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(2), where the bias of the walk is quantified by a parameter beta > 1, it has been conjectured (and partly proved) that there exists a critical value beta(c) > 1 such that the walk has positive speed when beta < beta(c) and speed zero when beta > beta(c). In this paper, biased random walk on the infinite cluster of a certain translation invariant percolation process on Z(2) is considered. The example is shown to exhibit the opposite behavior to what is expected for Lid. percolation, in the sense that it has a critical value beta(c) such that, for beta > beta(c), the random walk has speed zero, while, for beta > beta(c), the speed is positive. Hence the monotonicity in beta that is part of the conjecture for i.i.d. percolation cannot be extended to general translation invariant percolation processes.