The Annals of Probability

Neighboring clusters in Bernoulli percolation

Adám Timár

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Abstract

We consider Bernoulli percolation on a locally finite quasi-transitive unimodular graph and prove that two infinite clusters cannot have infinitely many pairs of vertices at distance 1 from one another or, in other words, that such graphs exhibit “cluster repulsion.” This partially answers a question of Häggström, Peres and Schonmann.

Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2332-2343.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1171377445

Digital Object Identifier
doi:10.1214/009117906000000485

Mathematical Reviews number (MathSciNet)
MR2294984

Zentralblatt MATH identifier
1112.60085

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]
Secondary: 60B99: None of the above, but in this section

Keywords
Cluster repulsion percolation nonamenable touching clusters

Citation

Timár, Adám. Neighboring clusters in Bernoulli percolation. Ann. Probab. 34 (2006), no. 6, 2332--2343. doi:10.1214/009117906000000485. https://projecteuclid.org/euclid.aop/1171377445


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References

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