Hostname: page-component-7dd5485656-zlgnt Total loading time: 0 Render date: 2025-10-29T07:19:28.733Z Has data issue: false hasContentIssue false
  • English
  • Français

On the critical threshold for continuum AB percolation

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

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: david.dereudre@univ-lille.fr
** Postal address: Department of Mathematical Sciences, University of Bath, BathBA2 7AY, UK.
Get access Rights & Permissions [Opens in a new window]

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

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 4 , December 2018 , pp. 1228 - 1237
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Esary, J. D. and Proschan, F. (1963). Coherent structures of non-identical components. Technometrics 5, 191–209.Google Scholar
[2]Franceschetti, M., Pensure, M. D. and Rosoman, T. (2010). Strict inequalities of critical probabilities on Gilbert’s continuum percolation graph. Preprint. Available at https://arxiv.org/abs/1004.1596.Google Scholar
[3]Franceschetti, M., Pensure, M. D. and Rosoman, T. (2011). Strict inequalities of critical values in continuum percolation. J. Statist. Phys. 142, 460–486.Google Scholar
[4]Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
[5]Iyer, S. K. and Yogeshwaran, D. (2012). Percolation and connectivity in AB random geometric graphs. Adv. Appl. Prob. 44, 21–41.Google Scholar
[6]Last, G. and Penrose, M. (2018). Lectures on the Poisson Process. Cambridge University Press.Google Scholar
[7]Lorenz, C. D. and Ziff, R. M. (2001). Precise determination of the critical percolation threshold for the three dimensional “Swiss cheese” model using a growth algorithm. J. Chem. Phys. 114, 3659–3661.Google Scholar
[8]Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
[9]Penrose, M. D. (2014). Continuum AB percolation and AB random geometric graphs. J. Appl. Prob. 51A, 333–344.Google Scholar
[10]Pinti, P. C. and Win, Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inform. Theory 58, 1716–1730.Google Scholar
[11]Quintanilla, J. A. and Ziff, R. M. (2007). Asymmetry of percolation thresholds of fully penetrable disks with two different radii. Phys. Rev. E 76, 051115.Google Scholar
[12]Sarkar, A. and Haenggi, M. (2013). Percolation in the Secrecy Graph. Discrete Appl. Math. 161, 2120–2132.Google Scholar