Branching processes, and random-cluster measures on trees

  • Geoffrey R. Grimmett

    University of Cambridge, UK
  • Svante Janson

    Uppsala Universitet, Sweden


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 of a branching process. What is the probability that every infinite path of , beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?

Cite this article

Geoffrey R. Grimmett, Svante Janson, Branching processes, and random-cluster measures on trees. J. Eur. Math. Soc. 7 (2005), no. 2, pp. 253–281

DOI 10.4171/JEMS/28