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Oriented Percolation in One–dimensional 1/|x−y|2 Percolation Models

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Abstract

We consider independent edge percolation models on ℤ, with edge occupation probabilities

$$p_{\{x,y\}}=\left\{\begin{array}{l@{\quad}l}p&\mathrm{if}\ \vert x-y\vert =1,\\1-\exp \{-\beta /\left\vert x-y\right\vert ^{2}\}&\mathrm{otherwise.}\end{array}\right.$$

We prove that oriented percolation occurs when β>1 provided p is chosen sufficiently close to 1, answering a question posed in Newman and Schulman (Commun. Math. Phys. 104:547, 1986). The proof is based on multi-scale analysis.

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Authors and Affiliations

  1. Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970, São Paulo, SP, Brasil

    D. H. U. Marchetti

  2. Centrum Wiskunde & Informatica (CWI), Science Park 123, 1098 XG, Amsterdam, Netherlands

    V. Sidoravicius

  3. Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Jardim Botânico, 22460-320, Rio de Janeiro, RJ, Brasil

    V. Sidoravicius

  4. Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brasil

    M. E. Vares

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  1. D. H. U. Marchetti
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  2. V. Sidoravicius
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  3. M. E. Vares
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Correspondence to D. H. U. Marchetti.

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Marchetti, D.H.U., Sidoravicius, V. & Vares, M.E. Oriented Percolation in One–dimensional 1/|x−y|2 Percolation Models. J Stat Phys 139, 941–959 (2010). https://doi.org/10.1007/s10955-010-9966-z

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  • DOI: https://doi.org/10.1007/s10955-010-9966-z

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