Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster

2021/05/19 by Stein Andreas Bethuelsen, Bethuelsen, Stein Andreas, Matthias Birkner +5 · 1 citation
Mathematics · Physics and Astronomy · #60J10 #60K35 #60K37 #82B43 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2105.09030

openalex publication_date 2021/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+1 with d ≥ 3 being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.

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