Box-Covariances of Hyperuniform Point Processes

TL;DR

This paper characterizes the covariance structure of point counts in boxes for hyperuniform processes, revealing how it depends on box overlap and introducing a new interpolating structure that leads to a limiting Gaussian process.

Contribution

It provides a complete covariance characterization for hyperuniform point processes and introduces a novel interpolating covariance structure beyond standard assumptions.

Findings

Covariance depends solely on box face overlap under standard assumptions

A new interpolating covariance structure is identified beyond standard assumptions

A limiting Gaussian process for point counts is established, varying with integrability conditions

Abstract

In this work, we present a complete characterization of the covariance structure of number statistics in boxes for hyperuniform point processes. Under a standard integrability assumption, the covariance depends solely on the overlap of the faces of the box. Beyond this assumption, a novel interpolating covariance structure emerges. This enables us to identify a limiting Gaussian 'coarse-grained' process, counting the number of points in large boxes as a function of the box position. Depending on the integrability assumption, this process may be continuous or discontinuous, e.g. in d=1 it is given by an increment process of a fractional Brownian motion.