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. When $J_p=0$, by means of a construction of a special class of limiting Gibbs measures, it is shown how these equations are related with the surface energy of the Hamiltonian. This relation reduces the problem of describing the limit Gibbs measures to find of solutions of a nonlinear functional equation. Moreover, the set of ground states of the one-level model is completely described. Using this fact, one finds Gibbs measures (pure phases) associated with the translation-invariant ground states. The critical temperature is exactly found and the phase diagram is presented. The free energies corresponding to translations-invariant Gibbs measures are found. Certain physical quantities are calculated as well.