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2019 The speed of critically biased random walk in a one-dimensional percolation model
Jan-Erik Lübbers, Matthias Meiners
Electron. J. Probab. 24(none): 1-29 (2019). DOI: 10.1214/19-EJP277

Abstract

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.

Citation

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Jan-Erik Lübbers. Matthias Meiners. "The speed of critically biased random walk in a one-dimensional percolation model." Electron. J. Probab. 24 1 - 29, 2019. https://doi.org/10.1214/19-EJP277

Information

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

zbMATH: 1419.82025
MathSciNet: MR3933202
Digital Object Identifier: 10.1214/19-EJP277

Subjects:
Primary: 60K37, 82B43

JOURNAL ARTICLE
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