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Probability Theory and Related Fields

, Volume 158, Issue 1–2, pp 477–512 | Cite as

On the Gibbs states of the noncritical Potts model on \(\mathbb Z ^2\)

  • Loren Coquille
  • Hugo Duminil-Copin
  • Dmitry Ioffe
  • Yvan VelenikEmail author
Article

Abstract

We prove that all Gibbs states of the \(q\)-state nearest neighbor Potts model on \(\mathbb Z ^2\) below the critical temperature are convex combinations of the \(q\) pure phases; in particular, they are all translation invariant. To achieve this goal, we consider such models in large finite boxes with arbitrary boundary condition, and prove that the center of the box lies deeply inside a pure phase with high probability. Our estimate of the finite-volume error term is of essentially optimal order, which stems from the Brownian scaling of fluctuating interfaces. The results hold at any supercritical value of the inverse temperature \(\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)\).

Keywords

Potts model Gibbs states DLR equation Aizenman–Higuchi theorem Translation invariance Interface fluctuations Pure phases 

Mathematics Subject Classification (2010)

60K35 82B20 82B24 

Notes

Acknowledgments

H. D.-C. was supported by the EU Marie-Curie RTN CODY, the ERC AG CONFRA, as well as by the Swiss National Science Foundation. The research of D.I. was supported by the Israeli Science Foundation (grant No 817/09). L.C. and Y.V. were partially supported by the Swiss National Science Foundation.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Loren Coquille
    • 1
  • Hugo Duminil-Copin
    • 1
  • Dmitry Ioffe
    • 2
  • Yvan Velenik
    • 1
    Email author
  1. 1.Département de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Faculty of IE&MTechnionHaifaIsrael

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