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The Asymptotic Size of the Largest Component in Random Geometric Graphs with Some Applications

Part of: Special processes Equilibrium statistical mechanics Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Ge Chen ,
Changlong Yao  and
Tiande Guo
Ge Chen*
Affiliation:
Chinese Academy of Sciences
Changlong Yao*
Affiliation:
Chinese Academy of Sciences
Tiande Guo*
Affiliation:
Chinese Academy of Sciences
*
∗ Postal address: National Center for Mathematics and Interdisciplinary Sciences and Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China. Email address: chenge@amss.ac.cn
∗∗ Postal address: Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, P. R. China. Email address: deducemath@126.com
∗∗∗ Postal address: School of Mathematical Science, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China. Email address: tdguo@ucas.ac.cn
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Abstract

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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.

Keywords

Random geometric graphpercolationlargest componentPoisson-Boolean modelnumber of open clusters

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60D05: Geometric probability and stochastic geometry 82B43: Percolation

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 307 - 324
Copyright
© Applied Probability Trust 

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