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

TL;DR

This paper explores the relationships between hyperuniformity, Coulomb energy, and Wasserstein distance in 2D point processes, establishing implications, counterexamples, and equivalences among these properties.

Contribution

It proves new implications and equivalences among hyperuniformity, Coulomb energy, and Wasserstein distance, and introduces an adapted screening construction for Coulomb gases in Wasserstein space.

Findings

i) Coulomb energy implies Wasserstein distance in 2D processes.

ii) Wasserstein distance implies hyperuniformity, under bounded density.

iii) Coulomb energy is equivalent to a form of hyperuniformity slightly stronger than standard hyperuniformity.

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.