Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 3944
author:  Dayue Chen and Yuval Peres 

title:  The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview 
keywords:  Cayley graphs, percolation, random walks, speed, anchored expansion constant. 
abstract: 
Benjamini, Lyons and Schramm (1999) considered properties
of an infinite graph
G
, and the simple random walk on it, that are
preserved by random perturbations. To address problems
raised by those authors, we study simple random walk on the
infinite percolation cluster in Cayley graphs of certain
amenable groups known as ``lamplighter groups''. We prove
that zero speed for 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. 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;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.

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reference:  Dayue Chen and Yuval Peres (2003), The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview , in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 3944 
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