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. Axelsson-Fisk and H\"aggstr\"om 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 $\lambda \geq \lambda_{\mathrm{c}}/2$.