## Electronic Journal of Probability

### Universality for the random-cluster model on isoradial graphs

#### Abstract

We show that the canonical random-cluster measure associated to isoradial graphs is critical for all $q \geq 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1 \leq q \leq 4$ and discontinuous for $q > 4$. For $1 \leq q \leq 4$, the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in $1+1$ dimensions.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 96, 70 pp.

Dates
Received: 18 November 2017
Accepted: 12 September 2018
First available in Project Euclid: 19 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1537322681

Digital Object Identifier
doi:10.1214/18-EJP223

Mathematical Reviews number (MathSciNet)
MR3858924

Zentralblatt MATH identifier
06964790

Subjects
Primary: 60 82

#### Citation

Duminil-Copin, Hugo; Li, Jhih-Huang; Manolescu, Ioan. Universality for the random-cluster model on isoradial graphs. Electron. J. Probab. 23 (2018), paper no. 96, 70 pp. doi:10.1214/18-EJP223. https://projecteuclid.org/euclid.ejp/1537322681

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