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On the critical threshold for continuum AB percolation

Part of: Geometric probability and stochastic geometry Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  16 January 2019

David Dereudre  and
Mathew Penrose
David Dereudre*
Affiliation:
Université de Lille
Mathew Penrose*
Affiliation:
University of Bath
*
* Postal address: Laboratoire Paul Painlevé, Université de Lille, 59655 Villeneuve d’Ascq, France. Email address: [email protected]
** Postal address: Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK.
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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. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

Keywords

Continuum percolationbipartite geometric graphconnectivity thresholdenhancement

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1228 - 1237
Copyright
Copyright © Applied Probability Trust 2018 

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