Voronoi and Voids Statistics for Super-homogeneous Point Processes

TL;DR

This paper investigates the statistical properties of Voronoi cells and voids in super-homogeneous point processes, revealing that large voids can exist despite the uniformity at large scales, and provides exact and heuristic tools for their analysis.

Contribution

It derives exact joint statistics for Voronoi cell sizes and introduces sum rules and heuristic conditions to classify super-homogeneous point processes.

Findings

Large Voronoi cells can exist in super-homogeneous systems.

Derived exact joint statistics for Voronoi cell sizes.

Proposed heuristic conditions for classifying super-homogeneous processes.

Abstract

We study the Voronoi and void statistics of super-homogeneous (or hyperuniform) point patterns in which the infinite-wavelength density fluctuations vanish. Super-homogeneous or hyperuniform point patterns arise in one-component plasmas, primordial density fluctuations in the Universe, and in jammed hard-particle packings. We specifically analyze a certain one-dimensional model by studying size fluctuations and correlations of the associated Voronoi cells. We derive exact results for the complete joint statistics of the size of two Voronoi cells. We also provide a sum rule that the correlation matrix for the Voronoi cells must obey in any space dimension. In contrast to the conventional picture of super-homogeneous systems, we show that infinitely large Voronoi cells or voids can exist in super-homogeneous point processes in any dimension. We also present two heuristic conditions to identify and classify any super-homogeneous point process in terms of the asymptotic behavior of the void size distribution.