Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-06T10:21:53.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Bounded-hop percolation and wireless communication

Part of: Special processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  24 October 2016

Christian Hirsch
Christian Hirsch*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
* Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany. Email address: christian.hirsch@wias-berlin.de
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.

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.

Keywords

Ad hoc networkchemical distanceconnection probabilitycontinuum percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 833 - 845
Copyright
Copyright © Applied Probability Trust 2016 

References

[1] Antal, P. (1994).Trapping problems for the simple random walk. Doctoral Thesis, ETH Zürich.Google Scholar
[2] Antal, P. and Pisztora, Á. (1986).On the chemical distance for supercritical Bernoulli percolation.Ann. Prob. 24,1036–1048.Google Scholar
[3] Benjamini, I.,Lyons, R.,Peres, P. Y. and Schramm, O. (1999).Group-invariant percolation on graphs. Geom. Funct. Anal. 9,29–66.Google Scholar
[4] Biskup, M. and König, W. (2001).Long-time tails in the parabolic Anderson model with bounded potential.Ann. Prob. 29,636–682.Google Scholar
[5] Deuschel, J.-D. and Pisztora, A. (1996).Surface order large deviations for high-density percolation.Prob. Theory Relat. Fields 104,467–482.CrossRefGoogle Scholar
[6] Grimmett, G. (1999).Percolation, 2nd edn.Springer,Berlin.Google Scholar
[7] Kesten, H. (1986).Aspects of first passage percolation.. InÉcole d’été de probabilités de Saint-Flour, XIV—1984 (Lecture Notes Math. 1180)Springer,Berlin, pp.125–264.Google Scholar
[8] Krengel, U. (1985).Ergodic Theorems.De Gruyter,Berlin.Google Scholar
[9] Lawler, G. F. (1980).A self-avoiding random walk.Duke Math. J. 47,655–693.Google Scholar
[10] Meester, R. and Roy, R. (1994).Uniqueness of unbounded occupied and vacant components in Boolean models.Ann. Appl. Prob. 4,933–951.Google Scholar
[11] Penrose, M. D. (1991).On a continuum percolation model.Adv. Appl. Prob. 23,536–556.Google Scholar
[12] Penrose, M. (2003).Random Geometric Graphs.Oxford University Press.Google Scholar
[13] Schneider, R. and Weil, W. (2008).Stochastic and Integral Geometry.Springer,Berlin.Google Scholar
[14] Timár, Á. (2013).Boundary-connectivity via graph theory.Proc. Amer. Math. Soc. 141,475–480.CrossRefGoogle Scholar
[15] 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.CrossRefGoogle Scholar