We 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 the present authors earlier as a corollary of a 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.
Itai Benjamini. Russell Lyons. Yuval Peres. Oded Schramm. "Critical Percolation on Any Nonamenable Group has no Infinite Clusters." Ann. Probab. 27 (3) 1347 - 1356, July 1999. https://doi.org/10.1214/aop/1022677450