Electronic Communications in Probability

Indicable groups and $p_c<1$

Aran Raoufi and Ariel Yadin

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A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z} $ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups.

The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 13, 10 pp.

Received: 30 June 2016
Accepted: 27 December 2016
First available in Project Euclid: 31 January 2017

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Zentralblatt MATH identifier

Primary: 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B26: Phase transitions (general)

percolation Cayley graphs phase transition

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Raoufi, Aran; Yadin, Ariel. Indicable groups and $p_c&lt;1$. Electron. Commun. Probab. 22 (2017), paper no. 13, 10 pp. doi:10.1214/16-ECP40. https://projecteuclid.org/euclid.ecp/1485831618

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