Phase transition for the p-adic Ising-Vannimenus model on the Cayley tree

In this present paper, we consider the p-adic Ising Vannimenus model on the Cayley tree of order two. A new measure-theoretical approach (in the p-adic sense) to investigate such a model is proposed. The main result of this paper is to establish the existence of the phase transition for the model. By the phase transition we mean the existence of at least two non-trivial p-adic quasi Gibbs measures, such that one is bounded and the second one is unbounded (note that in the p-adic probability, unlike a real setting, the probability measures could even be unbounded). To prove the main result, we investigate a nonlinear recurrence equation via the methods of p-adic analysis. Note that the methods used in the paper are not valid in a real setting.