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 $q\ge 1$. Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for $1\le q\le 4$ and discontinuous for $q>4$. For $1\le q\le 4$, 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.

arXiv comment69 pages, 33 figures

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• Universal Critical Behavior of Percolation in Orientationally Ordered Janus Particles and Other Anisotropic Systems
• Dimerization and Néel order in different quantum spin chains through a shared loop representation
• Convergence of Fermionic Observables in the Massive Planar FK-Ising Model
• Structure of Gibbs measures for planar FK-percolation and Potts models
• Dimerization in quantum spin chains with $O(n)$ symmetry
• Cube moves for $s$-embeddings and $\alpha$-realizations
• Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function