CONTINUUM AB PERCOLATION AND AB RANDOM GEOMETRIC GRAPHS

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 and mu. We show for d >= 2 that if. 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 -> infinity for the connectivity threshold of this graph.