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(-Theta(n^{1/3})).


Joint work with Tom Hutchcroft.
org: STAUFFER Alexandre