Lattice models with competing interactions and their phase diagrams

A lattice model is the model defined on countable graph $G = (V, L)$ without loops and multiple edges. Initially in statistical physics have been considered the $d$-dimensional integer lattice (cubic lattice or crystal lattice), denoted $Z^d$. In recent years models on a Cayley tree has been studied extensively because it turns out that there are physically interesting solutions correspond to the attractors of the mapping. This simplifies the numerical work considerably and detailed study of the whole phase diagram becomes feasible. Apart from the intrinsic interest attached to the study of models on trees, it is possible to argue that the results obtained on trees provide a useful guide to the more involved study of their counterparts on crystal lattices.
In this presentation firstly we discuss the origins and development of the Ising model.
Secondary, formulate few problems about phase transitions on some graphs.
Lastly, we consider models with competing interactions and discuss their phase diagrams.