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Continuum AB percolation and AB random geometric graphs

Part of:Graph theoryGeometric probability and stochastic geometrySpecial processesEquilibrium statistical mechanics

Published online by Cambridge University Press:  30 March 2016

Mathew D. Penrose
Mathew D. Penrose*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: m.d.penrose@bath.ac.uk.
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Abstract

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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.

Keywords

Bipartite geometric graphcontinuum percolationconnectivity threshold

MSC classification

Primary:60D05: Geometric probability and stochastic geometry05C80: Random graphs
Secondary:82B43: Percolation05C40: Connectivity60K35: Interacting random processes; statistical mechanics type models; percolation theory

Information

Type
Part 7. Stochastic geometry
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014, pp. 333 - 344
Copyright
Copyright © Applied Probability Trust 2014 

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