# Seminari del Dipartimento

 Probabilita'Anchored expansion in supercritical percolation on nonamenable graphs.Jonathan Hermon15-10-2019 - 14:30Largo San Leonardo Murialdo,1 - Pal.C - Aula 211 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