Revista Matemática Iberoamericana

On the cluster size distribution for percolation on some general graphs

Antar Bandyopadhyay , Jeffrey Steif , and Ádám Timár

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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.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 529-550.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1275671310

Mathematical Reviews number (MathSciNet)
MR2677006

Zentralblatt MATH identifier
1203.60142

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Keywords
amenability Cayley graphs cluster size distribution exponential decay percolation sub-exponential decay

Citation

Bandyopadhyay, Antar; Steif, Jeffrey; Timár, Ádám. On the cluster size distribution for percolation on some general graphs. Rev. Mat. Iberoamericana 26 (2010), no. 2, 529--550. https://projecteuclid.org/euclid.rmi/1275671310


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