Mathematical Models Described by Gibbs Measures and Phase Transitions on Cayley Tree

Abstract

In this thesis, five models are studied on lattice spin systems which are considered in statistical mechanics. A probability measure that's called Gibbs Measure is defined for these models on Cayley tree (or Bethe lattice) and the existence of phase transition is proved by using two approaches: Markov Random field method and Partition Function method. The first and second models are related to Ising-Vannimenus Model with three different competing interactions on semi-infinite Cayley tree, the analysis of these models is done by using Markov Random field method making use of the Kolmogorov consistency conditions. To achieve this, we constructed the set of recurrence equations that corresponds to the mentioned models and satisfied consistency condition, then we analyzed these equations and determined the conditions on the temperature and the coupling constants in which the phase transition exists. The third set of models (3 models) are called Potts Models, which are a generalization for Ising model, we have constructed the recurrence equations for these models by using Partition function method. In the same way as the previous models, the phase transition conditions are determined. Since we got a high order polynomials with complicated factors, we used Wolfarm Mathematica for equations analysis.