Universality for the random-cluster model on isoradial graphs

TL;DR

This paper establishes the criticality and phase transition properties of the random-cluster model on isoradial graphs, showing universality of critical behavior and arm exponents across different lattice types, including quantum models.

Contribution

It proves the criticality and universality of phase transition types and arm exponents for the random-cluster model on all isoradial graphs, extending known results to a broad class of lattices.

Findings

The random-cluster measure is critical for all q ≥ 1 on isoradial graphs.

Phase transition is continuous for 1 ≤ q ≤ 4 and discontinuous for q > 4.

Arm exponents are universal across all isoradial graphs, including triangular and hexagonal lattices.

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.