Anchored expansion, percolation and speed

Details Anchored expansion, percolation and speed Anchored expansion, percolation and speed

2003-03-26

arxiv.org/abs/math/0303321v4

Annals of Probability 2004, Vol. 32, No. 4, 2978-2995

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000586 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000586

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56-84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virag [Geom. Funct. Anal. 10 (2000) 1588-1605]. We prove that if G has a positive anchored expansion constant, then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1; a better estimate for the parameters p where this holds is in the Appendix. We also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail. We then study a simple random walk in the infinite percolation cluster in Cayley graphs of certain amenable groups known as ``lamplighter groups.'' We prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter p. For p large enough, we also establish the converse.

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