Revista Matemática Iberoamericana
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On the cluster size distribution for percolation on some general graphsAntar Bandyopadhyay, Jeffrey E. Steif and Ádám Timár (1) Indian Statistical Institute, New Delhi, India
(2) Chalmers University of Technology, Gothenburg, Sweden
(3) Universität Bonn, Germany
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.
Keywords: Amenability, Cayley graphs, cluster size distribution, exponential decay, percolation, sub-exponential decay
Bandyopadhyay Antar, Steif Jeffrey, Timár Ádám: On the cluster size distribution for percolation on some general graphs. Rev. Mat. Iberoamericana 26 (2010), 529-550. doi: 10.4171/RMI/608