On the speed of biased random walk in translation invariant percolation

Abstract

For biased random walk on the infinite cluster in supercritical i.i.d.\ percolation on ${\mathbb{Z}}^2$, 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 ${\mathbb{Z}}^2$ 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 ${\beta }_c$ 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.