Journal of Applied Probability

Bounded-hop percolation and wireless communication

Christian Hirsch

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Motivated by an application in wireless telecommunication networks, we consider a two-type continuum-percolation problem involving a homogeneous Poisson point process of users and a stationary and ergodic point process of base stations. Starting from a randomly chosen point of the Poisson point process, we investigate the distribution of the minimum number of hops that are needed to reach some point of the base station process. In the supercritical regime of continuum percolation, we use the close relationship between Euclidean and chemical distance to identify the distributional limit of the rescaled minimum number of hops that are needed to connect a typical Poisson point to a point of the base station process as its intensity tends to 0. In particular, we obtain an explicit expression for the asymptotic probability that a typical Poisson point connects to a point of the base station process in a given number of hops.

Article information

J. Appl. Probab. Volume 53, Number 3 (2016), 833-845.

First available in Project Euclid: 13 October 2016

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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]

Ad hoc network chemical distance connection probability continuum percolation


Hirsch, Christian. Bounded-hop percolation and wireless communication. J. Appl. Probab. 53 (2016), no. 3, 833--845.

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  • Antal, P. (1994). Trapping problems for the simple random walk. Doctoral Thesis, ETH Zürich.
  • Antal, P. and Pisztora, Á. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Prob. 24, 1036–1048.
  • Benjamini, I., Lyons, R., Peres, Y. and Schramm, O. (1999). Group-invariant percolation on graphs. Geom. Funct. Anal. 9, 29–66.
  • Biskup, M. and König, W. (2001). Long-time tails in the parabolic Anderson model with bounded potential. Ann. Prob. 29, 636–682.
  • Deuschel, J.-D. and Pisztora, A. (1996). Surface order large deviations for high-density percolation. Prob. Theory Relat. Fields 104, 467–482.
  • Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.
  • Kesten, H. (1986). Aspects of first passage percolation. In École d'été de probabilités de Saint-Flour, XIV–-1984 (Lecture Notes Math. 1180), Springer, Berlin, pp. 125–264.
  • Krengel, U. (1985). Ergodic Theorems. De Gruyter, Berlin.
  • Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47, 655–693.
  • Meester, R. and Roy, R. (1994). Uniqueness of unbounded occupied and vacant components in Boolean models. Ann. Appl. Prob. 4, 933–951.
  • Penrose, M. D. (1991). On a continuum percolation model. Adv. in Appl. Prob. 23, 536–556.
  • Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
  • Schneider, R. and Weil, W. (2008) Stochastic and Integral Geometry. Springer, Berlin.
  • Timár, Á. (2013). Boundary-connectivity via graph theory. Proc. Amer. Math. Soc. 141, 475–480.
  • Yao, C.-L., Chen, G. and Guo, T.-D. (2011). Large deviations for the graph distance in supercritical continuum percolation. J. Appl. Prob. 48, 154–172.