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Large charge fluctuations in the hierarchical Coulomb gas

Alon Nishry, Oren Yakir

TL;DR

This work extends the Jancovici-Lebowitz-Manificat tail law to the two-dimensional hierarchical Coulomb gas, establishing precise large-deviation tails for charge fluctuations across all scales. Leveraging the model’s hierarchical self-similarity, the authors develop a martingale exposure framework, sharp MGFs, and boundary-geometry analyses to prove the JLM law with a piecewise-linear rate function $\varphi(\alpha)$ that matches the anticipated phases: moderate deviations, large deviations, and overcrowding. The results advance understanding of tail phenomena for 2D Coulomb-type systems beyond the classical $\beta=2$ setting and connect to broader tail behaviors in related 2D point processes, including GEF zeros and hierarchical spin models. The methods combine concentration on dyadic cubes, careful control of boundary contributions, and change-of-measure arguments for moderate deviations, yielding a robust toolkit for analyzing hierarchical interacting particle systems.

Abstract

The two-dimensional one-component plasma (OCP) is a model of electrically charged particles which are embedded in a uniform background of the opposite charge, and interact through a logarithmic potential. More than 30 years ago, Jancovici, Lebowitz and Manificat discovered an asymptotic law for probabilities of large charge fluctuations in the OCP. We prove that this law holds for the hierarchical counterpart of the OCP. The hierarchical model was recently introduced by Chatterjee, and is inspired by Dyson's hierarchical model of the Ising ferromagnet.

Large charge fluctuations in the hierarchical Coulomb gas

TL;DR

This work extends the Jancovici-Lebowitz-Manificat tail law to the two-dimensional hierarchical Coulomb gas, establishing precise large-deviation tails for charge fluctuations across all scales. Leveraging the model’s hierarchical self-similarity, the authors develop a martingale exposure framework, sharp MGFs, and boundary-geometry analyses to prove the JLM law with a piecewise-linear rate function that matches the anticipated phases: moderate deviations, large deviations, and overcrowding. The results advance understanding of tail phenomena for 2D Coulomb-type systems beyond the classical setting and connect to broader tail behaviors in related 2D point processes, including GEF zeros and hierarchical spin models. The methods combine concentration on dyadic cubes, careful control of boundary contributions, and change-of-measure arguments for moderate deviations, yielding a robust toolkit for analyzing hierarchical interacting particle systems.

Abstract

The two-dimensional one-component plasma (OCP) is a model of electrically charged particles which are embedded in a uniform background of the opposite charge, and interact through a logarithmic potential. More than 30 years ago, Jancovici, Lebowitz and Manificat discovered an asymptotic law for probabilities of large charge fluctuations in the OCP. We prove that this law holds for the hierarchical counterpart of the OCP. The hierarchical model was recently introduced by Chatterjee, and is inspired by Dyson's hierarchical model of the Ising ferromagnet.
Paper Structure (32 sections, 19 theorems, 318 equations, 2 figures)

This paper contains 32 sections, 19 theorems, 318 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Charge fluctuations
  3. The JLM Law
  4. The hierarchical Coulomb gas
  5. Related works
  6. Structure of the proof
  7. Preliminaries
  8. Basic facts fromCha19
  9. Discrepancy at top level of the tree
  10. Sub-Gaussian tails and overcrowding in dyadic squares
  11. A few geometric claims
  12. Upper bound for 1 < alpha < 2
  13. The martingale term
  14. The boundary term
  15. Proof of Theorem \ref{['thm:Micro_JLM']}: the upper bound for 1 < alpha < 2
  16. ...and 17 more sections

Key Result

Theorem 1

The Jancovici-Lebowitz-Manificat law holds for the hierarchical Coulomb gas model in two-dimensions, at all scales. More precisely, for all $z\in(0,1)^2$, $\alpha>\tfrac{1}{2}$ and ${\varepsilon}>0$, there exists $R_0=R_0(\alpha,\beta,z,{\varepsilon})$ large enough so that for all $R>R_0$ we have, uniformly as $n\to\infty$. Here $\varphi$ is the piecewise linear function defined by eq:def_of_phi_

Figures (2)

  • Figure 1: Illustration of the hierarchical interacting potential $w(x,y)$.
  • Figure 2: The JLM intuition for $1<\alpha<2$: the discrepancy will likely occur in a small annuli (in green) of width $R^{\alpha-1}$ around the boundary of the disk. For $\tfrac{1}{2}<\alpha<1$ the width of the annuli is $\mathcal{O}(1)$.

Theorems & Definitions (57)

  • Theorem 1
  • Remark
  • Proposition 2: chatterjee
  • proof
  • Claim 3
  • proof
  • Claim 4: chatterjee
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 47 more