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

Journal article

We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least two, 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.

Authors: Fribergh, A; Hammond, A

• Alternative title: Phase transition of the speed
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Communications on Pure and Applied Mathematics
• Volume: 67
• Issue: 2
• Pages: 173-245
• Publication date: 2013-01-01
• Edition: Accepted Manuscript
• DOI: 10.1002/cpa.21491
• EISSN: 1097-0312
• ISSN: 0010-3640
• Language: English
• Keywords: random walk in random conductances; anisotropy; trap models; trap geometry; dynamic renormalization
• Subjects: Statistical mechanics,structure of matter (mathematics)