THE LINK BETWEEN HYPERUNIFORMITY, COULOMB ENERGY AND WASSERSTEIN DISTANCE TO LEBESGUE FOR TWO-DIMENSIONAL POINT PROCESSES

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 Lebesgue measure and (iii) hyperuniformity. In dimension 2, we prove that (i) implies (ii), which is known to imply (iii), and we provide simple counterexamples 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 of Sandier and Serfaty (2015) for Coulomb gases (of which we present an adaptation to 2-Wasserstein space which might be of independent interest), and on the recent necessary and sufficient conditions given by Sodin, Wennman and Yakir (2013) for the existence of stationary “electric” fields compatible with a given stationary point process.