Martedi' 15 Ottobre 2019 ORE 14:30 Dipartimento di Matematica e Fisica Universita' degli Studi Roma Tre Largo San Leonardo Murialdo,1 - Pal.C - Aula 211 Speaker: Jonathan Hermon Titolo: Anchored expansion in supercritical percolation on nonamenable graphs. Abstract: 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.