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Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with \({1 \le q \le 4}\)

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Abstract

This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic q-state Potts model on \({\mathbb{Z}^2}\) is continuous for \({q \in \{2,3,4\}}\), in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions.

The proof uses the random-cluster model with cluster-weight \({q \ge 1}\) (note that q is not necessarily an integer) and is based on two ingredients:

  • The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights \({1\le q\le 4}\), which is derived studying parafermionic observables on a discrete Riemann surface.

  • A new result proving the equivalence of several properties of critical random-cluster models:

    • the absence of infinite-cluster for wired boundary conditions,

    • the uniqueness of infinite-volume measures,

    • the sub-exponential decay of the two-point function for free boundary conditions,

    • a Russo–Seymour–Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions.

The result has important consequences toward the study of the scaling limit of the random-cluster model with \({q \in [1,4]}\). It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.

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Authors and Affiliations

  1. Universite de Geneva, Geneva, Switzerland

    Hugo Duminil-Copin & Vincent Tassion

  2. IMPA, Rio de Janeiro, Brazil

    Vladas Sidoravicius

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  1. Hugo Duminil-Copin
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  2. Vladas Sidoravicius
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Correspondence to Hugo Duminil-Copin.

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Communicated by F. Toninelli

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Duminil-Copin, H., Sidoravicius, V. & Tassion, V. Continuity of the Phase Transition for Planar Random-Cluster and Potts Models with \({1 \le q \le 4}\) . Commun. Math. Phys. 349, 47–107 (2017). https://doi.org/10.1007/s00220-016-2759-8

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