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A simple bound on fluctuations in the 3D Coulomb gas

Alex Cohen, Felipe Hernández

TL;DR

This work proves a new bound on macroscopic fluctuations in the 3D Coulomb gas at low temperature: smooth linear statistics $\langle \mu_X, \varphi\rangle$ fluctuate on the scale $N^{1-2/d}$, with $d=3$ giving $N^{1/3}$. The authors reduce fluctuations to controlling the $L^1$ norm of the Coulomb potential $P\mu_X$ via the identity $|\langle \varphi, \mu_X\rangle| = |\langle \Delta \varphi, P\mu_X\rangle|$, and use a Glauber-like resampling mechanism to argue that large negative potential regions would be corrected by particle moves. The torus and Euclidean-space (with confining potential) cases are treated, leveraging minimum-energy bounds (à la Serfaty) and Frostman-type energy minimization to obtain exponential moment and $L^1$ control, which in turn yield high-probability fluctuation bounds. In particular, for $d=3$ the results imply $\operatorname{Var}(\langle \varphi, \mu_X\rangle) = O(N^{2/3})$ and, for $C^2$ observables supported in the equilibrium support, fluctuations are $O(N^{1/3})$, illustrating hyperuniform-type behavior at macroscopic scales.

Abstract

The Coulomb gas models an interacting system of $N$ negatively charged particles. We give a new proof that, at sufficiently low temperature, smooth linear statistics $\sum_j \varphi(x_j)$ are bounded by $C N^{1-2/d}$.

A simple bound on fluctuations in the 3D Coulomb gas

TL;DR

This work proves a new bound on macroscopic fluctuations in the 3D Coulomb gas at low temperature: smooth linear statistics fluctuate on the scale , with giving . The authors reduce fluctuations to controlling the norm of the Coulomb potential via the identity , and use a Glauber-like resampling mechanism to argue that large negative potential regions would be corrected by particle moves. The torus and Euclidean-space (with confining potential) cases are treated, leveraging minimum-energy bounds (à la Serfaty) and Frostman-type energy minimization to obtain exponential moment and control, which in turn yield high-probability fluctuation bounds. In particular, for the results imply and, for observables supported in the equilibrium support, fluctuations are , illustrating hyperuniform-type behavior at macroscopic scales.

Abstract

The Coulomb gas models an interacting system of negatively charged particles. We give a new proof that, at sufficiently low temperature, smooth linear statistics are bounded by .

Paper Structure

This paper contains 13 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Result on the torus $\mathbb{R}^d / \mathbb{Z}^d$
  3. Result on $\mathbb{R}^d$
  4. Organization of the Paper
  5. Acknowledgements
  6. Coulomb gas on the torus
  7. The probabilistic estimate
  8. Proof of Lemma \ref{['lem:Lone-moment-torus']}
  9. The minimum energy configuration
  10. Setup
  11. Fluctuations of the potential
  12. The $L^1$ bound
  13. Application to fluctuations

Key Result

Theorem 1.1

For $d\geq 3$ there exists some absolute constant $C_d$ such that for $A\geq C_d$ the following bound holds for $X$ sampled from the Gibbs measure $\mathbb{P}_{N,\beta}$ and $\phi\in C^2(\mathbb{T}^d)$:

Theorems & Definitions (12)

  • Theorem 1.1: Bound for fluctuations on $\mathbb{T}^d$
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Proposition 3.2: Serfaty*Corollary 5.5
  • Proposition 3.3: Serfaty*Corollary 5.26
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 2 more