An application of Catalan Numbers on Cayley tree of order 2: Single Polygon Counting

Abstract. In this paper, we consider a problem on finding the number of different single connected component containing a fixed root for a given number of vertices on semi-infinite Cayley tree. The solution of this problem is the well known Catalan numbers. The result is then extended to the complete graph. Then, we gave a suitable estimate for the given problem

On Ground States and Phase Transition for λ\lambda-Model with the Competing Potts Interactions on Cayley Trees

In this paper, we consider the $\lambda$-model with nearest neighbor interactions and with competing Potts interactions on the Cayley tree of order-two. We notice that if $\lambda$-function is taken as a Potts interaction function, then this model contains as a particular case of Potts model with competing interactions on Cayley tree. In this paper, we first describe all ground states of the model. We point out that the Potts model with considered interactions was investigated only numerically, without rigorous (mathematical) proofs. One of the main points of this paper is to propose a measure-theoretical approach for the considered model in more general setting. Furthermore, we find certain conditions for the existence of Gibbs measures corresponding to the model, which allowed to establish the existence of the phase transition.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1704.0193

λ-model with competing Potts interactions on Cayley tree of order 2

In this paper, we consider the λ-model on Cayley tree for order two with Potts competing nearest-neighbor and prolonged next-nearest neighbor-interactions. We described the construction of the Gibbs measure for the considered model. We proved the existence of the translation-invariant limiting Gibbs measures for the model

On connected sub-tree with fixed nodes in Cayley tree of Order 2

In this paper we found an exact formula for a finite sub-tree counting problem. Note that the formulas, which correspond to two extremal cases, are Catalan Triangle introduced by Shapiro and ballot Catalan triangles. The general formula could be expressed as a linear combination of these Catalan triangles

Exact solution of an Ising model with competing interactions on a Cayley tree

The exact solution of an Ising model with competing restricted interactions on the Cayley tree, and in the absence of an external field is presented. A critical curve is defined where it is possible to get phase transitions above it, and a single Gibbs state is obtained elsewhere

Rock-Paper-Scissors Lattice Model

In this work, we introduce Rock-Paper-Scissors lattice model on Cayley tree of second order generated by Rock-Paper-Scissors game. In this strategic 2-player game, the rule is simple: rock beats scissors, scissors beat paper, and paper beats rock. A payoff matrix A of this game is a skewsymmetric. It is known that quadratic stochastic operator generated by this matrix is non-ergodic transformation. The Hamiltonian of Rock-Paper-Scissors Lattice Model is defined by this skewsymmetric payoff matrix A . In this paper, we discuss a connection between three fields of research: evolutionary games, quadratic stochastic operators, and lattice models of statistical physics. We prove that a phase diagram of the Rock-Paper-Scissors model consists of translation-invariant and periodic Gibbs measure with period 3

On a class of non-ergodic Lotka–Volterra operator

In present paper, we study dynamical systems generated by a class of Lotka–Volterra operators. It is proved that such class of operator has non-ergodic property