M ay 2 00 3 Anchored Expansion , Percolation and Speed

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. 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 (2000). 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 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 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. showed that simple random walk on the infinite cluster of supercritical Bernoulli percolation in Z d is transient for d ≥ 3; in other words, in Euclidean lattices, transience is preserved when the whole lattice is replaced by an infinite percolation cluster. Benjamini, Lyons and Schramm (1999), abbreviated as BLS (1999) hereafter, initiated a systematic study of the properties of a transitive graph G that are preserved under random perturbations such as passing from G to an infinite percolation cluster. They conjectured that positivity of the speed for simple random walk is preserved, and proved this for nonamenable Cayley graphs. Our results (see Theorems 1.5 and 1.6 below) lend further support to this conjecture.