On the Cluster Size Distribution for Percolation on Some General Graphs

TL;DR

This paper investigates the decay rate of cluster sizes in percolation on Cayley graphs and relates the positivity of this rate in supercritical regimes to the graph's amenability, offering insights into percolation behavior on various graphs.

Contribution

It establishes a connection between the exponential decay rate of cluster sizes and the amenability of the underlying graph, extending understanding of percolation on general graphs.

Findings

Exponential decay rate exists for Cayley graphs.

Positive decay rate in supercritical regime linked to non-amenability.

Provides criteria for cluster size distribution behavior on various graphs.

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.