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

Abstract

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.

30 pages, 2 figures

References in corpus (1)

• Majorizing multiplicative cascades for directed polymers in random media

Cited by in corpus (4)

• Minimal supporting subtrees for the free energy of polymers on disordered trees
• Limiting results for the free energy of directed polymers in random environment with unbounded jumps
• Existence of an intermediate phase for oriented percolation
• Rate of convergence for polymers in a weak disorder