## The Annals of Probability

### Infinite clusters in dependent automorphism invariant percolation on trees

Olle Häggström

#### Abstract

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 information

Source
Ann. Probab., Volume 25, Number 3 (1997), 1423-1436.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404518

Digital Object Identifier
doi:10.1214/aop/1024404518

Mathematical Reviews number (MathSciNet)
MR1457624

Zentralblatt MATH identifier
0895.60098

#### Citation

Häggström, Olle. Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab. 25 (1997), no. 3, 1423--1436. doi:10.1214/aop/1024404518. https://projecteuclid.org/euclid.aop/1024404518

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