We study two sets of models: independent percolation models in half spaces Zᵈ⁻¹ x Z₊, and Ising/Potts models as well as the Fortuin-Kasteleyn (FK) random cluster models on branching planes T x Z, where Z is the one-dimensional lattice, Z₊ = {0,1,2,...} and T is a Bethe lattice. We prove that for independent percolation in half spaces, the infinite cluster is unique whenever it exists. For the Ising/Potts models on branching planes, there are (at least) two phase transitions; that is, there exist(s) a unique Gibbs state, tree-like nonunique Gibbs states or plane-like nonunique Gibbs states corresponding to high temperature, intermediate temperature or low temperature. In the low temperature plus phase, the plus infinite cluster is unique and it "traps" the space T x Z and prevents co-existence of the minus infinite cluster. For the FK random cluster models (which are dependent percolation models) on T x Z, the number of infinite (open) clusters may be zero, infinity or one depending on the value of p--the probability of each bond being open. This is an extension of Grimmett and Newman's results for independent percolation on T x Z. We also prove that both the independent percolation model and the FK random cluster models satisfy a finite island property when p is close to 1. Chapter 1 is an introduction. Chapter 2 contains the proof of the uniqueness theorem for independent percolation in half spaces. The proof utilizes only a large deviation estimate and translation invariance of the models along the hyperplane Zᵈ⁻¹ x {0}. The Ising/Potts models and the FK random cluster models on the branching planes are studied in Chapter 3. The methods are to use the FK representation of Ising/Potts systems as dependent percolation models to carry over Grimmett and Newman's results for independent percolation to the Ising/Potts models. However, in order to prove the plane-like behavior of the Ising/Potts models, the corresponding results for independent percolation are not sufficient and this led us to investigate independent percolation again and prove a new finite island property. Chapters 2 and 3 are independent. Readers with basic knowledge of percolation and Ising models can omit chapter 1 and read chapters 2 and 3 directly.
| Identifer | oai:union.ndltd.org:arizona.edu/oai:arizona.openrepository.com:10150/185296 |
| Date | January 1990 |
| Creators | Wu, Chuntao. |
| Contributors | Newman, Charles M., Faris, William G., Kennedy, Thomas G. |
| Publisher | The University of Arizona. |
| Source Sets | University of Arizona |
| Language | English |
| Detected Language | English |
| Type | text, Dissertation-Reproduction (electronic) |
| Rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. |
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