Gibbs measures for SOS models on a Cayley tree

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, beta(1)(c), (= T-c(STISG)) is an element of (0, infinity) such that for all 0 <= beta <= beta(1)(c), there exists a unique symmetric TISGM mu* and for all beta > beta(1)(c) there are exactly three symmetric TISGMs: mu(+)* (a "bottom" symmetric TISGM), mu(m)* (a "middle" symmetric TISGM) and mu(-)* (a "top" symmetric TISGM). For beta >beta(1)(c) 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).