Regularity of the Speed of Biased Random Walk in a One-Dimensional Percolation Model

Abstract

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed v ¯ of the walk. Namely, there exists some critical value λc> 0 such that v ¯ > 0 if λ∈ (0 , λc) and v ¯ = 0 if λ≥ λc. We show that the speed v ¯ is continuous in λ on (0 , ∞) and differentiable on (0 , λc/ 2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of v ¯ on (0 , λc/ 2) , we require and prove a central limit theorem for the biased random walk. Additionally, we prove that the central limit theorem fails to hold for λ≥ λc/ 2.