I consider p-Bernoulli bond percolation on transitive, nonamenable, planar graphs with one end and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i.e. , where is the critical probability and -the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in ∂ℍ². This result is similar to some results in [Lal].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-6,
author = {Jan Czajkowski},
title = {Clusters in middle-phase percolation on hyperbolic plane},
journal = {Banach Center Publications},
volume = {95},
year = {2011},
pages = {99-113},
zbl = {1256.60035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-6}
}
Jan Czajkowski. Clusters in middle-phase percolation on hyperbolic plane. Banach Center Publications, Tome 95 (2011) pp. 99-113. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc96-0-6/