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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 $\alpha$ isdeterministic, it can be expressed in terms of the free energy of a polymermodel, 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 numberof such paths is $n^{-1/2} W e^{n\alpha(\rho)}(1+o(1))$ for some nondegeneraterandom variable $W$. We build on connections with the model of directedpolymers in random environment, and we use techniques and results developed inthis context.

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