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July 2010 The growth of the infinite long-range percolation cluster
Pieter Trapman
Ann. Probab. 38(4): 1583-1608 (July 2010). DOI: 10.1214/09-AOP517

Abstract

We consider long-range percolation on ℤd, where the probability that two vertices at distance r are connected by an edge is given by p(r) = 1 − exp[−λ(r)] ∈ (0, 1) and the presence or absence of different edges are independent. Here, λ(r) is a strictly positive, nonincreasing, regularly varying function. We investigate the asymptotic growth of the size of the k-ball around the origin, $|\mathcal{B}_{k}|$, that is, the number of vertices that are within graph-distance k of the origin, for k → ∞, for different λ(r). We show that conditioned on the origin being in the (unique) infinite cluster, nonempty classes of nonincreasing regularly varying λ(r) exist, for which, respectively:

• $|\mathcal{B}_{k}|^{1/k}\to\infty$> almost surely;

• there exist 1 < a1 < a2 < ∞ such that $\lim_{k\to \infty}\mathbb{P}(a_{1}< |\mathcal{B}_{k}|^{1/k}< a_{2})=1$;

• $|\mathcal{B}_{k}|^{1/k}\to1$ almost surely.

This result can be applied to spatial SIR epidemics. In particular, regimes are identified for which the basic reproduction number, R0, which is an important quantity for epidemics in unstructured populations, has a useful counterpart in spatial epidemics.

Citation

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Pieter Trapman. "The growth of the infinite long-range percolation cluster." Ann. Probab. 38 (4) 1583 - 1608, July 2010. https://doi.org/10.1214/09-AOP517

Information

Published: July 2010
First available in Project Euclid: 8 July 2010

zbMATH: 1196.60171
MathSciNet: MR2663638
Digital Object Identifier: 10.1214/09-AOP517

Subjects:
Primary: 60K35 , 92D30
Secondary: 82B28

Keywords: Chemical distance , epidemics , Long-range percolation

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 4 • July 2010
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