The Annals of Probability

Percolation on nonunimodular transitive graphs

Ádám Timár

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Abstract

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy clusters, this result has already been established, but it also follows from one of our results. We give a general necessary condition for nonunimodular graphs to have a phase with infinitely many heavy clusters. We present an invariant spanning tree with pc=1 on some nonunimodular graph. Such trees cannot exist for nonamenable unimodular graphs. We show a new way of constructing nonunimodular graphs that have properties more peculiar than the ones previously known.

Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2344-2364.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1171377446

Digital Object Identifier
doi:10.1214/009117906000000494

Mathematical Reviews number (MathSciNet)
MR2294985

Zentralblatt MATH identifier
1114.60083

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 60C05: Combinatorial probability

Keywords
Nonunimodular percolation critical percolation light clusters heavy clusters

Citation

Timár, Ádám. Percolation on nonunimodular transitive graphs. Ann. Probab. 34 (2006), no. 6, 2344--2364. doi:10.1214/009117906000000494. https://projecteuclid.org/euclid.aop/1171377446


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