Translation-Invariant Gibbs Measures for the Blum-Kapel Model on a Cayley Tree

2016 ◽  
Vol 15 (2) ◽  
pp. 239-255 ◽  
Author(s):  
Nosir Khatamov ◽  
◽  
Rustam Khakimov ◽  
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
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.

scholarly journals GIBBS MEASURES FOR SOS MODELS ON A CAYLEY TREE

2006 ◽  
Vol 09 (03) ◽  
pp. 471-488 ◽  
Author(s):  
U. A. ROZIKOV ◽  
Y. M. SUHOV
Keyword(s):  
Ising Model ◽  
Normal Subgroup ◽  
Finite Index ◽  
Cayley Tree ◽  
Nearest Neighbor ◽  
Gibbs Measures ◽  
Natural Generalization ◽  
Mirror Reflection ◽  
Sos Model ◽  
Translation Invariant

We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0, 1,…, m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalization of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and "splitting" (S) Gibbs measures (GMs). (Splitting GMs have a particular Markov-type property specific for a tree.) Furthermore, we focus on symmetric TISGMs, with respect to a "mirror" reflection of the spins. [For the Ising model (where m = 1), such measures are reduced to the "disordered" phase obtained for free boundary conditions, see Refs. 9, 10.] For m = 2, in the antiferromagnetic (AFM) case, a symmetric TISGM (and even a general TISGM) is unique for all temperatures. In the ferromagnetic (FM) case, for m = 2, the number of symmetric TISGMs and (and the number of general TISGMs) varies with the temperature: this gives an interesting example of phase transition. Here we identify a critical inverse temperature, [Formula: see text] such that [Formula: see text], there exists a unique symmetric TISGM μ* and [Formula: see text] there are exactly three symmetric TISGMs: [Formula: see text] (a "bottom" symmetric TISGM), [Formula: see text] (a "middle" symmetric TISGM) and [Formula: see text] (a "top" symmetric TISGM). For [Formula: see text] we also construct a continuum of distinct, symmertric SGMs which are non-TI. Our second result gives complete description of the set of periodic Gibbs measures for the SOS model on a Cayley tree. A complete description of periodic GMs means a characterisation of such measures with respect to any given normal subgroup of finite index in the representation group of the tree. We show that (i) for an FM SOS model, for any normal subgroup of finite index, each periodic SGM is in fact TI. Further, (ii) for an AFM SOS model, for any normal subgroup of finite index, each periodic SGM is either TI or has period two (i.e. is a chess-board SGM).

Non-Translation-Invariant Gibbs Measures of an SOS Model on a Cayley Tree

2020 ◽  
Vol 86 (3) ◽  
pp. 315-324
Author(s):  
Muzaffar M. Rahmatullaev ◽  
B.U. Abraev
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model ◽  
Translation Invariant

scholarly journals Translation-invariant p-adic quasi-Gibbs measures for the Ising–Vannimenus model on a Cayley tree

2016 ◽  
Vol 187 (1) ◽  
pp. 583-602 ◽  
Author(s):  
F. M. Mukhamedov ◽  
M. Kh. Saburov ◽  
O. Kh. Khakimov
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Translation Invariant

scholarly journals Positive Fixed Points of Cubic Operators on R² and Gibbs Measures

2019 ◽  
pp. 663-673
Author(s):  
Yusup Kh. Eshkabilov ◽  
Shohruh D. Nodirov
Keyword(s):  
Fixed Points ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Nearest Neighbour ◽  
Translation Invariant

One model with nearest neighbour interactions of spins with values from the set [0; 1] on the Cayley tree of order three is considered in the paper. Translation-invariant Gibbs measures for the model are studied. Results are proved by using properties of the positive fixed points of a cubic operator in the cone R²₊

Extremality of the Translation-Invariant Gibbs Measures for the Potts Model on the Cayley Tree

2018 ◽  
Vol 196 (1) ◽  
pp. 1043-1058
Author(s):  
U. A. Rozikov ◽  
R. M. Khakimov ◽  
F. Kh. Khaidarov
Keyword(s):  
Potts Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Translation Invariant
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