New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

2021 ◽  
Vol 24 (4) ◽  
Author(s):  
F. H. Haydarov
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree

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.

On uniqueness of Gibbs measure for -adic countable state Potts model on the Cayley tree

2009 ◽  
Vol 71 (11) ◽  
pp. 5327-5331 ◽  
Author(s):  
Andrei Khrennikov ◽  
Farrukh Mukhamedov
Keyword(s):  
Gibbs Measure ◽  
Potts Model ◽  
Cayley Tree ◽  
Countable State

Nonuniqueness of a Gibbs measure for a model on the Cayley tree

2011 ◽  
Vol 167 (2) ◽  
pp. 668-679 ◽  
Author(s):  
U. A. Rozikov ◽  
G. T. Madgoziev
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree

scholarly journals Extremality of Disordered Phase of λ-Model on Cayley Trees

Algorithms ◽  
2022 ◽  
Vol 15 (1) ◽  
pp. 18
Author(s):  
Farrukh Mukhamedov
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Gibbs Measure ◽  
Cayley Tree ◽  
Communication Theory ◽  
Arbitrary Order ◽  
Reconstruction Problem ◽  
Free Boundary Conditions ◽  
Cayley Trees ◽  
Disordered Phase

In this paper, we consider the λ-model for an arbitrary-order Cayley tree that has a disordered phase. Such a phase corresponds to a splitting Gibbs measure with free boundary conditions. In communication theory, such a measure appears naturally, and its extremality is related to the solvability of the non-reconstruction problem. In general, the disordered phase is not extreme; hence, it is natural to find a condition for their extremality. In the present paper, we present certain conditions for the extremality of the disordered phase of the λ-model.

scholarly journals Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

2012 ◽  
Vol 147 (4) ◽  
pp. 779-794 ◽  
Author(s):  
Y. K. Eshkabilov ◽  
F. H. Haydarov ◽  
U. A. Rozikov
Keyword(s):  
Gibbs Measure ◽  
Cayley Tree
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