2018/08/09 by Lübbers, Jan-Erik, Meiners, Matthias
#60K37 #82B43 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1808.03171
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 ℤd, namely, for some critical value λc >0 of the bias, it holds that the asymptotic linear speed v of the walk is strictly positive if the bias λ is strictly smaller than λc, whereas v=0 if λ≥ λc. We show that at the critical bias λ= λc, 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 ℤd. 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.