Hyperuniformity and optimal transport of point processes
Raphaël Lachièze-Rey, D. Yogeshwaran
TL;DR
This work advances the understanding of optimal transport between stationary random measures by linking hyperuniformity to transport efficiency. Using Fourier-analytic bounds on restricted samples (rooted in the structure factor and RPCM), the authors derive finite-volume transport bounds and show how these yield an invariant matching in the infinite-volume limit; in 2D HU settings, a mild logarithmic integrability condition on the reduced pair correlation measure guarantees a planar $L^2$-perturbed lattice, while in higher dimensions integrable RPCM suffices for an $L^2$-perturbed lattice without HU. The results extend to random measures and recover known Poisson-case bounds, offering a unified, nonPDE approach that highlights the role of second-order spatial statistics in transport problems. The study connects deterministic and probabilistic transport perspectives, with concrete implications for determinantal processes, Ginibre ensembles, Gaussian entire function zeros, and Coulomb gases, and provides sharp, finite-sample bounds that inform both theory and applications in spatial statistics and materials science.
Abstract
We examine optimal matchings or transport between two stationary random measures. It covers allocation from the Lebesgue measure to a point process and matching a point process to a regular (shifted) lattice. The main focus of the article is the impact of hyperuniformity(reduced variance fluctuations in point processes) to optimal transport: in dimension 2, we show that the typical matching cost has finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are L2-perturbed lattices. Our method also retrieves known sharp bounds in finite windows for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems. Further, in three dimensions onwards, all point processes with an integrable pair correlation measure are L2-perturbed lattices without requiring hyperuniformity.