Continuum AB percolation and AB random geometric graphs

TL;DR

This paper investigates the percolation properties of bipartite random geometric graphs in higher dimensions, establishing conditions for supercriticality and analyzing connectivity thresholds as point intensities grow large.

Contribution

It extends known results for $d=2$ to higher dimensions, showing supercriticality conditions and deriving a law of large numbers for connectivity thresholds.

Findings

Supercriticality for $d \\geq 2$ when $\\lambda$ is supercritical for the one-type graph with doubled radius.

Existence of $\\mu$ ensuring supercritical bipartite graphs in higher dimensions.

Strong law of large numbers for connectivity thresholds as $\\lambda \to \infty$.

Abstract

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. We show for $d \geq 2$ that if $\lambda$ is supercritical for the one-type random geometric graph with distance parameter $2r$, there exists $\mu$ such that $(\lambda,\mu)$ is supercritical (this was previously known for $d=2$). For $d=2$ we also consider the restriction of this graph to points in the unit square. Taking $\mu = \tau \lambda$ for fixed $\tau$, we give a strong law of large numbers as $\lambda \to \infty$, for the connectivity threshold of this graph.