TY - JOUR

T1 - Gibbs measures for SOS models on a cayley tree

AU - Rozikov, U. A.

AU - Suhov, Y. M.

N1 - Funding Information:
U.A.R. thanks Cambridge Colleges Hospitality Scheme for supporting the visit to Cambridge in July 2002. Y.M.S. worked in association with the ESF/RSDES Programme “Phase Transitions and Fluctuation Phenomena for Random Dynamics in Spatially Extended Systems” and was supported by the INTAS Grant 0265 Mathematics of Stochastic Networks. Both authors thank IHES, Bures-sur-Yvette, and the IGS programme at the Isaac Newton Institute, University of Cambridge, for support and hospitality.

PY - 2006/9

Y1 - 2006/9

N2 - 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 777, = 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, βc1 (= TcSTISG) ∈ (0, ∞) such that ∀ 0 ≤ β ≤ βc1, there exists a unique symmetric TISGM μ* and ∀ β > βc1 there are exactly three symmetric TISGMs: μ +* (a "bottom" symmetric TISGM), μm * (a "middle" symmetric TISGM) and μ-* (a "top" symmetric TISGM). For β > βc 1we 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 Tl or has period two (i.e. is a chess-board SGM).

AB - 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 777, = 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, βc1 (= TcSTISG) ∈ (0, ∞) such that ∀ 0 ≤ β ≤ βc1, there exists a unique symmetric TISGM μ* and ∀ β > βc1 there are exactly three symmetric TISGMs: μ +* (a "bottom" symmetric TISGM), μm * (a "middle" symmetric TISGM) and μ-* (a "top" symmetric TISGM). For β > βc 1we 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 Tl or has period two (i.e. is a chess-board SGM).

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U2 - 10.1142/S0219025706002494

DO - 10.1142/S0219025706002494

M3 - Article

AN - SCOPUS:33748633164

SN - 0219-0257

VL - 9

SP - 471

EP - 488

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

IS - 3

ER -