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We prove the sharpness of the phase transition for speed in the biased randomwalk on the supercritical percolation cluster on Z^d. That is, for each d atleast 2, and for any supercritical parameter p > p_c, we prove the existence ofa critical strength for the bias, such that, below this value, the speed ispositive, and, above the value, it is zero. We identify the value of thecritical bias explicitly, and, in the sub-ballistic regime, we find thepolynomial order of the distance moved by the particle. Each of theseconclusions is obtained by investigating the geometry of the traps that aremost effective at delaying the walk. A key element in proving our results is tounderstand that, on large scales, the particle trajectory is essentiallyone-dimensional; we prove such a `dynamic renormalization' statement in a muchstronger form than was previously known.

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