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. Axelsson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value ${\lambda }_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $\lambda \in (0,{\lambda }_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $\lambda >{\lambda }_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $\lambda$ on the interval $(0,{\lambda }_{\mathrm{c}})$ and differentiable on $(0,{\lambda }_{\mathrm{c}}/2)$. Moreover, we characterize the derivative as a covariance. For the proof of the differentiability of $\overline{\mathrm{v}}$ on $(0,{\lambda }_{\mathrm{c}}/2)$, 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 $\lambda \ge {\lambda }_{\mathrm{c}}/2$.

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