Clustering comparison of point processes with applications to random geometric models

TL;DR

This paper reviews methods for comparing clustering in point processes, focusing on tools like void probabilities, moment measures, and stochastic orderings, and discusses applications to random geometric models and topological data analysis.

Contribution

It introduces a comprehensive framework for comparing clustering in point processes using various probabilistic tools and applies these methods to analyze properties of random geometric models.

Findings

Void probabilities and moment measures effectively capture clustering differences.

Comparison tools help extend Poisson process results to sub-Poisson processes.

Applications include percolation, coverage, and topological properties of geometric models.

Abstract

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the Boolean model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some "sub-Poisson" processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.