The Potts model on a Cayley tree in the presence of competing two binary interactions and magnetic field is considered. We exactly solve a problem of phase transitions for the model, namely we calculate critical surface such that there is a phase transition above it and a single Gibbs state found elsewhere. | Turk J Math 31 (2007) , 229 – 238. ¨ ITAK ˙ c TUB Potts Model with two Competing Binary Interactions Nasir Ganikhodjaev, Hasan Akın and Seyit Temir Abstract The Potts model on a Cayley tree in the presence of competing two binary interactions and magnetic field is considered. We exactly solve a problem of phase transitions for the model,namely we calculate critical surface such that there is a phase transition above it,and a single Gibbs state found elsewhere. Key Words: Potts model, competing interactions, Gibbs measure. 1. Introduction Lattice spin systems is a large class of systems considered in statistical mechanics. Some of them have a real physical meaning, others are studied as suitable simplified models of more complicated systems. The structure of the lattice plays an important role in investigation of spin systems. For example, in order to study a phase transition problem for a system on Zd and on a Cayley tree, there are two different methods: Pirogov-Sinai theory on Zd , Markov random field theory and recurrent equations of this theory on Cayley tree. In [2-9], Gibbs measures are described for several models on a Cayley tree, using the Markov random field theory. The Potts model was introduced as a generalization of the Ising model. The idea came from the representation of the Ising model as interacting spins which can be either parallel or antiparallel. An obvious generalization was to extend the number of directions of the spins. Such a model was proposed by as a PhD thesis for his student in 1952. At present, the Potts model encompasses a number of problems in statistical physics and lattice theory. It has been a subject of increasing intense research interest in recent years [10]. It includes the ice-rule vertex and bond percolation models as special cases. In this paper we consider Mathematics Subject Classification: 82B20, 82B26 229 ˙ GANIKHODJAEV, AKIN, TEMIR a Potts model with competing two binary interactions and external .
