On $q$- Component Models on Cayley Tree: The General Case

In the paper we generalize results of paper [12] for a $q$- component models on a Cayley tree of order $k\geq 2$. We generalize them in two directions: (1) from $k=2$ to any $k\geq 2;$ (2) from concrete examples (Potts and SOS models) of $q-$ component models to any $q$- component models (with nearest neighbor interactions). We give a set of periodic ground states for the model. Using the contour argument which was developed in [12] we show existence of $q$ different Gibbs measures for $q$-component models on Cayley tree of order $k\geq 2$.