Translation-invariant Gibbs measures for the Ising–Potts model on a second-order Cayley tree

We consider a mixed-type model given by the three-state Ising–Potts model on a Cayley tree. A criterion for the existence of limit Gibbs measures for this model on an arbitrary-order Cayley tree is obtained. Translation-invariant Gibbs measures on a second-order Cayley tree are studied. The existence of a phase transition is proved: a range of parameter values is found in which there are one to seven Gibbs measures for the three-state Ising–Potts model.