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Positivity

, Volume 23, Issue 2, pp 291–301 | Cite as

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

  • Golibjon BotirovEmail author
  • Benedikt Jahnel
Article
  • 28 Downloads

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.

Keywords

Cayley trees Hammerstein operators Splitting Gibbs measures Phase transitions 

Mathematics Subject Classification

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

Notes

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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Weierstrass Institute BerlinBerlinGermany
  2. 2.National University of UzbekistanTashkentUzbekistan

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