Geodesic
Branching processes, and random-cluster measures on trees
Journal of the European Mathematical Society, Tome 7 (2005) no. 2, pp. 253-281

Voir la notice de l'article provenant de la source EMS Press

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?
DOI : 10.4171/jems/28
Classification : 60-XX, 82-XX, 00-XX
Keywords: Branching process, random-cluster measure, mean-field model 
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Geoffrey R. Grimmett; Svante Janson. Branching processes, and random-cluster measures on trees. Journal of the European Mathematical Society, Tome 7 (2005) no. 2, pp. 253-281. doi: 10.4171/jems/28

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