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.

**Date**: 7 May 2019, 12:00 (Tuesday, 2nd week, Trinity 2019)**Venue**: Mathematical Institute

Woodstock Road OX2 6GGSee location on maps.ox**Speaker**: Jonathan Hermon (University of Cambridge)**Organising department**: Department of Statistics**Organisers**: Christina Goldschmidt (Department of Statistics, University of Oxford), James Martin (Department of Statistics, University of Oxford)**Part of**: Probability seminar**Booking required?**: Not required**Audience**: Public- Editor: Christina Goldschmidt