The number of open paths in an oriented $ρ$-percolation model

Details The number of open paths in an oriented $ρ$-percolation model The number of open paths in an oriented $ρ$-percolation model

2007-07-05

arxiv.org/abs/0707.0818v1

Journal of Statistical Physics, v. 131, p. 357-379, 2008

Mathematics

Probability

30 pages, 2 figures

Scientific paper

10.1007/s10955-008-9506-2

We study the asymptotic properties of the number of open paths of length $n$ in an oriented $\rho$-percolation model. We show that this number is $e^{n\alpha(\rho)(1+o(1))}$ as $n \to \infty$. The exponent $\alpha$ is deterministic, it can be expressed in terms of the free energy of a polymer model, and it can be explicitely 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 $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ 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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