  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
(1999) in the Bernoulli percolation case and confirms a conjecture of Schonmann
(2001). We also prove that every infinite cluster of Bernoulli percolation on a
nonamenable quasi-transitive graph is transient almost surely.
