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.