The Annals of Probability

Anchored expansion, percolation and speed

Dayue Chen, Yuval Peres, and Gábor Pete

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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 Virág [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.

Article information

Ann. Probab., Volume 32, Number 4 (2004), 2978-2995.

First available in Project Euclid: 8 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B99: None of the above, but in this section
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60J15

Cayley graphs percolation random walks speed anchored expansion constant


Chen, Dayue; Peres, Yuval; Pete, Gábor. Anchored expansion, percolation and speed. Ann. Probab. 32 (2004), no. 4, 2978--2995. doi:10.1214/009117904000000586.

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