We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size $n$ decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.
"On the cluster size distribution for percolation on some general graphs." Rev. Mat. Iberoamericana 26 (2) 529 - 550, June, 2010.