On the Cluster Size Distribution for Percolation on Some General Graphs

2008/05/23 by Antar Bandyopadhyay, Bandyopadhyay, Antar, Jeffrey E. Steif +3
Mathematics · #60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Limits and Structures in Graph Theory #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.0805.3620

openalex publication_date 2008/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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