Critical Percolation on Any Nonamenable Group Has No Innnite Clusters

We show that independent percolation on any Cayley graph of a nonamenable group has no innnite components at the critical parameter. This result was obtained in Benjamini, Lyons, Peres, and Schramm (1997) 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. The main long-standing open question in percolation theory is to show that critical per-colation in Z d has no innnite components for all d 2. The work of Harris (1960) and Kesten (1980) established the two-dimensional case; Hara and Slade (1994) proved it for d 19. Recently, a study of percolation on other graphs, such as Cayley graphs, was initiated. The relevant deenitions appear below. Here is our main theorem. Theorem 1.1. Let X be a Cayley graph of a nitely generated nonamenable group, and consider either site or bond percolation on X. Then at the corresponding critical value p = p c , almost surely there is no innnite component. In particular, there are no innnite components for critical percolation on any lattice in hyperbolic space, nor on a (k-regular tree) Z. The latter was previously known when k 7 (Wu 1993). Wu's proof goes along the lines of the high-dimensional Euclidean proof and uses the triangle condition. The key tool in our proof is a technique of \mass transport". This tool is especially valuable in nonamenable settings, where it can sometimes replace the ergodic theorem for approximation of expectations by suitable spatial averages.