Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A new computation of the critical point for the planar random-cluster model with $q\ge1$

Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion

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Abstract

We present a new computation of the critical value of the random-cluster model with cluster weight $q\ge1$ on $\mathbb{Z}^{2}$. This provides an alternative approach to the result in (Probab. Theory Related Fields 153 (2012) 511–542). We believe that this approach has several advantages. First, most of the proof can easily be extended to other planar graphs with sufficient symmetries. Furthermore, it invokes RSW-type arguments which are not based on self-duality. And finally, it contains a new way of applying sharp threshold results which avoid the use of symmetric events and periodic boundary conditions. Some of the new methods presented in this paper have a larger scope than the planar random-cluster model, and may be useful to investigate sharp threshold phenomena for more general dependent percolation processes in arbitrary dimensions.

Résumé

Nous proposons une nouvelle preuve du fait que le point critique du modèle de percolation de Fortuin–Kasteleyn sur le réseau carré vaut $\sqrt{q}/(1+\sqrt{q})$ lorsque $q\ge1$. Cette preuve est une alternative à la stratégie implémentée dans (Probab. Theory Related Fields 153 (2012) 511–542). Cette approche a plusieurs avantages. Tout d’abord, la grande majorité des arguments peuvent être généralisés aux autres graphes planaires (ayant suffisamment de symétries). De plus, elle n’invoque pas d’argument de type RSW basés sur l’auto-dualité du modèle. Enfin, elle repose sur une nouvelle façon d’appliquer les théorèmes de seuil qui n’utilise pas la symmétrie des évǹements de croisement et les conditions de bord périodiques. Certaines de ces nouvelles méthodes ont un champs d’application qui dépasse largement le cas du modèle de percolation de Fortuin–Kasteleyn et pourrait être utile pour prouver la décroissance exponentielles d’autres modèles de percolation dépendante.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 422-436.

Dates
Received: 27 May 2016
Revised: 21 November 2016
Accepted: 22 November 2016
First available in Project Euclid: 19 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1519030834

Digital Object Identifier
doi:10.1214/16-AIHP809

Mathematical Reviews number (MathSciNet)
MR3765895

Zentralblatt MATH identifier
06880060

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

Keywords
Phase transition Random-cluster model Potts model Critical point Sharp phase transition

Citation

Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent. A new computation of the critical point for the planar random-cluster model with $q\ge1$. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 422--436. doi:10.1214/16-AIHP809. https://projecteuclid.org/euclid.aihp/1519030834


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References

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