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

Abstract

Abstract. Ising model with competing nearest-neighbors and prolonged next-nearestneighbors interactions on a Cayley tree has long been studied but there are still many problems untouched. This paper tackles new Gibbs measures of Ising-Vannimenus model with competing nearest-neighbors and prolonged next-nearest-neighbors interactions on a Cayley tree (or Bethe lattice) of order three. By using a new approach, we describe the translation-invariant Gibbs measures for the model. We show that some of the measures are extreme Gibbs distributions. In this paper we take up with trying to determine when phase transition does occur.

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

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In this paper, we consider the Ising-Vanniminus model on an arbitrary order Cayley tree. We generalize the results conjectured in [3,4] for an arbitrary order Cayley tree. We establish existence and a full classification of translation invariant Gibbs measures with memory of length 2 associated with the model on arbitrary order Cayley tree.

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“…We investigate translation invariant Gibbs measures associated with Ising-Vannimenus model on arbitrary order Cayley tree. We generalize the results obtained in [3,4]. We use the Markov random field method to describe the Gibbs measures.…”

“…It was established the existence of the phase transition, for the model in terms of finitely correlated states, which describes ground states of the model. Up to this day many authors have studied the existence of phase transition by means of the recurrence equations corresponding to the Ising-Vanniminus model on Cayley tree of order two and three [3,4,22,28,33]. Recently, Ganikhodjaev [29] has studied the existence of phase transition for Ising model on the semi-infinite Cayley tree of second order with competing interactions up to third-nearest-neighbor generation with spins belonging to the different branches of the tree.…”

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

“…If we take J SL = 0, then the equation is the same as the dynamical system in [28]. The recurrence equations obtained in the present paper totally differ from [1,28,34,39]. Note that for the Ising model associated with the Hamiltonian (3.4) on the chandelier lattices of order k, in contrast to the symmetry of arbitrary order Cayley tree [6,39], if k > 3, then the chandelier lattice of order k is not symmetry.…”

“…It is well known that the comprehension of phase transitions is one of the most interesting, perhaps the central, problems of equilibrium statistical mechanics [7]. By the phase transition we mean the existence of at least two distinct Gibbs measures associated with the given model [1,8,18,28]. We will investigate the existence of translation invariant Gibbs measures on a wide class of the TCL, restricted only to the memory of length 2.…”