Electronic Journal of Probability

The phase transitions of the random-cluster and Potts models on slabs with $q \geq 1$ are sharp

Ioan Manolescu and Aran Raoufiï

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We prove sharpness of the phase transition for the random-cluster model with $q \geq 1$ on graphs of the form $\mathcal{S} := \mathcal{G} \times S$, where $\mathcal{G} $ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q \geq 1$, there exists some parameter $p_c = p_c(\mathcal{S} , q)$, below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 63, 25 pp.

Received: 7 April 2016
Accepted: 25 July 2017
First available in Project Euclid: 23 July 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

random-Cluster model Potts model sharp phase transition percolation models

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Manolescu, Ioan; Raoufiï, Aran. The phase transitions of the random-cluster and Potts models on slabs with $q \geq 1$ are sharp. Electron. J. Probab. 23 (2018), paper no. 63, 25 pp. doi:10.1214/17-EJP86. https://projecteuclid.org/euclid.ejp/1532332836

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