The scaling-limit conjecture for fuzzy Potts clusters and cluster measures

Fix $q\in(1,4)$, set $r=1/q$, and let $\omega^\delta$ be the fuzzy Potts configuration on $\delta\mathbb{Z}^2$. For lattice approximations $z_i^\delta$ of $z_1,\ldots,z_n\in\mathbb{C}$, let $P_n^\delta(z_1,\ldots,z_n)$ be the probability that they all lie in the same red cluster, and let $\pi_\delta$ be the probability of a red nearest-neighbor path from the origin to the unit circle. Define

$$P_n(z_1,\ldots,z_n):=\lim_{\delta\to0}\pi_\delta^{-n}P_n^\delta(z_1,\ldots,z_n).$$

Let $\{\mathcal{C}_i^\delta\}$ be the red clusters and let $\mathcal{M}_i^\delta$ be counting measure on the vertices of $\mathcal{C}_i^\delta$, normalized by $\delta^2\pi_\delta^{-1}$. Fuzzy Potts scaling-limit conjecture. The displayed limit exists, and under the weak Hausdorff topology,

$$(\omega^\delta,\{\mathcal{M}_i^\delta\})\overset{d}{\longrightarrow}(\omega,\{C\cdot\mathcal{M}_i\})\qquad\text{as }\delta\to0,$$

where $\omega$ is the continuum fuzzy Potts configuration, $\mathcal{M}_i$ is the Miller--Schoug measure on the continuum cluster $\mathcal{C}_i$, and $C>0$ is a constant. This conjecture supplies the scaling-limit interpretation of the continuum Green's functions and connectivity probabilities used in the paper; the theorem preceding it computes the resulting constant $R(q)$, but does not establish this discrete-to-continuum convergence.