Revista Matemática Iberoamericana


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Volume 26, Issue 2, 2010, pp. 529–550
DOI: 10.4171/RMI/608

On the cluster size distribution for percolation on some general graphs

Antar Bandyopadhyay[1], Jeffrey E. Steif[2] and Ádám Timár[3]

(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