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On the Truncated Long-Range Percolation on Z 2

Part of:Equilibrium statistical mechanicsSpecial processes

Published online by Cambridge University Press:  14 July 2016

Bernardo Nunes Borges de Lima and
Artëm Sapozhnikov
Bernardo Nunes Borges de Lima*
Affiliation:
UFMG
Artëm Sapozhnikov*
Affiliation:
CWI
*
∗ Postal address: UFMG, Av. Antônio Carlos 6627, C.P. 702, 30123-970 Belo Horizonte, Brazil. Email address: bnblima@mat.ufmg.br
∗∗ Postal address: CWI, Kruislaan 413, NL-1098SJ Amsterdam, The Netherlands. Email address: artem.sapozhnikov@cwi.nl
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Abstract

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We consider an independent long-range bond percolation on Z 2. Horizontal and vertical bonds of length n are independently open with probability p_n ∈ [0, 1]. Given ∑n=1 ∞∏i=1 n (1 − p i ) < ∞, we prove that there exists an infinite cluster of open bonds of length less than or equal to N for some large but finite N. The result gives a partial answer to the truncation problem.

Keywords

Long-range percolationtruncationrenewal theoryrenormalizationmixed percolation

MSC classification

Secondary:60K35: Interacting random processes; statistical mechanics type models; percolation theory82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008, pp. 287 - 291
Copyright
Copyright © Applied Probability Trust 2008 

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