Phase transitions for a model with uncountable spin space on the Cayley tree: The general case
- Botirov, Golibjon
- Jahnel, Benedikt
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.
- Positivity, 23 (2019), pp. 291--301 (published online on 17.08.2018), DOI 10.1007/s11117-018-0606-1 .