Weakly periodic p-adic generalized Gibbs measures for some models

A central problem in statistical mechanics is the study of infinite-volume Gibbs (p-adic generalized Gibbs) measures corresponding to a given Hamiltonian. This problem includes studying phase transitions, which occur for a Hamiltonian if at least two distinct Gibbs measures exist. However, a complete analysis of the set of all Gibbs measures for a particular Hamiltonian is a difficult problem. In this context, weakly periodic Gibbs (p-adic generalized Gibbs) measures play an important role, as they generalize translation-invariant and q-periodic Gibbs (p-adic generalized Gibbs) measures. In this talk, the existence of weakly periodic p-adic generalized Gibbs measures on the Cayley tree was demonstrated. Such measures were described for the Ising model on the Cayley tree, their boundedness was examined, and the occurrence of phase transitions for the considered model was studied. Furthermore, weakly periodic p-adic generalized Gibbs measures for the Ising model with an external field on the Cayley tree corresponding to fixed normal subgroups were investigated. Finally, by analyzing translation-invariant p-adic generalized Gibbs measures for the Ising model with an external field on the Cayley tree of order three, the existence of phase transitions for this model was established.