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November 2006 Percolation on nonunimodular transitive graphs
Ádám Timár
Ann. Probab. 34(6): 2344-2364 (November 2006). DOI: 10.1214/009117906000000494


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.


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Ádám Timár. "Percolation on nonunimodular transitive graphs." Ann. Probab. 34 (6) 2344 - 2364, November 2006.


Published: November 2006
First available in Project Euclid: 13 February 2007

zbMATH: 1114.60083
MathSciNet: MR2294985
Digital Object Identifier: 10.1214/009117906000000494

Primary: 60K35 , 82B43
Secondary: 60B99 , 60C05

Keywords: Critical percolation , heavy clusters , light clusters , Nonunimodular , percolation

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 6 • November 2006
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