We study dependent bond percolation on the homogeneous tree $T_n$
of order $n \geq 2$ under the assumption of automorphism invariance. Excluding
a trivial case, we find that the number of infinite clusters a.s. is either 0
or $\infty$. Furthermore, each infinite cluster a.s. has either 1, 2 or
infinitely many topological ends, and infinite clusters with infinitely many
topological ends have a.s. a branching number greater than 1. We also show that
if the marginal probability that a single edge is open is at least $2/(n + 1)$,
then the existence of infinite clusters has to have positive probability.
Several concrete examples are considered.
@article{1024404518,
author = {H\"aggstr\"om, Olle},
title = {Infinite clusters in dependent automorphism invariant
percolation on trees},
journal = {Ann. Probab.},
volume = {25},
number = {4},
year = {1997},
pages = { 1423-1436},
language = {en},
url = {http://dml.mathdoc.fr/item/1024404518}
}
Häggström, Olle. Infinite clusters in dependent automorphism invariant
percolation on trees. Ann. Probab., Tome 25 (1997) no. 4, pp. 1423-1436. http://gdmltest.u-ga.fr/item/1024404518/