Finch, Sam (2010) The random cluster model on the tree : Markov chains, random connections and boundary conditions. PhD thesis, University of Warwick.

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Abstract

The Fortuin-Kasteleyn random cluster model was adapted to the tree by Häggström [38] who considered a Gibbs specification corresponding to "wired boundary conditions" on the tree. Grimmett and Janson [36] generalized this idea by considering boundary conditions defined by equivalence relations on the set of rays of the tree. In this thesis we continue this project by defining a new object, a "random connection;" a type of random equivalence relation that allows us to redefine what is meant by a cluster of edges on the tree. Our definition is general enough to include Grimmett and Janson's boundary conditions. The random connection approach allows us to reconnect the random cluster model on the tree with Bernoulli bond percolation and we define two critical probabilities for bond percolation on a tree associated with each random connection that allow us to identify three behavioral phases of the associated random cluster model. We consider some examples of random connections defined by equivalence relations, including the "open" boundary conditions described in [36] where we are able to describe the behaviour of the random cluster model exactly and "Mandelbrot" bondary conditions described by a map from the boundary of a tree to the unit square that defines fractal percolation. In addition we adapt work of Zachary [66] to the wired random cluster model on a tree so as to prove a conjecture of Häggström concerning uniqueness of the Gibbs measure for large bond strengths.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Trees (Graph theory), Boundary value problems, Markov processes
Date:	September 2010
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Extent:	iv, 189 leaves : ill., charts
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/47385

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