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We define a DNA as a sequence of $$\pm 1$$ 's and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which 'lives' on the same Cayley tree. This hierarchy has property that each vertex of the Cayley tree belongs only to one of DNA. Then we give a model (energy, Hamiltonian) of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) on a set of admissible configurations. To study thermodynamic properties of the model of DNAs we describe corresponding translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two. We show that there is a critical temperature $$T_{\mathrm{c}}$$ such that (i) if temperature $$T>T_{\mathrm{c}}$$ then there exists unique TIGM; (ii) if $$T=T_{\mathrm{c}}$$ then there are two TIGMs; (iii) if $$T

We propose a model on the Cayley tree and prove that a uncountable set of Ĝ-periodic Gibbs measures exists for this model, in contrast to models studied previously. [ABSTRACT FROM AUTHOR]

We consider fertile hard-core (HC) models with three states on a homogeneous Cayley tree. It is known that four types of such models exist. For these models, we describe the translation-invariant and periodic HC Gibbs measures. We also construct a uncountable set of nonperiodic Gibbs measures. [ABSTRACT FROM AUTHOR]

We introduce the concept of a weakly periodic Gibbs measure. For the Ising model, we describe a set of such measures corresponding to normal subgroups of indices two and four in the group representation of a Cayley tree. In particular, we prove that for a Cayley tree of order four, there exist critical values T c < T cr of the temperature T > 0 such that there exist five weakly periodic Gibbs measures for 0 < T < T c or T > T cr , three weakly periodic Gibbs measures for T = T c , and one weakly periodic Gibbs measure for T c < T ≤ T cr . [ABSTRACT FROM AUTHOR]

We introduce a natural generalization of the notion of harmonic functions on a Cayley tree and use some properties of the group representation of the Cayley tree to describe the set of harmonic functions periodic with respect to normal subgroups of finite index. [ABSTRACT FROM AUTHOR]

In the Letter we investigate aq-component models on a Cayley tree. The main goal of the Letter is to develop a ‘‘contour’’ argument on Cayley tree. We define contours and study some properties of these contours. Using a contour argument we show existence ofqdifferent Gibbs measures for severalq-component models. [ABSTRACT FROM AUTHOR]