Gibbs measures of the Ising model with mixed spin-1 and spin-1/2 on a Cayley tree

In the present paper, the Ising model with mixed spin-(1,1/2) is considered on the second order Cayley tree. A construction of splitting Gibbs measures corresponding the model is given which allows to establish the existence of the phase transition (non-uniqueness of Gibbs measures). We point out that, in the phase transition region, the considered model has three translation-invariant Gibbs measures in the ferromagnetic and anti-ferromagnetic regimes, while the classical Ising model does not possesses such Gibbs measures in the anti-ferromagnetic regime. It turns out that the considered model, like the Ising model, exhibits a disordered Gibbs measure. Therefore, non-extremity and extremity of such disordered Gibbs measures is investigated by means of tree-indexed Markov chains.