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Mathematical Physics, Analysis and Geometry

, Volume 17, Issue 3–4, pp 323–331 | Cite as

Phase Transition and Critical Values of a Nearest-Neighbor System with Uncountable Local State Space on Cayley Trees

  • Benedikt JahnelEmail author
  • Christof Külske
  • Golibjon I. Botirov
Article

Abstract

We consider a ferromagnetic nearest-neighbor model on a Cayley tree of degree \(k\geqslant 2\) with uncountable local state space [0,1] where the energy function depends on a parameter 𝜃 ∈[0, 1). We show that for \(0\leqslant \theta \leqslant \frac {5}{3k}\) the model has a unique translation-invariant Gibbs measure. If \(\frac {5}{3k}<\theta <1\) there is a phase transition, in particular there are three translation-invariant Gibbs measures.

Keywords

Cayley tree Hammerstein’s integral operator Bifurcation analysis Gibbs measures Phase transition 

Mathematics Subject Classification (2010)

82B20 82B26 82B27 

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Benedikt Jahnel
    • 1
    Email author
  • Christof Külske
    • 1
  • Golibjon I. Botirov
    • 2
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany
  2. 2.Faculty of Physics and MathematicsBukhara State UniversityBukharaUzbekistan

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