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 exp(nα(ρ)(1+o(1)) as n→∞. The exponent $\alpha$ 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 1/√n W exp(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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