Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T05:59:28.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuum AB percolation and AB random geometric graphs

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

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 geometry 05C80: Random graphs
Secondary: 82B43: Percolation 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory
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 

References

Chayes, L., and Schonmann, R. H. (2000). Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Prob. 10, 1182–1196.Google Scholar
Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
Iyer, S. K., and Yogeshwaran, D. (2012). Percolation and connectivity in AB random geometric graphs. Adv. Appl. Prob. 44, 21–41.Google Scholar
Lorenz, C. D., and Ziff, R. M. (2000). Precise determination of the critical percolation threshold for the three-dimensional “Swiss cheese” model using a growth algorithm. J. Chem. Phys. 114, 3659–3661.CrossRefGoogle Scholar
Meester, R., and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Pinto, P. C., and Win, Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inform. Theory 58, 1716–1730.Google Scholar
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, 6 pp.Google Scholar
Sarkar, A., and Haenggi, M. (2013). Percolation in the secrecy graph. Discrete Appl. Math. 161, 2120–2132.Google Scholar