Let G be a transitive nonamenable graph, and consider supercritical Bernoulli bond percolation on G . We prove that the probability that the origin lies in a finite cluster of size n decays exponentially in n . We deduce that:

1. Every infinite cluster has anchored expansion (a relaxation of having positive Cheeger constant), and so is nonamenable in some weak sense. This answers positively a question of Benjamini, Lyons, and Schramm (1997).

2. Various observables, including the percolation probability and the truncated susceptibility (which was not even known to be finite!) are analytic functions of p throughout the entire supercritical phase.

3. A RW on an infinite cluster returns to the origin at time 2n with probability exp(-Θ(n^{1/3})).

Joint work with Tom Hutchcroft.