Journal of Applied Probability

Continuum AB percolation and AB random geometric graphs

Mathew D. Penrose


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.

Article information

J. Appl. Probab., Volume 51A (2014), 333-344.

First available in Project Euclid: 2 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C80: Random graphs [See also 60B20]
Secondary: 82B43: Percolation [See also 60K35] 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Bipartite geometric graph continuum percolation connectivity threshold


Penrose, Mathew D. Continuum AB percolation and AB random geometric graphs. J. Appl. Probab. 51A (2014), 333--344. doi:10.1239/jap/1417528484.

Export citation


  • Chayes, L. and Schonmann, R. H. (2000). Mixed percolation as a bridge between site and bond percolation. Ann. Appl. Prob. 10, 1182–1196.
  • Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.
  • Iyer, S. K. and Yogeshwaran, D. (2012). Percolation and connectivity in AB random geometric graphs. Adv. Appl. Prob. 44, 21–41.
  • 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.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.
  • Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
  • Pinto, P. C. and Win, Z. (2012). Percolation and connectivity in the intrinsically secure communications graph. IEEE Trans. Inform. Theory 58, 1716–1730.
  • 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, 6pp.
  • Sarkar, A. and Haenggi, M. (2013). Percolation in the secrecy graph. Discrete Appl. Math. 161, 2120–2132.