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

TL;DR

This paper investigates the scaling limits and arm exponents of the fuzzy Potts model derived from critical FK percolation in the plane, establishing conformal invariance and connecting discrete and continuum models.

Contribution

It proves the convergence of the fuzzy Potts model to conformal loop ensembles and computes arm exponents based on this convergence.

Findings

Convergence of the fuzzy Potts model to conformal loop ensembles.

Arm exponents in the discrete model match those in the continuum.

Explicit arm exponents for the fuzzy Potts model are determined.

Abstract

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. Based on discrete considerations, we also show that the arm exponents for this coloring in the discrete model are identical to the ones of the continuum model. Using the values of these arm exponents in the continuum, we determine the arm exponents for the fuzzy Potts model.