The Annals of Probability

Infinite clusters in dependent automorphism invariant percolation on trees

Olle Häggström

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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.

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

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 05C05: Trees 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Percolation trees automorphism invariance topological ends branching number


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

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