Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The link between hyperuniformity, Coulomb energy, and Wasserstein distance to Lebesgue for two-dimensional point processes

Martin Huesmann, Thomas Leblé

TL;DR

The paper analyzes stationary planar point processes through three lenses—finite regularized Coulomb energy, finite 2-Wasserstein distance to Lebesgue, and hyperuniformity—and proves a sharp chain of implications in 2D: HU_ star quals Coul quals Wass_2 implies HU, while the converses fail in general. A key contribution is a Wasserstein-analog of the screening technique, enabling a Coulomb-energy-to-Wasserstein bound and a reverse W2-to-Coulomb bound under a uniform density cap; these are tied together by a spectral (SC) condition that is shown to be equivalent to HU_ star via results of Sodin-Wennman-Yakir. The work also provides counterexamples to demonstrate the optimality of the implications, including Wass_2 inite Coulomb energy failure and HU inite Wasserstein distance failure, and discusses density-bounded refinements that recover parts of the chain. Overall, the results illuminate how regularized Coulomb interactions, optimal transport costs, and hyperuniform fluctuations interact in the infinite-volume, planar setting, with potential implications for Coulomb-gas models and hyperuniform materials in physics.

Abstract

We investigate the interplay between three possible properties of stationary point processes: i) Finite Coulomb energy with short-scale regularization, ii) Finite $2$-Wasserstein transportation distance to the Lebesgue measure and iii) Hyperuniformity. In dimension $2$, we prove that i) implies ii), which is known to imply iii), and we provide simple counter-examples to both converse implications. However, we prove that ii) implies i) for processes with a uniformly bounded density of points, and that i) - finiteness of the regularized Coulomb energy - is equivalent to a certain property of quantitative hyperuniformity that is just slightly stronger than hyperuniformity itself. Our proof relies on the classical link between $H^{-1}$-norm and $2$-Wasserstein distance between measures, on the screening construction for Coulomb gases (of which we present an adaptation to $2$-Wasserstein space which might be of independent interest), and on recent necessary and sufficient conditions for the existence of stationary "electric" fields compatible with a given stationary point process.

The link between hyperuniformity, Coulomb energy, and Wasserstein distance to Lebesgue for two-dimensional point processes

TL;DR

The paper analyzes stationary planar point processes through three lenses—finite regularized Coulomb energy, finite 2-Wasserstein distance to Lebesgue, and hyperuniformity—and proves a sharp chain of implications in 2D: HU_ star quals Coul quals Wass_2 implies HU, while the converses fail in general. A key contribution is a Wasserstein-analog of the screening technique, enabling a Coulomb-energy-to-Wasserstein bound and a reverse W2-to-Coulomb bound under a uniform density cap; these are tied together by a spectral (SC) condition that is shown to be equivalent to HU_ star via results of Sodin-Wennman-Yakir. The work also provides counterexamples to demonstrate the optimality of the implications, including Wass_2 inite Coulomb energy failure and HU inite Wasserstein distance failure, and discusses density-bounded refinements that recover parts of the chain. Overall, the results illuminate how regularized Coulomb interactions, optimal transport costs, and hyperuniform fluctuations interact in the infinite-volume, planar setting, with potential implications for Coulomb-gas models and hyperuniform materials in physics.

Abstract

We investigate the interplay between three possible properties of stationary point processes: i) Finite Coulomb energy with short-scale regularization, ii) Finite -Wasserstein transportation distance to the Lebesgue measure and iii) Hyperuniformity. In dimension , we prove that i) implies ii), which is known to imply iii), and we provide simple counter-examples to both converse implications. However, we prove that ii) implies i) for processes with a uniformly bounded density of points, and that i) - finiteness of the regularized Coulomb energy - is equivalent to a certain property of quantitative hyperuniformity that is just slightly stronger than hyperuniformity itself. Our proof relies on the classical link between -norm and -Wasserstein distance between measures, on the screening construction for Coulomb gases (of which we present an adaptation to -Wasserstein space which might be of independent interest), and on recent necessary and sufficient conditions for the existence of stationary "electric" fields compatible with a given stationary point process.
Paper Structure (50 sections, 29 theorems, 166 equations)

This paper contains 50 sections, 29 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Hyperuniformity
  4. Wasserstein distances for stationary point processes
  5. Summary.
  6. Infinite-volume Coulomb energy
  7. The importance of screened fields
  8. The role of abstract fields
  9. Truncation of the field.
  10. Connection with the literature
  11. Notation, plan of the paper
  12. From Coulomb energy to finite Wasserstein distance
  13. The basic W2-H-1 inequality
  14. Screening of electric fields in large boxes
  15. An approximate sequence of point processes
  16. ...and 35 more sections

Key Result

Theorem 1

In any dimension $\mathsf{d} \geq 2$, $\texttt{Coul}$ implies $\texttt{Wass}_2$. More precisely, there exists a constant $C$ depending only on the dimension such that, for all stationary point processes $\mathbf{X}^{{\mathbin{\vcenter{\hbox{$\bullet$}}}}}$: thus effectively bounding the $2$-Wasserstein distance to Lebesgue (defined in Section sec:DefWass) by the regularized Coulomb energy $\maths

Theorems & Definitions (70)

  • Theorem 1: $\texttt{Coul} \implies \texttt{Wass}_2$
  • Theorem 2: $\texttt{Wass}_2 \centernot \implies \texttt{Coul}$, $\texttt{HU} \centernot \implies \texttt{Wass}_2$
  • Theorem 3: $\texttt{Wass}_2$ plus density bound implies $\texttt{Coul}$
  • Definition 1.1: $\texttt{HU}_\star$
  • Theorem 4: $\texttt{HU}_\star \iff \texttt{Coul}$
  • Remark 1.2
  • Remark 1.3
  • Claim 1.4: Properties of the truncated field
  • Definition 1.5
  • Lemma 1.6: Two properties of the regularized energy
  • ...and 60 more