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A Short Proof of the Discontinuity of Phase Transition in the Planar Random-Cluster Model with \(q>4\)

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Abstract

The goal of this paper is to provide a short proof of the discontinuity of phase transition for the random-cluster model on the square lattice with parameter \(q>4\). This result was recently shown in Duminil-Copin et al. (arXiv:1611.09877, 2016) via the so-called Bethe ansatz for the six-vertex model. Our proof also exploits the connection to the six-vertex model, but does not rely on the Bethe ansatz. Our argument is soft (in particular, it does not rely on a computation of the correlation length) and only uses very basic properties of the random-cluster model [for example, we do not even need the Russo–Seymour–Welsh machinery developed recently in Duminil-Copin et al. (Commun Math Phys 349(1):47–107, 2017)].

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Acknowledgements

We are grateful to Alexander Glazman, and Ron Peled who made available to us a draft of their paper [13]. We also thank them, Hugo Duminil-Copin and Marcin Lis for several useful comments on an earlier draft of this paper. We also thank the referee for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Victoria, Victoria, BC, V8W 2Y2, Canada

    Gourab Ray

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Yinon Spinka

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  1. Gourab Ray
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  2. Yinon Spinka
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Correspondence to Gourab Ray.

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Communicated by H. Duminil-Copin

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Gourab Ray: Supported in part by NSERC 50311-57400 and University of Victoria start-up 10000-27458.

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Ray, G., Spinka, Y. A Short Proof of the Discontinuity of Phase Transition in the Planar Random-Cluster Model with \(q>4\). Commun. Math. Phys. 378, 1977–1988 (2020). https://doi.org/10.1007/s00220-020-03827-9

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