Open Access
2023 Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster
Stein Andreas Bethuelsen, Matthias Birkner, Andrej Depperschmidt, Timo Schlüter
Author Affiliations +
Electron. J. Probab. 28: 1-54 (2023). DOI: 10.1214/23-EJP924


We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+1 with d3 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.

Funding Statement

SAB, MB and TS were supported by the DFG SPP Priority Programme 1590 “Probabilistic Structures in Evolution”, SAB through grant GA 582/7-2 and MB and TS through grants BI 1058/3-2. We also thank DFG Priority Programme 2265 Random Geometric Systems for support through Project 443869423.


We would like to thank Noam Berger for many useful discussions and for various versions of [2]. We thank Nina Gantert for useful discussions and for her constant interest in this work. We thank the anonymous referee for their careful reading and insightful comments which helped to improve the presentation and correct several inaccuracies in proof details.


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Stein Andreas Bethuelsen. Matthias Birkner. Andrej Depperschmidt. Timo Schlüter. "Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster." Electron. J. Probab. 28 1 - 54, 2023.


Received: 16 July 2021; Accepted: 16 February 2023; Published: 2023
First available in Project Euclid: 1 March 2023

Digital Object Identifier: 10.1214/23-EJP924

Primary: 60J10 , 60K35 , 60K37 , 82B43

Keywords: environment viewed from the particle , Oriented percolation , quenched local limit theorem in random environment , random walk in dynamical random environment , supercritical cluster

Vol.28 • 2023
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