On the Cluster Size Distribution for Percolation on Some General Graphs

Abstract : 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.

[Joint work with Jeffrey Steif and \'{A}d\'{a}m Tim\'{a}r]