Phase transition for the speed of the biased random walk on the supercritical percolation cluster

TL;DR

This paper establishes a precise phase transition for the speed of biased random walks on supercritical percolation clusters, identifying a critical bias value and analyzing the walk's behavior in different regimes.

Contribution

It explicitly determines the critical bias for speed transition and analyzes the walk's geometry and delay traps, advancing understanding of biased walks on percolation clusters.

Findings

Existence of a critical bias separating positive speed and zero speed regimes.

Explicit identification of the critical bias value.

Polynomial order of displacement in the sub-ballistic regime.

Abstract

We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a `dynamic renormalization' statement in a much stronger form than was previously known.