2021/10/02 by M. M. Rahmatullaev, Rahmatullaev, M. M., Obid Sh. Karshiboev +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2110.00730
openalex publication_date 2021/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a nearest-neighbor solid-on-solid (SOS) model, with several spin values 0,1,…,m, m≥2, and nonzero external field, on a Cayley tree of degree k (with k+1 neighbors). We are aiming to extend the results of \citers where the SOS model is studied with (mainly) three spin values and zero external field. 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.) For m=2, in the antiferromagnet (AFM) case, a TISGM is unique for all temperatures with an external field. In the ferromagnetic (FM) case, for m=2, the number of TISGMs varies with the temperature and the external field: this gives an interesting example of phase transition. Our second result gives a classification of all TISGMs of the Three-State SOS-Model on the Cayley tree of degree two with the presence of an external field. We show uniqueness in the case of antiferromagnetic interactions and the existence of up to seven TISGMs in the case of ferromagnetic interactions, where the number of phases depends on the interaction strength and external field.