Branching Processes, and Random-Cluster Measures on Trees
Probability
2007-05-23 v1 Mathematical Physics
math.MP
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 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?
Keywords
Cite
@article{arxiv.math/0410311,
title = {Branching Processes, and Random-Cluster Measures on Trees},
author = {Geoffrey Grimmett and Svante Janson},
journal= {arXiv preprint arXiv:math/0410311},
year = {2007}
}