Branching processes, and random-cluster measures on trees

Grimmett, Geoffrey, and Janson, Svante. "Branching processes, and random-cluster measures on trees." Journal of the European Mathematical Society 007.2 (2005): 253-281. <http://eudml.org/doc/277635>.

@article{Grimmett2005,
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 subtree?},
author = {Grimmett, Geoffrey, Janson, Svante},
journal = {Journal of the European Mathematical Society},
keywords = {branching process; random-cluster measure; mean-field model; mean-field model},
language = {eng},
number = {2},
pages = {253-281},
publisher = {European Mathematical Society Publishing House},
title = {Branching processes, and random-cluster measures on trees},
url = {http://eudml.org/doc/277635},
volume = {007},
year = {2005},
}

TY - JOUR
AU - Grimmett, Geoffrey
AU - Janson, Svante
TI - Branching processes, and random-cluster measures on trees
JO - Journal of the European Mathematical Society
PY - 2005
PB - European Mathematical Society Publishing House
VL - 007
IS - 2
SP - 253
EP - 281
AB - 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 subtree?
LA - eng
KW - branching process; random-cluster measure; mean-field model; mean-field model
UR - http://eudml.org/doc/277635
ER -