Lattice models on Cayley tree with competing interactions

The existence of competing interactions lies at the heart of a variety of original phenomena in magnetic systems, ranging from the spin-glass transitions found in many disordered materials to the modulated phases with an infinite number of commensurate regions, that are observed in certain models with periodic interactions. The similarity of results obtained for models defined on Cayley trees and on crystal lattices is a strong motivation for the study of models on trees, since the statistical mechanics on trees presents many simplifying aspects that are absent in models defined on crystal lattices. This suggests that more complicated models should be studied on trees first, with the hope to discover new phases or unusual types of behaviour. The important point is that statistical mechanics on trees involve nonlinear recursion equationsand are naturally connected to the rich world of dynamical systems, a world presently under intense investigation. We consider the following kinds of bonds on the Cayley tree: the first-, second- and third nearest neighbors, where spins for second- and third nearest neighbors can belong to the same branch of the tree as well as different branches. In our talk we review well-known results about models on Cayley tree with competing interactions on the first-, second- and third nearest neighbors, and present some new results