Abstract
We consider a directed random walk on the backbone of the infinite cluster gener-
ated by supercritical oriented percolation, or equivalently the space-time embedding
of the “ancestral lineage” of an individual in the stationary discrete-time contact pro-
cess. We prove a law of large numbers and an annealed central limit theorem (i.e.,
averaged over the realisations of the cluster) using a regeneration approach. Fur-
thermore, we obtain a quenched central limit theorem (i.e. for almost any realisation
of the cluster) via an analysis of joint renewals of two independent walks on the same
cluster.
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