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

TL;DR

This paper extends the analysis of q-component models on Cayley trees, establishing the existence of multiple Gibbs measures and characterizing ground states for any number of components and tree order.

Contribution

It generalizes previous results to any Cayley tree order and any q-component model with nearest neighbor interactions, broadening the scope of known phase behaviors.

Findings

Existence of q different Gibbs measures for the models.

Identification of periodic ground states.

Generalization from specific models to all q-component models.

Abstract

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 $k\geq 2$.