ensembles Associated with Regular Closed Sets

TL;DR

This paper studies a two-dimensional charged particle system influenced by a boundary region, analyzing its limiting behavior and showing it concentrates near the equilibrium measure as the number of particles grows.

Contribution

It provides a detailed description of the weak* limits of the joint intensities of the particle process and characterizes its concentration near the equilibrium measure.

Findings

Particles tend to accumulate near the boundary of the region K when charge exceeds N.

The joint intensities converge to a limit described by potential theory.

It is exponentially unlikely to observe deviations from the equilibrium measure.

Abstract

A two-dimensional statistical model of N charged particles interacting via logarithmic repulsion in the presence of an oppositely charged regular closed region K whose charge density is determined by its equilibrium potential at an inverse temperature \beta is investigated. When the charge on the region, s, is greater than N, the particles accumulate in a neighborhood of the boundary of K, and form a point process in the complex plane. We describe the weak* limits of the joint intensities of this point process and show that it is exponentially likely to find the process in a neighborhood of the equilibrium measure for K.