Construction of an uncountable number of limiting Gibbs measures in the inhomogeneous ising model

1999 ◽  
Vol 118 (1) ◽  
pp. 77-84 ◽  
Author(s):  
U. A. Rozikov
Keyword(s):  
Ising Model ◽  
Gibbs Measures ◽  
Inhomogeneous Ising Model

Corner transfer matrix study of an inhomogeneous Ising model

1992 ◽  
Vol 504 (2) ◽  
pp. 125-133 ◽  
Author(s):  
Ingo Peschel ◽  
Roland Wunderling
Keyword(s):  
Ising Model ◽  
Transfer Matrix ◽  
Inhomogeneous Ising 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).

A subshift of finite type that is equivalent to the Ising model

1995 ◽  
Vol 15 (3) ◽  
pp. 543-556 ◽  
Author(s):  
Olle Häggström
Keyword(s):  
Ising Model ◽  
Finite Type ◽  
Gibbs Measures ◽  
Ergodic Measure ◽  
Maximal Entropy ◽  
Subshift Of Finite Type ◽  
Translation Invariant ◽  
Measures Of Maximal Entropy ◽  
Natural Way ◽  
Measure Of Maximal Entropy

AbstractFor the Ising model with rational parameters we show how to construct a subshift of finite type that is equivalent to this Ising model, in that the translation invariant Gibbs measures for the Ising model and the measures of maximal entropy for the subshift of finite type can be identified in a natural way. This is generalized to the non-translation invariant case as well. We also show how to construct, given any H > 0, an ergodic measure of maximal entropy for a subshift of finite type and a continuous factor, such that the factor has entropy H.

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