On set of p-adic Gibbs measures for the countable state 1D SOS model

Previous studies mainly focused on the p-adic Potts model with countable spin values, demonstrating that this model has only one p-adic Gibbs measure. Furthermore, it was shown that the model exhibits a phase transition in the set of generalized Gibbs measures. A challenge remained to find a countable spin p-adic model where the set of all p-adic Gibbs measures would include at least two elements. In this paper, we have examined the one-dimensional p-adic SOS model and demonstrated that the set of all p-adic Gibbs measures has continuum cardinality. This phenomenon has not been observed in countable state p-adic Potts models. Our result addresses the aforementioned problem affirmatively. To establish this finding, we employed a p-adic dynamical system related to the p-adic Gibbs measure through the renormalization group technique. Our analysis confirms the occurrence of a phase transition for the model in question.