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

References

• [1] M. Aizenman and D. J. Barsky. Sharpness of the phase transition in percolation models. Comm. Math. Phys., 108(3):489–526, 1987.
• [2] D. Basu and A. Sapozhnikov. Crossing probabilities for critical bernoulli percolation on slabs. preprint arXiv:1512.05178, 2016.
• [3] V. Beffara and H. Duminil-Copin. The self-dual point of the two-dimensional random-cluster model is critical for $q\geq 1$. Probab. Theory Related Fields, 153(3-4):511–542, 2012.
• [4] H. Duminil-Copin. Parafermionic observables and their applications to planar statistical physics models, volume 25 of Ensaios Matematicos. Brazilian Mathematical Society, 2013.
• [5] H. Duminil-Copin, J.H. Li, and I. Manolescu. Random-cluster model with critical weights on isoradial graphs. preprint, 2015.
• [6] H. Duminil-Copin and I. Manolescu. The phase transitions of the planar random-cluster and Potts models with q $\geq$ 1 are sharp. to appear in Probab. Theory Related Fields, 2014. preprint arXiv:1409.3748.
• [7] H. Duminil-Copin, V. Sidoravicius, and V. Tassion. Absence of infinite cluster for critical bernoulli percolation on slabs. preprint, arXiv:1401.7130, 2014.
• [8] H. Duminil-Copin, V. Sidoravicius, and V. Tassion. Continuity of the phase transition for planar random-cluster and potts models with $1\le q\le 4$. preprint, arXiv:1505.04159, 2015.
• [9] H. Duminil-Copin and V. Tassion. Grimmet marstrand for the potts model. In preparation.
• [10] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for bernoulli percolation on $\mathbb{Z} ^d$. ArXiv e-prints arXiv:1502.03051.
• [11] B. Graham and G. R. Grimmett. Sharp thresholds for the random-cluster and Ising models. Ann. Appl. Probab., 21(1):240–265, 2011.
• [12] G. R. Grimmett. The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
• [13] G. R. Grimmett and J. M. Marstrand. The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A, 430(1879):439–457, 1990.
• [14] G. R. Grimmett and M.S.T. Piza. Decay of correlations in random-cluster models. Communications in mathematical physics, 189(2):465–480, 1997.
• [15] M. V. Menshikov. Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR, 288(6):1308–1311, 1986.
• [16] C. Newman, V. Tassion, and W. Wu. Critical percolation and the minimal spanning tree in slabs. Communications on Pure and Applied Mathematics, 70(11):2084–2120, 2017.