2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161548We 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.22 pagesProbabilityMathematical Physics60K35, 82B43text