scholarly journals Gradient Gibbs measures for the SOS model with countable values on a Cayley tree

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Florian Henning ◽  
Christof Külske ◽  
Arnaud Le Ny ◽  
Utkir A. Rozikov
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model

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 Gradient Gibbs measures for the SOS model with integer spin values on a Cayley tree

2020 ◽  
Vol 2020 (9) ◽  
pp. 093102
Author(s):  
G I Botirov ◽  
F H Haydarov
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model ◽  
Integer Spin

Gibbs measures for the SOS model with four states on a Cayley tree

2006 ◽  
Vol 149 (1) ◽  
pp. 1312-1323 ◽  
Author(s):  
U. A. Rozikov ◽  
Sh. A. Shoyusupov
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model

scholarly journals Periodic Gibbs Measures for the Potts-SOS Model on a Cayley Tree

2019 ◽  
Vol 199 (1) ◽  
pp. 586-592 ◽  
Author(s):  
M. A. Rasulova
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model

scholarly journals Extremality of translation-invariant Gibbs measures for the Potts–SOS model on the Cayley tree

2021 ◽  
Vol 2021 (7) ◽  
pp. 073201
Author(s):  
M M Rahmatullaev ◽  
M A Rasulova
Keyword(s):  
Cayley Tree ◽  
Gibbs Measures ◽  
Sos Model ◽  
Translation Invariant

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 Translation Invariance of the Periodic Gibbs Measures for the Potts Model on the Cayley Tree

2019 ◽  
Vol 199 (2) ◽  
pp. 726-735
Author(s):  
R. M. Khakimov ◽  
M. T. Makhammadaliev
Keyword(s):  
Potts Model ◽  
Cayley Tree ◽  
Gibbs Measures ◽  
Translation Invariance

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.

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