Electronic Journal of Probability

The speed of critically biased random walk in a one-dimensional percolation model

Jan-Erik Lübbers and Matthias Meiners

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We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z} ^d$, namely, for some critical value $\lambda _{\mathrm{c} }>0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm {v}} $ of the walk is strictly positive if the bias $\lambda $ is strictly smaller than $\lambda _{\mathrm{c} }$, whereas $\overline{\mathrm {v}} =0$ if $\lambda \geq \lambda _{\mathrm{c} }$.

We show that at the critical bias $\lambda = \lambda _{\mathrm{c} }$, the displacement of the random walk from the origin is of order $n/\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\mathbb{Z} ^d$.

Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 23, 29 pp.

Received: 10 August 2018
Accepted: 9 February 2019
First available in Project Euclid: 23 March 2019

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Digital Object Identifier

Primary: 82B43: Percolation [See also 60K35] 60K37: Processes in random environments

biased random walk critical bias ladder graph percolation

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Lübbers, Jan-Erik; Meiners, Matthias. The speed of critically biased random walk in a one-dimensional percolation model. Electron. J. Probab. 24 (2019), paper no. 23, 29 pp. doi:10.1214/19-EJP277. https://projecteuclid.org/euclid.ejp/1553306439

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