WIAS Preprint No. 2490, (2018)

Phase transitions for a model with uncountable spin space on the Cayley tree: The general case


  • Botirov, Golibjon
  • Jahnel, Benedikt
    ORCID: 0000-0002-4212-0065

2010 Mathematics Subject Classification

  • 82B05 82B20 60K35


  • Cayley trees, Hammerstein operators, splitting Gibbs measures, phase transitions




In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14, Bo17]. The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value θ c such that for θ≤θ c there is a unique translation-invariant splitting Gibbs measure. For θ c < θ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.

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