## Electronic Journal of Probability

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

#### Abstract

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

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

Dates
Accepted: 25 July 2017
First available in Project Euclid: 23 July 2018

https://projecteuclid.org/euclid.ejp/1532332836

Digital Object Identifier
doi:10.1214/17-EJP86

Mathematical Reviews number (MathSciNet)
MR3835469

Zentralblatt MATH identifier
06924675

#### Citation

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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