Heavy Bernoulli-percolation clusters are indistinguishable

Abstract

We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (Ann. Probab.27 (1999) 1809–1836) in the Bernoulli percolation case and confirms a conjecture of Schonmann (Comm. Math. Phys. 219 (2001) 271–322). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.