On the components of random geometric graphs in the dense limit

Mathew D. Penrose, Xiaochuan Yang

Published 2025-01-05Version 1

Consider the geometric graph on $n$ independent uniform random points in a connected compact region $A$ of ${\bf R}^d, d \geq 2$ with smooth boundary, or in the unit square, with distance parameter $r_n$. Let $K_n$ be the number of components of this graph, and $R_n$ the number of vertices not in the giant component. Let $S_n$ be the number of isolated vertices. We show that if $r_n$ is chosen so that $nr_n^d$ tends to infinity but slowly enough that ${\bf E}[S_n]$ also tends to infinity, then $K_n$, $R_n$ and $S_n$ are all asymptotic to $\mu_n$ in probability as $n \to \infty$ where (with $|A|$, $\theta_d$ and $|\partial A|$ denoting the volume of $A$, the volume of the unit $d$-ball, and the perimeter of $A$ respectively) $\mu_n := ne^{-\pi n r_n^d/|A|}$ if $d=2$ and $\mu_n := ne^{-\theta_d n r_n^d/|A|} + \theta_{d-1}^{-1} |\partial A| r_n^{1-d} e^{- \theta_d n r_n^d/(2|A|)}$ if $d\geq 3$. We also give variance asymptotics and central limit theorems for $K_n$ and $R_n$ in this limiting regime when $d \geq 3$, and for Poisson input with $d \geq 2$. We extend these results (substituting ${\bf E}[S_n]$ for $\mu_n$) to a class of non-uniform distributions on $A$.