Existence of weakly periodic Gibbs measures for the Ising model on the Cayley tree of order three

One of the main problems of the Ising model Hamiltonian is to describe all limiting Gibbs measures corresponding to this Hamiltonian. It is well known that for the Ising model, such measures form a nonempty convex compact subset in the set of all probability measures. The problem of completely describing the elements of this set is far from being completely solved. For the Ising model on the Cayley tree of order three translation-invariant and periodic Gibbs measures are studied, but weakly periodic Gibbs measures have not been studied yet. Therefore, it is interesting to study weakly periodic Gibbs measures which is non-periodic. The paper is devoted to the study of weakly periodic Gibbs measures for the Ising model on a Cayley tree of order three ($k = 3$). It is known that the weakly periodic Gibbs measure for the Ising model depends on the choice of the normal subgroup of the group representation of the Cayley tree. In this paper, we consider one normal subgroup of index four of the group representation of a Cayley tree. With respect to this normal subgroup, the existence of weakly periodic Gibbs measures for the Ising model on a Cayley tree of order three is proved. More precisely, the fact that under some conditions on parameters the existence of at least four weakly periodic (non-periodic) Gibbs measures is proved.