GIBBS MEASURES OF THE ISING MODEL ON A CAYLEY TREE
Keywords:
Keywords: Ising model, Cayley tree, Gibbs measures, statistical mechanics, probability theory, spin systems, boundary conditions, phase transition, critical temperature, Hamiltonian, equilibrium states, recursive equations, magnetization, thermodynamic limit.Abstract
Abstract: This thesis examines the Gibbs measures of the Ising model on a Cayley tree, focusing on their mathematical properties, existence, uniqueness, and relationship with phase transitions. The Ising model is one of the fundamental models in statistical mechanics and probability theory. It describes systems consisting of interacting spins that can take one of two possible values. The Cayley tree provides a useful mathematical structure for investigating these systems because its branching geometry allows the analysis of spin interactions through recursive equations. Particular attention is given to the Hamiltonian of the Ising model, the definition of Gibbs measures, boundary conditions, and the influence of temperature on equilibrium states. The thesis also discusses the role of critical temperature in the appearance of multiple translation-invariant Gibbs measures. The study demonstrates that the structure of the Cayley tree and the choice of boundary conditions are essential for understanding the equilibrium behavior of the Ising model. These mathematical results contribute to the broader study of statistical mechanics, probabilistic models, and complex interacting systems.
