Directed random walk on the backbone of an oriented percolation cluster - Birkner, Matthias; Cerny, Jiri; Depperschmidt, Andrej; Gantert, Nina (Electronic Journal of Probability, 2013)
We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the "ancestral lineage" of an individual in the stationary discrete-time contact process. 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. Furthermore, 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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Keywords: Central limit theorem in random environment; Dynamical random environment; Oriented percolation; Random walk; Supercritical cluster