Branching Processes, and Random-Cluster Measures on Trees

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Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?

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Probability, Mathematical Physics, 60K35, 60J80, 82B20

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http://salesiana.dossiersoluciones.com/handle/123456789/72875

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0410311
 http://arxiv.org/abs/math/0410311

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arXiv