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 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.