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Branching Processes, and Random-Cluster Measures on Trees

Geoffrey Grimmett·Svante Janson
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Abstract

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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Geoffrey Grimmett, Svante Janson. 2004-10-13. Branching Processes, and Random-Cluster Measures on Trees. https://arxiv.org/abs/math/0410311

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