Sharpness for Inhomogeneous Percolation on Quasi-Transitive Graphs

TL;DR

This paper establishes the sharpness of the phase transition in inhomogeneous percolation on quasi-transitive graphs, showing finite expected cluster size and exponential decay of one-arm events in the subcritical regime.

Contribution

It extends the proof of phase transition sharpness to inhomogeneous quasi-transitive graphs, generalizing previous homogeneous results.

Findings

Expected cluster size is finite in the subcritical regime.

Probability of one-arm event decays exponentially in the subcritical regime.

Sharp phase transition holds for inhomogeneous quasi-transitive graphs.

Abstract

In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive graphs by Duminil-Copin and Tassion [Comm. Math. Phys., 2016], and the result generalizes previous results of Antunovi\'c and Veseli\'c [J. Stat. Phys., 2008] and Menshikov [Dokl. Akad. Nauk 1986].