The Annals of Probability
- Ann. Probab.
- Volume 34, Number 6 (2006), 2344-2364.
Percolation on nonunimodular transitive graphs
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.
Ann. Probab., Volume 34, Number 6 (2006), 2344-2364.
First available in Project Euclid: 13 February 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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