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

Abstract

In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in Botirov (Positivity 21(3):955–961, 2017), Eshkabilov et al. (J Stat Phys 147(4):779–794, 2012), Eshkabilov and Rozikov (Math Phys Anal Geom 13:275–286, 2010), Botirov et al. (Lobachevskii J Math 34(3):256–263 2013) and Jahnel et al. (Math Phys Anal Geom 17:323–331 2014). 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 \(\theta _{\mathrm{c}}\) such that for \(\theta \le \theta _{\mathrm{c}}\) there is a unique translation-invariant splitting Gibbs measure. For \(\theta _{\mathrm{c}}<\theta \) 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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Acknowledgements

Golibjon Botirov thanks the DAAD program for the financial support and the Weierstrass Institute Berlin for its hospitality. Benedikt Jahnel thanks the Leibniz program ’Probabilistic methods for mobile ad-hoc networks’ for the support.

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Correspondence to Golibjon Botirov.

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Botirov, G., Jahnel, B. Phase transitions for a model with uncountable spin space on the Cayley tree: the general case. Positivity 23, 291–301 (2019). https://doi.org/10.1007/s11117-018-0606-1

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Keywords

  • Cayley trees
  • Hammerstein operators
  • Splitting Gibbs measures
  • Phase transitions

Mathematics Subject Classification

  • 82B05
  • 82B20 (primary)
  • 60K35 (secondary)