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October 2004 Anchored expansion, percolation and speed
Dayue Chen, Yuval Peres, Gábor Pete
Ann. Probab. 32(4): 2978-2995 (October 2004). DOI: 10.1214/009117904000000586

Abstract

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.

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Dayue Chen. Yuval Peres. Gábor Pete. "Anchored expansion, percolation and speed." Ann. Probab. 32 (4) 2978 - 2995, October 2004. https://doi.org/10.1214/009117904000000586

Information

Published: October 2004
First available in Project Euclid: 8 February 2005

zbMATH: 1069.60093
MathSciNet: MR2094436
Digital Object Identifier: 10.1214/009117904000000586

Subjects:
Primary: 60B99
Secondary: 60J15 , 60K35 , 60K37

Keywords: anchored expansion constant , Cayley graphs , percolation , Random walks , Speed

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 4 • October 2004
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