  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.
