2011/03/07 by Alexander Fribergh, Fribergh, Alexander, Alan Hammond +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1103.1371
openalex publication_date 2011/03/07 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We prove the sharpness of the phase transition for speed in the biased random\nwalk on the supercritical percolation cluster on Zd. That is, for each d at\nleast 2, and for any supercritical parameter p > pc, we prove the existence of\na critical strength for the bias, such that, below this value, the speed is\npositive, and, above the value, it is zero. We identify the value of the\ncritical bias explicitly, and, in the sub-ballistic regime, we find the\npolynomial order of the distance moved by the particle. Each of these\nconclusions is obtained by investigating the geometry of the traps that are\nmost effective at delaying the walk. A key element in proving our results is to\nunderstand that, on large scales, the particle trajectory is essentially\none-dimensional; we prove such a `dynamic renormalization' statement in a much\nstronger form than was previously known.\n