Translation-invariant generalized P-adic Gibbs measures for the Ising model on Cayley trees

In the present paper, we analyze the occurrence of a phase transition by the use of a p-adic probability theory. To carry out this research, generalized p-adic Gibbs measures are investigated, for the Ising model on the Cayley tree owing to the fact that this specific model has broad theoretical and practical applications. To explore translation-invariant generalized p-adic Gibbs measures, we describe the set of fixed points of the Ising-Potts mapping which appears by means of renormalization group. This description allows us to establish the phase transition. In the real setting, the phase transition yields the singularity of the limiting Gibbs measures. However, we show that the generalized p-adic Gibbs measures do not exhibit the mentioned type of singularity; this phenomena is called a strong phase transition.