On the Cluster Size Distribution for Percolation on Some General Graphs
  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. 
Antar Bandyopadhyay, Jeffrey Steif, Adam Timar