On the three state Potts model with competing interactions on the Bethe lattice

In the present paper the three state Potts model with competing binary interactions (with couplings J and J_p) on the second order Bethe lattice is considered. The recurrent equations for the partition functions are derived. For J_p = 0, by means of a construction of a special class of limiting Gibbs measures, it is shown how these equations are related to the surface energy of the Hamiltonian. This relation reduces the problem of describing the limit Gibbs measures to that of finding solutions of a nonlinear functional equation. Moreover, the set of ground states of the one level model is completely described. Using this fact, we find Gibbs measures (pure phases) associated with the translation-invariant ground states. The critical temperature is found exactly and the phase diagram is presented. The free energies corresponding to translation-invariant Gibbs measures are found. Certain physical quantities are calculated as well.