Shokirjon Shoyusupov (Namangan Institute of Engineering and Technology, Uzbekistan)
Description of Gibbs measures for some nonsymmetrical models on Cayley tree
My talk is about the description of Gibbs measures for SOS (solid-on-solid) models with four states and HC (hard core) with three states on Cayley tree. These measures are translational-invariant or periodical. Also we describe some classes of non-periodical and limitary Gibbs measures. We use the methods of Markov random fields and recurrent equations of this theory in our research. Besides that we apply methods of measure theory and contractive maps. And we have got the following results:
- It is proved that for antiferromagnetic SOS models with four states on Cayley tree of order translation-invariant Gibbs measure is unique for all values of temperature ; for ferromagnetic SOS models if and (where is critical value of temperature) then there are three translation-invariant Gibbs measures.
- Description of all periodic Gibbs measures for SOS models on Cayley tree is given.
- Using translation-invariant measures an uncountable set of non-periodic Gibbs measures are described.
- For three types of HC models on Cayley tree the translation-invariant Gibbs measures are described.
- For the type "hinge" of HC model all periodic Gibbs measures are given. Moreover, for this type an uncountable set of non-periodic Gibbs measures is constructed.