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

Abstract

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.

Gibbs measures with memory of length 2 on an arbitrary-order Cayley tree

“…One of the main purposes of equilibrium statistical mechanics consists in describing all limit Gibbs distributions corresponding to a given Hamiltonian [9]. One of the methods used for the description of Gibbs measures on Cayley trees is Markov random field theory and recurrent equations of this theory [27,28,30,33,35,38,40]. The approach we use here is based on the theory of Markov random fields on trees and recurrent equations of this theory.…”

Phase transition and Gibbs measures of Vannimenus model on semi-infinite Cayley tree of order three

“…In [24] the authors have studied the problem of phase transition for models considered by Vannimenus [17]. Mukhamedov et al [29] have proved the existence of the phase transition for the Vannimenus model [17] in the p-adic setting. Ganikhodjaev [30] has considered the Ising model on the semi-infinite Cayley tree of second order with competing interactions up to the third-nearest-neighbors with spins belonging to the different branches of the tree and for this model investigated the problem of phase transition.…”

Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice

“…The number of translation-invariant splitting Gibbs measures associated with the Ising model on a Cayley tree can only be one or more than one, depending on temperature [26]. The p-adic counterpart of the Ising-Vanniminus model on the Cayley tree of order two was first studied in [27]. There was proposed a measure-theoretical approach to investigate the model in the p-adic setting.…”