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

Coquille, Loren ; Duminil-Copin, Hugo ; Ioffe, Dmitry ; Velenik, Yvan

In: Probability Theory and Related Fields, 2014, vol. 158, no. 1-2, p. 477-512

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    Title
    • On the Gibbs states of the noncritical Potts model on $$\mathbb Z ^2$$
    Author
    • Coquille, Loren. Département de Mathématiques, Université de Genève, Geneva, Switzerland
    • Duminil-Copin, Hugo. Département de Mathématiques, Université de Genève, Geneva, Switzerland
    • Ioffe, Dmitry. Faculty of IE&M, Technion, Haifa, Israel
    • Velenik, Yvan. Département de Mathématiques, Université de Genève, Geneva, Switzerland
    Document Type
    • Postprint
    Language
    • English
    Published in
    • Probability Theory and Related Fields, 2014, vol. 158, no. 1-2, p. 477-512. Springer Berlin Heidelberg
    Other Version
    OAI-PMH Identifier
    • oai:doc.rero.ch:326404
    Summary
    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 $$\beta >\beta _c (q) = \log \left(1+\sqrt{q}\right)$$ .