Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters

TL;DR

This paper proves that at the critical point, percolation on any quasi-transitive graph with exponential growth does not produce infinite clusters, extending previous results to new classes of graphs.

Contribution

It establishes the absence of infinite clusters at criticality for all quasi-transitive graphs with exponential growth, including amenable and nonunimodular cases.

Findings

Critical percolation has no infinite clusters on these graphs.

The result applies to both amenable and nonunimodular graphs.

It generalizes earlier findings to broader graph classes.

Abstract

We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.