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.
The Interface of the FK-representation of the Quantum Ising Model Converges to the SLE(16/3) to appear in Probability Theory and Related Fields.
We study the interface in the FK-representation of the 1D quantum
Ising model and show that in the limit, it converges to the SLE(16/3) curve.
Personal notes
PhD : Universality of the Random-Cluster Model and Applications to Quantum Systems under the supervision of H. Duminil-Copin and S. Smirnov, 12th December 2017.