# Branching processes, and random-cluster measures on trees

Geoffrey Grimmett; Svante Janson

Journal of the European Mathematical Society (2005)

- Volume: 007, Issue: 2, page 253-281
- ISSN: 1435-9855

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topGrimmett, 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 -

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