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Wed 21 Aug 2019, 3:00pm SPECIAL
Probability Seminar
MATH 126
Supercritical percolation on nonamenable graphs
MATH 126
Wed 21 Aug 2019, 3:00pm-4:00pm


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 almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997).
  2. Various observables including the percolation probability and the truncated susceptibility are analytic functions of p throughout the entire supercritical phase.
Joint work with Jonathan Hermon.