Abstract. We show that the nearest neighbors Ising model on the Cayley tree exhibits new temperature driven phase transitions. These transitions holds at various inverse temperatures different from the critical one. They are depicted by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points. We also consider the model in presence of an external field and compute the free energies of translation invariant periodic boundary conditions.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We study periodic Gibbs measures of the model with period two. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures.

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. [0, 1]) set of spin values on the Cayley tree of order two. We reduce the problem of describing the "splitting Gibbs measures" of the model to the analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of translation-invariant Gibbs measures.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for models with four competing interactions and uncountable (i.e. [0, 1]) set of spin values on a Cayley tree. We prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of a fixed point.

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In the present paper we continue the investigation from Bovier and Külske (1996 J. Stat. Phys. 83 751-59) and consider the SOS (solid-on-solid) model on the Cayley tree of order k 2. In the ferromagnetic SOS case on the Cayley tree, we find three solutions to a class of period-4 height-periodic boundary law equations and these boundary laws define up to three periodic gradient Gibbs measures.

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In this paper, we give a characterization of the normal subgroups of index 2 s (2n+1), s ∈ {1, 2}, n ∈ N and of the subgroups of index three of the group representation of the Cayley tree.