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

We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelson-Fisk and Haggstrom established for this model a phase transition for the asymptotic linear speed (v) over bar of the walk. Namely, there exists some critical value lambda(c) > 0 such that (v) over bar > 0 if lambda is an element of (0, lambda(c)) and (v) over bar = 0 if lambda >= lambda(c). We show that the speed (v) over bar is continuous in lambda on (0, infinity) and differentiable on (0, lambda(c)/2). Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of (v) over bar on (0, lambda(c)/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 lambda >= lambda(c)/2.