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Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Part of:Special processesEquilibrium 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 H λ 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(H λ; 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 theory82B43: Percolation

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 1 , March 2011, pp. 154 - 172
Copyright
Copyright © Applied Probability Trust 2011 

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