Four competing interactions for models with an uncountable set of spin values on a Cayley tree

Abstract

In this paper we consider four competing interactions (external field, nearest neighbor, second neighbors and triples of neighbors) of models with uncountable (i.e. [0, 1]) set of spin values on the Cayley tree of order two. We reduce the problem of describing the "splitting Gibbs measures" of the model to the analysis of solutions to some nonlinear integral equation and study some particular cases for Ising and Potts models. Also we show that periodic Gibbs measures for given models are either translation-invariant or periodic with period two and we give examples of the non-uniqueness of translation-invariant Gibbs measures.Mathematics Subject Classifications (2010). 82B05, 82B20 (primary); 60K35 (secondary)

Lyapunov operator L with degenerate kernel and Gibbs measures

“…In [13] it's considered splitting Gibbs measures for four competing interactions i.e. (J • J 1 • J 3 • α = 0) of models with uncountable set of spin values on the Cayley tree of order two.…”

“…In this paper, we provide a connection between Gibbs measures for the model which is defined in [14] and positive solutions of the Lyupanov integral equations. Also we study the fixed points of the Lyapunov operator with degenerate kernel.…”

“…In [14] it was described splitting Gibbs measures on Γ 2 were described by solutions to a nonlinear integral equation for the case J 2 3 + J 2 1 + J 2 + α 2 = 0 which is a generalization of the case J 3 = J = α = 0, J 1 = 0. Also, it was proved that periodic Gibbs measure for the Hamiltonian with four competing interactions is either translation-invariant or G (2) k -periodic, and given examples of non-uniqueness for Hamiltonian (2.1) in the case J 3 = 0, J = J 1 = α = 0.…”

Positive fixed points of Lyapunov operator

“…In [13] it is described that splitting Gibbs measures on Γ 2 by solutions to a nonlinear integral equation for the case J 2 3 + J 2 1 + J 2 + α 2 = 0 which a generalization of the case J 3 = J = α = 0, J 1 = 0. Also, it is proven that periodic Gibbs measure for Hamiltonian (1) with four competing interactions is either translation-invariant or G (2) k − periodic.…”