We show that for any Cayley graph, the probability (at any ) that the cluster of the origin has size 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.
Cite this article
Antar Bandyopadhyay, Jeffrey E. Steif, Ádám Timár, On the cluster size distribution for percolation on some general graphs. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 529–550DOI 10.4171/RMI/608