Continuum AB percolation and AB random geometric graphs

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 $d\ge 2$ 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.