The fuzzy Potts model in the plane: Scaling limits and arm exponents

Understand

We consider a critical Fortuin-Kasteleyn (FK) percolation with cluster weight $q \in [1,4)$ in the plane, and color its clusters in red (respectively blue) with probability $r \in (0,1)$ (respectively $1-r$), independently of each other.

• We study the resulting fuzzy Potts model, which corresponds to the critical Ising model in the special case $q=2$ and $r=1/2$.
• We show that under the assumption that the critical FK percolation converges to a conformally invariant scaling limit (which is known to hold for the FK-Ising model, i.e.
• $q=2$), the obtained coloring converges to variants of Conformal Loop Ensembles constructed, described and studied by Miller, Sheffield and Werner.