On $k$-clusters of high-intensity random geometric graphs

TL;DR

This paper analyzes the asymptotic probability and distribution of clusters of size k in high-dimensional random geometric graphs and Poisson Boolean models, revealing their mean, variance, and approximation behaviors in different density regimes.

Contribution

It provides new asymptotic formulas for cluster probabilities and distributions in high-dimensional random geometric graphs and Poisson models, covering both dense and sparse regimes.

Findings

Asymptotic probability of k-clusters in Poisson Boolean model determined.

Variance of cluster count asymptotic to its mean, enabling distribution approximations.

Results extend to both dense and sparse limiting regimes.

Abstract

Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime). We also show that the variance of $S_{n,k}$ is asymptotic to its mean, and prove Poisson and normal approximation results for $S_{n,k}$ in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points). We also give similar results in the so-called mildly sparse limiting regime where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $ becomes large.