On the cluster size distribution for percolation on some general graphs

Antar Bandyopadhyay; Jeffrey E. Steif; Ádám Timár

doi:10.4171/rmi/608

JournalsrmiVol. 26, No. 2pp. 529–550

On the cluster size distribution for percolation on some general graphs

Antar Bandyopadhyay
Indian Statistical Institute, New Delhi, India
Jeffrey E. Steif
Chalmers University of Technology, Gothenburg, Sweden
Ádám Timár
Universität Bonn, Germany

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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.

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. 529–550

DOI 10.4171/RMI/608