On the speed of biased random walk in translation invariant percolation

For biased random walk on the infinite cluster in supercritical i.i.d. percolation on Z, where the bias of the walk is quantified by a parameter β > 1, it has been conjectured (and partly proved) that there exists a critical value βc > 1 such that the walk has positive speed when β < βc and speed zero when β > βc. In this paper, biased random walk on the infinite cluster of a certain translation invariant percolation process on Z is considered. The example is shown to exhibit the opposite behavior to what is expected for i.i.d. percolation, in the sense that it has a critical value βc such that, for β < βc, the random walk has speed zero, while, for β > βc, the speed is positive. Hence the monotonicity in β that is part of the conjecture for i.i.d. percolation cannot be extended to general translation invariant percolation processes.