  We show that on a Cayley graph of a nonamenable group, almost surely the
infinite clusters of Bernoulli percolation are transient for simple random
walk, that simple random walk on these clusters has positive speed, and that
these clusters admit bounded harmonic functions. A principal new finding on
which these results are based is that such clusters admit invariant random
subgraphs with positive isoperimetric constant.
  We also show that percolation clusters in any amenable Cayley graph almost
surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a
Cayley graph admitting nonconstant harmonic Dirichlet functions, almost surely
the infinite clusters of $p$-Bernoulli percolation also have nonconstant
harmonic Dirichlet functions when $p$ is sufficiently close to 1. Many
conjectures and questions are presented.
