arXiv:1411.5886v1 [math-ph] 21 Nov 2014 EXTREMALITY OF TRANSLATION-INVARIANT PHASES FOR A FINITE-STATE SOS-MODEL ON THE BINARY TREE C. KUELSKE, U. A. ROZIKOV Abstract. We consider the SOS (solid-on-solid) model, with spin values 0, 1, 2, on the Cayley tree of order two (binary tree). We treat both ferromagnetic and antiferromagnetic coupling, with interactions which are proportional to the absolute value of the spin differences. We present a classification of all translation-invariant phases (splitting Gibbs measures) of the model: We show uniqueness in the case of antiferromagnetic interactions, and existence of up to seven phases in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength. Next we investigate whether these states are extremal or non-extremal in the set of all Gibbs measures, when the coupling strength is varied, whenever they exist. We show that two states are always extremal, two states are always non- extremal, while three of the seven states make transitions between extremality and non-extremality. We provide explicit bounds on those transition values, making use of algebraic properties of the models, and an adaptation of the method of Martinelli, Sinclair, Weitz. Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (sec- ondary) Key words. SOS model, temperature, Cayley tree, Gibbs measure, extreme measure, tree-indexed Markov chain, reconstruction problem. 1. Introduction A solid-on-solid (SOS) model is a spin system with spins taking values in (a subset of) the integers, and formal Hamiltonian H (σ)= −J  〈x,y〉 |σ(x) − σ(y)|, where J ∈ R is a coupling constant. An (infinite-volume) spin-configuration σ is a function from the vertices of the underlying graph to the local configuration space Φ ⊂ Z. The vertex will be the Cayley tree in our case, and for most of our analysis we will restrict to the binary tree. As usual, 〈x, y〉 denotes a pair of nearest neighbor vertices. For the local configuration space Φ we consider in the present paper the finite set Φ := {0, 1,...,m}, where m ≥ 1. Most of the times we will further specify to m = 2 for which we will present an (almost) complete analysis of the translation-invariant. 1