Extremality of the unique translation-invariant Gibbs measure for hard-core models on the Cayley tree of order $$k=3$$

2021 ◽  
Vol 206 (1) ◽  
pp. 97-108
Author(s):  
R. M. Khakimov ◽  
K. O. Umirzakova
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree ◽  
Hard Core ◽  
Translation Invariant

scholarly journals ON INHOMOGENEOUS p-ADIC POTTS MODEL ON A CAYLEY TREE

2005 ◽  
Vol 08 (02) ◽  
pp. 277-290 ◽  
Author(s):  
FARRUKH MUKHAMEDOV ◽  
UTKIR ROZIKOV
Keyword(s):  
Gibbs Measure ◽  
Potts Model ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Nearest Neighbors ◽  
Translation Invariant

We consider a nearest-neighbor inhomogeneous p-adic Potts (with q≥2 spin values) model on the Cayley tree of order k≥1. The inhomogeneity means that the interaction Jxy couplings depend on nearest-neighbors points x, y of the Cayley tree. We study (p-adic) Gibbs measures of the model. We show that (i) if q∉pℕ then there is unique Gibbs measure for any k≥1 and ∀ Jxy with | Jxy |< p-1/(p -1). (ii) For q∈p ℕ, p≥3 one can choose Jxy and k≥1 such that there exist at least two Gibbs measures which are translation-invariant.

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

2015 ◽  
Vol 18 (01) ◽  
pp. 1550006 ◽  
Author(s):  
U. A. Rozikov ◽  
F. H. Haydarov
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Sufficient Condition ◽  
Translation Invariant ◽  
Periodic Gibbs Measure

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 show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. 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 has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

Gibbs measures for the fertile three-state hard-core models on a Cayley tree

2015 ◽  
Vol 81 (1) ◽  
pp. 49-69 ◽  
Author(s):  
U. A. Rozikov ◽  
R. M. Khakimov
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Hard Core
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