Abstract
In the present paper we continue our investigations started in [Accardi L. , Ohno, H. , Mukhamedov, F. , Quantum Markov fields on graphs, Inf. Dim. Analysis, Quantum Probab. Related Topics (accepted) arxi v: 0911 . 1667]. In [Accardi L., Mukhamedov, F., Saburov M. On Quantum Markov chains on Cayley tree and associated chains with XY-model arxiv: 1004.3623] we provided a construction of forward and backward Quantum Markov Chains (QMC) defined on the Cayley tree, and established uniqueness of QMC associated with XY-model on a Cayley tree order 2. In the present paper we study the same model on a Cayley tree order 3. Surprisingly in this case, we establish a phase transition (i.e. existence of two distinct quantum Markov chains) for the considered model on the Cayley tree order 3.
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