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Gibbs Measures For SOS Models On a Cayley Tree

We consider a nearest-neighbor SOS model, spin values $0,1,..., m$, $m\geq 2$, on a Cayley tree of order $k$ . We mainly assume that $m=2$ and study translation-invariant (TI) and `splitting' (S) Gibbs measures (GMs). For $m=2$, in the anti-ferromagnetic (AFM) case, a symmetric TISGM is unique for all temperatures. In the ferromagnetic (FM) case, for $m=2$, the number of symmetric TISGMs varies with the temperature: here we identify a critical inverse temperature, $β^1_{\rm{cr}}$ ($=T_{\rm{cr}}^{\rm{STISG}}$) $\in (0,\infty)$ such that $\forall$ $0\leq β\leqβ^1_{\rm{cr}}$, there exists a unique symmetric TISGM $μ^*$ and $\forall$ $β>β^1_{\rm{cr}}$ there are exactly three symmetric TISGMs : $μ^*_+$, $μ^*_{\rm m}$ and $μ^*_-$ For $β>β^1_{\rm{cr}}$ 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. 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).