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Two-dimensional Coulomb gases with multiple outposts

Kohei Noda

Abstract

We study two-dimensional Coulomb gases in the presence of $m\in\mathbb{N}_{>0}$ outposts. An outpost is a connected component of the coincidence set that lies outside the droplet. The case $m=1$ was previously investigated by Ameur, Charlier, and Cronvall. They showed that, as the total number of particles in the Coulomb gas tends to infinity, the number of particles accumulating near the outpost remains of order one and converges in distribution to the Heine distribution. In this work, we extend this analysis to the case of an arbitrary but fixed number $m$ of outposts. We prove that the joint distribution of the numbers of particles near the outposts converges to a multidimensional Heine distribution. Our results reveal a interesting phenomenon: although the outposts are geometrically disconnected, the particle count near each outpost is strongly correlated with the particle counts near all other outposts, not only the nearest ones (provided the outposts are not separated by a component of the droplet).

Two-dimensional Coulomb gases with multiple outposts

Abstract

We study two-dimensional Coulomb gases in the presence of outposts. An outpost is a connected component of the coincidence set that lies outside the droplet. The case was previously investigated by Ameur, Charlier, and Cronvall. They showed that, as the total number of particles in the Coulomb gas tends to infinity, the number of particles accumulating near the outpost remains of order one and converges in distribution to the Heine distribution. In this work, we extend this analysis to the case of an arbitrary but fixed number of outposts. We prove that the joint distribution of the numbers of particles near the outposts converges to a multidimensional Heine distribution. Our results reveal a interesting phenomenon: although the outposts are geometrically disconnected, the particle count near each outpost is strongly correlated with the particle counts near all other outposts, not only the nearest ones (provided the outposts are not separated by a component of the droplet).
Paper Structure (6 sections, 11 theorems, 84 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. Main results
  3. Preliminaries
  4. Proofs of Theorem \ref{['theorem: case 1']} and \ref{['theorem: case 2']}
  5. Proof of Theorem \ref{['theorem: case 1']}
  6. Proof of Theorem \ref{['theorem: case 2']}

Key Result

Theorem 1.2

(ACC2023b) Consider the item:case1 when $m=1$. Let $N_n$ be the number of particles lying in a small but fixed neighborhood of $\{|z|=t_1\}$. As $n\to+\infty$, the random variables $N_n$ converge in distribution to $\mathrm{He}(\theta\rho,\rho^2)$, where $\theta=\sqrt{\frac{\Delta Q(b_0)}{\Delta Q(t

Figures (1)

  • Figure 1: Two cases of $S^{\ast}$

Theorems & Definitions (19)

  • Example 1.1
  • Definition 1.1: One-dimensional Heine distribution
  • Theorem 1.2
  • Definition 1.3: Multi-dimensional Heine distribution
  • Remark 1.4: Consistency with the one-dimensional Heine distribution
  • Lemma 1.5
  • proof
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • ...and 9 more