On the speed of biased random walk in translation invariant percolation

TL;DR

This paper demonstrates that in a specific translation invariant percolation model, the biased random walk exhibits a non-monotonic speed behavior contrary to the conjecture for i.i.d. percolation, with zero speed below a critical bias and positive speed above it.

Contribution

The paper provides a counterexample showing that the expected monotonic relationship between bias and speed does not hold in general translation invariant percolation models.

Findings

Speed is zero for bias below critical value

Speed becomes positive for bias above critical value

Contradicts previous conjectures for i.i.d. percolation

Abstract

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 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.