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A note on two-dimensional truncated long-range percolation

Part of: Special processes

Published online by Cambridge University Press:  19 February 2016

M. Menshikov ,
V. Sidoravicius  and
M. Vachkovskaia
M. Menshikov*
Affiliation:
University of Durham
V. Sidoravicius*
Affiliation:
Instituto de Matemática Pura e Aplicada, Rio de Janeiro
M. Vachkovskaia*
Affiliation:
University of São Paulo
*
∗ Postal address: Department of Mathematical Sciences, South Road, Durham DH1 3LE, UK. Email address: mikhail.menshikov@durham.ac.uk
∗∗ Postal address: IMPA, Estr. Dona Castorina 110, Rio de Janeiro, Brazil.
∗∗∗ Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Mail Box 66.281, CEP 05315-970, São Paulo, SP, Brazil.
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Abstract

We prove that for a class of anisotropic long-range percolation models for which connection probabilities p<x,z> satisfy some regularity properties, and such that ∑z∈Z2p<x,z> = ∞, percolation still will occur even if we truncate all edges whose length exceeds some constant (which in this case depends on the family of connectivity probabilities {p<x,z>). We also present an example of dependent long-range percolation model for which this is not true.

Keywords

Percolationoriented percolationbranching process

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 912 - 929
Copyright
Copyright © Applied Probability Trust 2001 

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References

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