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Geometric Functional Analysis and Probability Seminar

ThursdayJul 25, 201913:30
Geometric Functional Analysis and Probability SeminarRoom 155
Speaker:Jonathan Hermon Title:Anchored expansion in supercritical percolation on nonamenable graphs.Abstract:opens in new windowin html    pdfopens in new window

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 Tom Hutchcroft.