On comparison of clustering properties of point processes

TL;DR

This paper introduces a new comparison tool for spatial homogeneity of point processes based on void probabilities and factorial moments, showing its relevance for understanding clustering and percolation properties.

Contribution

The paper proposes a novel comparison method for point processes using void probabilities and factorial moments, extending the $dcx$ ordering framework and providing a spectrum of comparable processes.

Findings

Determinantal and permanental processes are comparable to Poisson processes in this framework.

The new tool is relevant for studying macroscopic, percolative properties of point processes.

A large monotone spectrum of point processes is constructed, from grids to Cox processes.

Abstract

In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results and preview further ones, showing that the new tool is relevant in the study of macroscopic, percolative properties of point processes. This new comparison is also implied by the directionally convex ($dcx$ ordering of point processes, which has already been shown to be relevant to comparison of spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point process as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity northe total independence property are not realistic assumptions.