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Let $G$ be a connected, locally finite, transitive graph, and considerBernoulli bond percolation on $G$. We prove that if $G$ is nonamenable and $p >p_c(G)$ then there exists a positive constant $c_p$ such that \[\mathbf{P}_p(n\leq |K| < \infty) \leq e^{-c_p n}\] for every $n\geq 1$, where $K$ is thecluster of the origin. We deduce the following two corollaries:  1. Every infinite cluster in supercritical percolation on a transitivenonamenable graph has anchored expansion almost surely. This answers positivelya question of Benjamini, Lyons, and Schramm (1997).  2. For transitive nonamenable graphs, various observables including thepercolation probability, the truncated susceptibility, and the truncatedtwo-point function are analytic functions of $p$ throughout the supercriticalphase.

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