Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree

Abstract

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We study periodic Gibbs measures of the model with period two. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model have not any periodic Gibbs measure with period two. We construct several models which have at least two periodic Gibbs measures.

Positive Fixed Points of Cubic Operators on R² and Gibbs Measures

“…Let us note that problems of studying Gibbs measures for models with a finite and countable set of spin values on a Cayley tree are reduced to the study of systems of algebraical or functional equations [3][4][5][6][7][8][9][10][11][12][13]. One of the main factors is that studying translation-invariant Gibbs measures for models with a continuum set of spin values is reduced to the study of positive fixed points of non-linear integral operator [14][15][16][17][18][19][20].…”

“…In the case of continuum set of spin values (i.e., [0,1]) various models with the nearest neighbour interactions on a Cayley tree were considered [14][15][16][17][18][19][20]. It was found that the existence of translation-invariant Gibbs measure for the models is equivalent to the existence of a positive fixed point of Hammerstein's nonlinear integral operator [17,20].…”

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

“…Paper [12] deals with a class of Gibbs measures which are periodic and also a Markov chain. It is shown that the period must be either 1 or 2.…”

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

“…On the other hand, on a Cayley tree Γ k of order k = 2, phase transitions were proven to exist. See [5]- [8], [14], [25]- [26]. We note that all of these papers were considered for the case J 3 = J = α = 0, J 1 = 0.…”