Continuum AB percolation and AB random geometric graphs

2014/05/12 by Mathew D. Penrose, Penrose, Mathew D.
Mathematics · #05C40 #05C80 #60D05 #82B43 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:05C40 #msc:05C80 #msc:60D05 #msc:82B43

paper · pdf · doi:10.48550/arxiv.1405.2717

arxiv created 2014/05/12 · arxiv updated 2014/05/13

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 λ,μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ,μ) 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 μ= τλ for fixed τ, we give a strong law of large numbers as λ→ ∞, for the connectivity threshold of this graph.

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