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

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.

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Correspondence to Benedikt Jahnel.

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Jahnel, B., Külske, C. & Botirov, G.I. Phase Transition and Critical Values of a Nearest-Neighbor System with Uncountable Local State Space on Cayley Trees. Math Phys Anal Geom 17, 323–331 (2014). https://doi.org/10.1007/s11040-014-9158-1

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Keywords

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

Mathematics Subject Classification (2010)

  • 82B20
  • 82B26
  • 82B27