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

TL;DR

This paper analyzes the critical bias in a one-dimensional percolation model, showing the walk's displacement scales as n/log n at criticality and providing detailed tail estimates for regeneration times.

Contribution

It establishes the order of displacement at critical bias and offers new tail estimates for regeneration times in biased random walks on percolation models.

Findings

Displacement at critical bias is of order n/log n.

Tail estimates for regeneration times are derived.

Fluctuation orders are characterized in different phases.

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.