Biased random walk in a one-dimensional percolation model

We consider random walk with a nonzero bias to the right, on the infinite cluster in the following percolation model: take i.i.d. bond percolation with retention parameter p$p$ on the so-called infinite ladder, and condition on the event of having a bi-infinite path from −∞$-\infty$ to ∞$\infty$. The random walk is shown to be transient, and to have an asymptotic speed to the right which is strictly positive or zero depending on whether the bias is below or above a certain critical value which we compute explicitly.