Gibbs Measures and Phase Transitions in Potts and Beach Models
The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statis...
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| Format: | Doctoral Thesis |
| Language: | English |
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KTH, Matematik
2004
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| Online Access: | http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3837 http://nbn-resolving.de/urn:isbn:91-7283-849-3 |
| Summary: | The theory of Gibbs measures belongs to the borderlandbetween statistical mechanics and probability theory. In thiscontext, the physical phenomenon of phase transitioncorresponds to the mathematical concept of non-uniqueness for acertain type of probability measures. The most studied model in statistical mechanics is thecelebrated Ising model. The Potts model is a natural extensionof the Ising model, and the beach model, which appears in adifferent mathematical context, is in certain respectsanalogous to the Ising model. The two main parts of this thesisdeal with the Potts model and the beach model,respectively. For theq-state Potts model on an infinite lattice, there areq+1 basic Gibbs measures: one wired-boundary measure foreach state and one free-boundary measure. For infinite trees,we construct "new" invariant Gibbs measures that are not convexcombinations of the basic measures above. To do this, we use anextended version of the random-cluster model together withcoupling techniques. Furthermore, we investigate the rootmagnetization as a function of the inverse temperature.Critical exponents to this function for different parametercombinations are computed. The beach model, which was introduced by Burton and Steif,has many features in common with the Ising model. We generalizesome results for the Ising model to the beach model, such asthe connection between phase transition and a certain agreementpercolation event. We go on to study aq-state variant of the beach model. Using randomclustermodel methods again we obtain some results on where in theparameter space this model exhibits phase transition. Finallywe study the beach model on regular infinite trees as well.Critical values are estimated with iterative numerical methods.In different parameter regions we see indications of both firstand second order phase transition. Keywords and phrases:Potts model, beach model,percolation, randomcluster model, Gibbs measure, coupling,Markov chains on infinite trees, critical exponent. |
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