Gibbs measures for HC-models on Cayley trees: review of results

In the talk we review all known results about Gibbs measures of HC-models with a finite number of states on Cayley trees. We discuss extremely properties of translation invariant Gibbs measures of these models. Moreover, weakly-periodic Gibbs measures for the HC-model with two states will be discussed, these measures defined depending on normal subgroups of index two and four. Besides, non-uniqueness of translation invariant Gibbs measures for HC-models with four states on a tree of Cayley of an order $k>4$ will be given.