The number of open paths in oriented percolation

Abstract

We study the number $N_{n}$ of open paths of length $n$ in supercritical oriented percolation on $\mathbb{Z}^{d}\times\mathbb{N}$, with $d\ge1$, and we prove the existence of the connective constant for the supercritical oriented percolation cluster: on the percolation event $\{\inf N_{n}>0\}$, $N_{n}^{1/n}$ almost surely converges to a positive deterministic constant.

The proof relies on the introduction of adapted sequences of regenerating times, on subadditive arguments and on the properties of the coupled zone in supercritical oriented percolation. This global convergence result can be deepened to give directional limits and can be extended to more general random linear recursion equations known as linear stochastic evolutions.