## Electronic Communications in Probability

### Bernoulli and self-destructive percolation on non-amenable graphs

#### Abstract

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement  of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as $p\searrow p_c$. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.

#### Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 40, 6 pp.

Dates
Accepted: 12 July 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316742

Digital Object Identifier
doi:10.1214/ECP.v19-2611

Mathematical Reviews number (MathSciNet)
MR3233202

Zentralblatt MATH identifier
1310.60137

Rights

#### Citation

Ahlberg, Daniel; Sidoravicius, Vladas; Tykesson, Johan. Bernoulli and self-destructive percolation on non-amenable graphs. Electron. Commun. Probab. 19 (2014), paper no. 40, 6 pp. doi:10.1214/ECP.v19-2611. https://projecteuclid.org/euclid.ecp/1465316742

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