WIAS Preprint No. 3100, (2024)

Gibbs measures for hardcore-SOS models on Cayley trees


Authors

  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065
  • Rozikov, Utkir

2020 Mathematics Subject Classification

  • 82B26 60K35

Keywords

  • Cayley tree, p-SOS model, Gibbs measure, tree-indexed Markov chain

Abstract

We investigate the finite-state p-solid-on-solid model, for p=∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1→3 →5 →6 →7. This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p>0 and, in the case of k=2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous.

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