TY - JOUR

T1 - Continuum AB percolation and AB random geometric graphs

AU - Penrose, Mathew D.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - 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 and 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 = τ λ for fixed τ , we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

AB - 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 and 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 = τ λ for fixed τ , we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

KW - Bipartite geometric graph

KW - Connectivity threshold

KW - Continuum percolation

UR - http://www.scopus.com/inward/record.url?scp=84918521565&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1405.2717

UR - http://dx.doi.org/10.1239/jap/1417528484

U2 - 10.1239/jap/1417528484

DO - 10.1239/jap/1417528484

M3 - Article

AN - SCOPUS:84918521565

VL - 51A

SP - 333

EP - 344

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

ER -