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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Part of: Special processes Equilibrium statistical mechanics

Published online by Cambridge University Press:  14 July 2016

Chang-Long Yao ,
Ge Chen  and
Tian-De Guo
Chang-Long Yao*
Affiliation:
Graduate University of Chinese Academy of Sciences
Ge Chen*
Affiliation:
Graduate University of Chinese Academy of Sciences
Tian-De Guo*
Affiliation:
Graduate University of Chinese Academy of Sciences
*
∗Postal address: School of Mathematical Science, Graduate University of Chinese Academy of Sciences, 100049, Beijing, P. R. China.
∗Postal address: School of Mathematical Science, Graduate University of Chinese Academy of Sciences, 100049, Beijing, P. R. China.
∗Postal address: School of Mathematical Science, Graduate University of Chinese Academy of Sciences, 100049, Beijing, P. R. China.
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Abstract

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Denote the Palm measure of a homogeneous Poisson process Hλ with two points 0 and x by P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0, x) / μ||x||2 ∉ (1 − ε, 1 + ε) | 0, x ∈ C∞) exponentially decreases when ||x||2 tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C∞ of the random geometric graph G(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.

Keywords

Continuum percolationchemical distanceshape theoremlarge deviation inequality

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B43: Percolation
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011 , pp. 154 - 172
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Prob. 24, 1036–1048.Google Scholar
[2] Baccelli, F. and Bordenave, C. (2007). The radial spanning tree of a poisson point process. Ann. Appl. Prob. 17, 305–359.Google Scholar
[3] Dousse, O., Mannersalo, P. and Thiran, P. (2004). Latency of wireless sensor networks with uncoordinated power saving mechanisms. In Proc. 5th ACM Internat. Symp. Mobile Adhoc Networking Computing (MOBIHOC '04), Association for Computing Machinery, New York, pp. 109–120.Google Scholar
[4] Garet, O. and Marchand, R. (2004). Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster. ESAIM Prob. Statist. 8, 169–199.Google Scholar
[5] Garet, O. and Marchand, R. (2007). Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Prob. 35, 833–866.Google Scholar
[6] Garet, O. and Marchand, R. (2010). Moderate deviations for the chemical distance in Bernoulli percolation. ALEA Latin Amer. J. Prob. Math. Statist. 7, 171–191.Google Scholar
[7] Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.Google Scholar
[8] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrscheinlichkeitsth. 66, 335–366.Google Scholar
[9] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin. (Statist. Lab., University of California, Berkeley), Springer, New York, pp. 61–110.Google Scholar
[10] Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170.Google Scholar
[11] Howard, C. D. and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Prob. 29, 577–623.Google Scholar
[12] Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285.Google Scholar
[13] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
[14] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
[15] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar
[16] Vahidi-Asl, M. Q. and Wierman, J. C. (1992). A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs, Vol. 2, John Wiley, New York, pp. 247–262.Google Scholar