Gibbs measures for SOS models on a cayley tree

U. A. Rozikov, Y. M. Suhov

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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).

Original languageEnglish (US)
Pages (from-to)471-488
Number of pages18
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Issue number3
StatePublished - Sep 2006

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Mathematical Physics
  • Applied Mathematics


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