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The Number of Open Paths in an Oriented ρ-Percolation Model

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Abstract

We study the asymptotic properties of the number of open paths of length n in an oriented ρ-percolation model. We show that this number is e nα(ρ)(1+o(1)) as n→∞. The exponent α is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitly computed in some range of the parameters. Moreover, in a restricted range of the parameters, we even show that the number of such paths is n −1/2 We nα(ρ)(1+o(1)) for some nondegenerate random variable W. We build on connections with the model of directed polymers in random environment, and we use techniques and results developed in this context.

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Authors and Affiliations

  1. UFR de Mathématiques, case 7012, Université Paris 7, 2, place Jussieu, 75251, Paris Cedex 05, France

    Francis Comets

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, rua do Matão 1010, 05508–090, São Paulo, SP, Brasil

    Serguei Popov

  3. Departamento de Estatística, Instituto de Matemática, Estatística e Computação Científica, Universidade de Campinas, Caixa Postal 6065, 13083–970, Campinas, SP, Brasil

    Marina Vachkovskaia

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  1. Francis Comets
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  2. Serguei Popov
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  3. Marina Vachkovskaia
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Correspondence to Francis Comets.

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Comets, F., Popov, S. & Vachkovskaia, M. The Number of Open Paths in an Oriented ρ-Percolation Model. J Stat Phys 131, 357–379 (2008). https://doi.org/10.1007/s10955-008-9506-2

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  • DOI: https://doi.org/10.1007/s10955-008-9506-2

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