Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let $G$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. We prove that if $G$ is nonamenable and $p > p_c(G)$thenthereexistsapositiveconstant$c_p$ such that $${\mathbf{P}}_p(n\le \mathrm{\mid }K\mathrm{\mid }<\mathrm{\infty })\le e^{-c_{p}n}$$ for every $n\ge 1$, where $K$ is the cluster of the origin. We deduce the following two corollaries:

1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
2. For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of $p$ throughout the supercritical phase.