Directed random walk on the backbone of an oriented percolation cluster (Q388968)

The authors consider a directed random walk on the backbone of the infinite open cluster generated by a supercritical oriented percolation random field on \(\mathbb{Z}^d \times \mathbb{Z}\), i.e., on the collection of all sites of the lattice \(\mathbb{Z}^d \times \mathbb{Z}\) connected to infinity by an open path. This process is equivalent to the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process on \(\mathbb{Z}^d\). Using a regeneration approach, they prove a law of large numbers and a central limit theorem for the random walk, the last in the annealed case (i.e., taking the average w.r.t.\ the realizations of the cluster). They also prove a central limit theorem for almost all the realizations of the cluster analysing the joint renewals of two independent random walks on the same cluster.