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Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster
Author(s) -
Fribergh Alexander,
Hammond Alan
Publication year - 2014
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21491
Subject(s) - supercritical fluid , percolation (cognitive psychology) , mathematics , random walk , statistical physics , phase transition , cluster (spacecraft) , directed percolation , condensed matter physics , statistics , critical exponent , physics , thermodynamics , computer science , biology , neuroscience , programming language
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 ≥ 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 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.