On p-adic ising model with competing interactions on the cayley tree - Mukhamedov, Farrukh; Akin, Hasan; Dogan, Mutlay (2014)
It is known that the Ising model is one of the most studied models in statistical mechanics. Since, this model is related to a number of outstanding problems in statistical and mathematical physics, and in graph theory. On the other hand, that most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. Therefore, in this direction a lot of investigations are devoted to the mathematical physics models over p-adic eld. In the present paper, we further develop the theory of statistical mechanics. Namely, we consider p-adic Ising model with competing next-nearest-neighbor interactions on the Cayley tree of order two. Note that usual p-adic Ising model on the tree was earlier studied by the rst author. A main aim of this work is the establishment of a phase transition phenomena for the mentioned model. Here the phase transition means the existence of two nontrivial p-adic Gibbs measures. To prove the occurrence of the phase transition we reduce the problem to the existence of at leat two solutions of nonlinear difference equations.