Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

Abstract

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on ℤd. That is, for each d ≥ 2, and for any supercritical parameter p > pc, 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 subballistic 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. © 2013 Wiley Periodicals, Inc.

Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

“…On Galton-Watson trees with leaves (see [2,3,10,13]) or on supercritical percolation clusters (see [5,8,16]), the speed is certainly not increasing since it eventually vanishes. In these models, the slowdown of the walk can be explained by the presence of dead ends in the environment, which act as powerful traps.…”

“…Let us explain what the difficulty of this problem is since, indeed, questions about the speed of random walks in random environments can be subtle (see [17,18,20] for general reviews of the subject). On Galton-Watson trees with leaves (see [2,3,10,13]) or on supercritical percolation clusters (see [5,8,16]), the speed is certainly not increasing since it eventually vanishes. In these models, the slowdown of the walk can be explained by the presence of dead ends in the environment, which act as powerful traps.…”

Biased Random Walk on the Trace of Biased Random Walk on the Trace of …

“…and hence we obtain from the finiteness of the moments of the geometric distribution and (15) that sup n≥0 Γ 2 q < ∞ for arbitrary q ≥ 1. Since the latter part of the argument also implies (13), the proof is complete.…”

Conditional decoupling of random interlacements

“…[1,16,7]. In particular, one remarkable fact is the following: the walk is ballistic (transient and with positive speed) in the direction of the drift if β > 0 is small enough; however, it moves only sublinearly fast (its displacement is only of order t a by time t with a ∈ (0, 1), as proved in [8]) for large values of β.…”