Critical percolation on any nonamenable group has no infinite clusters (Q1568289)

The authors show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by them earlier as a corollary of the general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is a ``mass-transport'' method, which is a technique of averaging in nonamenable settings.