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

TL;DR

This work analyzes the emergence of fixed-order $k$-clusters in high-density random geometric graphs via the Poisson Boolean model and its finite-window analogues. It derives sharp first- and second-order asymptotics for the number of $k$-clusters, including mean and variance, and establishes LLNs, concentration results, and distributional approximations (Poisson and normal) in mildly dense settings, with parallel results for both binomial and Poisson point ensembles. The authors introduce the quasi-gravitational energy functional $g$ to describe the limiting internal structure of large $k$-clusters and prove convergence in distribution of scaled cluster points to densities proportional to $e^{-g}$, capturing a compression phenomenon. They also extend the analysis to mildly sparse regimes and to nonuniform densities, showing that similar asymptotics hold under suitable conditions. The results employ Stein's method and local-dependence techniques to obtain explicit error bounds, offering rigorous probabilistic descriptions relevant to continuum percolation, random geometric graphs, and topological data analysis applications.

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.

Figures (3)

• Figure 1: The union of the disks is the set $D((z_1,z_2,z_3))$.
• Figure 2: Illustration of the proof of Lemma \ref{['l:QGE']}.
• Figure 3: Illustration for the proof of Lemma \ref{['l:geom']}. The small circle represents $B_{\delta_2 s}(z)$. The larger segment represents $B^*(y,s,\delta_2,e(y))$.

Theorems & Definitions (53)

• Remark 1.1
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['t:alpha']}
• proof : Proof of Theorem \ref{['t:ptprlim']}
• Lemma 3.1
• proof