Percolation and Connectivity in AB Random Geometric Graphs

TL;DR

This paper investigates percolation and connectivity in a generalized continuum AB random geometric graph model based on Poisson point processes, establishing existence results, bounds, and asymptotic properties.

Contribution

It extends the AB percolation model to continuum space, providing existence proofs, critical intensity bounds, and connectivity thresholds for the first time.

Findings

Percolation exists for all dimensions greater than 1.

Bounds for the critical intensity are derived.

Asymptotic behavior of the connectivity threshold is characterized.

Abstract

Given two independent Poisson point processes $\Phi^{(1)},\Phi^{(2)}$ in $R^d$, the continuum AB percolation model is the graph with points of $\Phi^{(1)}$ as vertices and with edges between any pair of points for which the intersection of balls of radius $2r$ centred at these points contains at least one point of $\Phi^{(2)}$. This is a generalization of the $AB$ percolation model on discrete lattices. We show the existence of percolation for all $d > 1$ and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when $d = 2$. To study the connectivity problem, we consider independent Poisson point processes of intensities $n$ and $cn$ in the unit cube. The $AB$ random geometric graph is defined as above but with balls of radius $r$. We derive a weak law result for the largest nearest neighbour distance and almost sure asymptotic bounds for the connectivity threshold.