Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs

TL;DR

This paper proves that on quasi-transitive graphs, in the subcritical phase, the expected cluster size is finite and the cluster size distribution decays exponentially, extending classical results to more general graphs.

Contribution

It extends known theorems about cluster size finiteness and exponential decay from transitive to quasi-transitive graphs in percolation theory.

Findings

Expected cluster size is finite in the subcritical regime.

Cluster size distribution decays exponentially.

Results apply to both edge and site percolation, including long-range.

Abstract

We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the Aizenman & Barsky method.