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

@article{Comets2007TheNO,
  title={The Number of Open Paths in an Oriented $\rho$-Percolation Model},
  author={Francis Comets and Serguei Yu. Popov and M. Vachkovskaia},
  journal={Journal of Statistical Physics},
  year={2007},
  volume={131},
  pages={357-379}
}
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 enα(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−1/2Wenα(ρ)(1+o(1)) for some nondegenerate random variable W. We build… 

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