# On Ground States and Phase Transition for $\lambda$-Model with the Competing Potts Interactions on Cayley Trees

## TL;DR

This paper rigorously analyzes the ground states and phase transitions of a $\lambda$-model with competing Potts interactions on Cayley trees, providing a measure-theoretical framework and establishing conditions for Gibbs measures.

## Contribution

It introduces a rigorous measure-theoretical approach to the $\lambda$-model with Potts interactions on Cayley trees, extending previous numerical studies.

## Key findings

- All ground states of the model are characterized.
- Conditions for the existence of Gibbs measures are established.
- Phase transition phenomena are rigorously demonstrated.

## Abstract

In this paper, we consider the $\lambda$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $\lambda$-function is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model in more general setting. Furthermore, we find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition.