Subject of inquiry: Gibbs measures for q-eomponenl model and Potts model with competing interactions on a Cayley tree.
Aim of the inquiry: We study Gibbs measures and periodic ground states of the Potts and q-componcnt models with competing interactions on a Cayley tree.
Methods of the inquiry: Methods of contours on a Cayley tree, methods of Pirogov-Sinay theory, measure theory and contractive maps.
The results achieved and their novelty: The obtained results arc new. They consist of the following:
• For q-componcnt models on a Cayley tree contours and ground states arc constructed.
• For q-componcnt models, at sufficiently low temperatures, by a contour method on a Cayley tree existence of at least q different Gibbs measures is proved.
• For a Potts model with competing interactions on a Cayley tree the set of periodic ground states is constructed.
• It is shown that the Pcicrls’s condition is satisfied for the Hamiltonian of the Potts model.
• At sufficiently low temperatures, for the Potts model with competing interactions and three spins existence of at least three Gibbs measures is proved.
• On parameters of a model with the interaction radius two a sufficient conditions arc found under which the periodic configurations arc the ground states of this model.
Practical value: the results of the dissertation work have theoretical character. They can be applied in problems of statistical physics.
Sphere of usage: results of the work can be used in measure theory, theory of phase transitions, theory of probability, theoretical and mathematical physcs.
