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

Abstract

We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and H\"aggstr\"om 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.