Advances in Applied Probability

The asymptotic size of the largest component in random geometric graphs with some applications

Ge Chen, Changlong Yao, and Tiande Guo

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we estimate the expectation of the size of the largest component in a supercritical random geometric graph; the expectation tends to a polynomial on a rate of exponential decay. We sharpen the expectation's asymptotic result using the central limit theorem. Similar results can be obtained for the size of the biggest open cluster, and for the number of open clusters of percolation on a box, and so on.

Article information

Adv. in Appl. Probab., Volume 46, Number 2 (2014), 307-324.

First available in Project Euclid: 29 May 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random geometric graph percolation largest component Poisson-Boolean model number of open clusters


Chen, Ge; Yao, Changlong; Guo, Tiande. The asymptotic size of the largest component in random geometric graphs with some applications. Adv. in Appl. Probab. 46 (2014), no. 2, 307--324. doi:10.1239/aap/1401369696.

Export citation


  • Glauche, I., Krause, W., Sollacher, R. and Greiner, M. (2003). Continuum percolation of wireless ad hoc communication networks. Physica A 325, 577–600.
  • Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.
  • Hekmat, R. and Van Mieghem, P. (2006). Connectivity in Wireless Ad Hoc Networks with a Log-normal Radio Model. Mobile Networks Appl. 11, 351–360.
  • Jiang, J., Zhang, S. and Guo, T. (2010). A convergence rate in a martingale CLT for percolation clusters. J. Graduate School Chinese Acad. Sci. 27, 577–583.
  • Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.
  • Penrose, M. D. (1991). On a continuum percolation model. Adv. Appl. Prob. 23, 536–556.
  • Penrose, M. D. (1995). Single linkage clustering and continuum percolation. J. Multivariate Anal. 53, 94–109.
  • Penrose, M. D. (2001). A central limit theorem with applications to percolation, epidemics and Boolean models. Ann. Prob. 29, 1515–1546.
  • Penrose, M. D. (2003). Random Geometric Graphs. Oxford University Press.
  • Penrose, M. D. and Pisztora, A. (1996). Large deviations for discrete and continuous percolation. Adv. Appl. Prob. 28, 29–52.
  • Pishro-Nik, H., Chan, K. and Fekri, F. (2009). Connectivity properties of large-scale sensor networks. Wireless Networks 15, 945–964.
  • Ta, X., Mao, G. and Anderson, B. D. O. (2009). On the giant component of wireless multihop networks in the presence of shadowing. IEEE Trans. Vehicular Tech. 58, 5152–5163.