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.
Publié le : 2006-11-14
Classification:
Nonunimodular,
percolation,
critical percolation,
light clusters,
heavy clusters,
60K35,
82B43,
60B99,
60C05
@article{1171377446,
author = {Tim\'ar, \'Ad\'am},
title = {Percolation on nonunimodular transitive graphs},
journal = {Ann. Probab.},
volume = {34},
number = {1},
year = {2006},
pages = { 2344-2364},
language = {en},
url = {http://dml.mathdoc.fr/item/1171377446}
}
Timár, Ádám. Percolation on nonunimodular transitive graphs. Ann. Probab., Tome 34 (2006) no. 1, pp. 2344-2364. http://gdmltest.u-ga.fr/item/1171377446/