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July 1997 Infinite clusters in dependent automorphism invariant percolation on trees
Olle Häggström
Ann. Probab. 25(3): 1423-1436 (July 1997). DOI: 10.1214/aop/1024404518


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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Olle Häggström. "Infinite clusters in dependent automorphism invariant percolation on trees." Ann. Probab. 25 (3) 1423 - 1436, July 1997.


Published: July 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0895.60098
MathSciNet: MR1457624
Digital Object Identifier: 10.1214/aop/1024404518

Primary: 60K35
Secondary: 05C05 , 60J80

Keywords: automorphism invariance , branching number , percolation , topological ends , trees

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 3 • July 1997
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