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

TL;DR

This work analyzes the fuzzy Potts model on critical FK percolation in the plane, combining a two-layer randomness (FK sampling plus cluster coloring) with a conjectured conformal scaling limit for FK percolation to CLE. The authors connect discrete arm exponents to their continuum CLE/BCLE counterparts, deriving the full set of arm exponents in the plane and half-plane, and proving convergence of discrete fuzzy Potts interfaces and loop encodings to their continuum colored-CLE descriptions under the FK-to-CLE conjecture (unconditional for FK-Ising, $q=2$). They develop discrete tools—arm separation and quasi-multiplicativity—for the fuzzy Potts model, then leverage continuum SLE/imaginary-geometry exponents to obtain the continuum arm exponents, finally transferring them back to the discrete setting. The results yield explicit exponent formulas in terms of $q$ and the coloring parameter $r$, with even-arm exponents independent of $r$ in the continuum, and special, unconditional cases recovered for Ising. This provides a rigorous link between two-layer random colorings, conformal loop ensembles, and arm-exponent theory in planar critical models, shaping the understanding of fuzzy Potts scaling limits and their universality.

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.

Figures (14)

• Figure 1: Left. The set $\cup_{C\in \mathcal{C}(\omega)} O^C_\omega$ associated to a bond percolation configuration $\omega$ is shaded in blue and the clusters $\mathcal{C}(\omega)$ are colored to obtain a configuration $\sigma$. Edges which are part of the boundary of the domain but are not open with respect to $\omega$ are drawn as a dashed line. Center. We consider a site percolation configuration $\sigma$ and draw a tile $X^+$ (as shown at the bottom) over each blue vertex and take the union to obtain the area shaded in blue. Right. The set in the center figure is intersected with the closure of the domain to obtain $O^+_\sigma$. This step corresponds to the definition of blue clusters in terms of weak paths which stay within the discrete domain. The curve in green shows the curve $\gamma^+_{\sigma,a,b}$ from the boundary point $a$ (at the bottom) to $b$ (at the top).
• Figure 2: This figure illustrates the relation between $\Xi\sim \mathop{\mathrm{BCLE}}\nolimits_\kappa(\rho)$ (in red), its false loops $\Xi^*\sim \mathop{\mathrm{BCLE}}\nolimits_\kappa(\kappa-6-\rho)$ (the boundaries of the regions shaded in light blue) and the curve $\gamma\sim \mathop{\mathrm{SLE}}\nolimits_\kappa(\rho,\kappa-6-\rho)$ (in green).
• Figure 3: The picture illustrates the constructions relevant for Theorems \ref{['thm:msw-free']} and \ref{['thm:msw-wired']}. The collection of $\Xi_B$ should be interpreted as the divide and color interfaces touching the boundary which are blue on the inside and red on the outside. $\Xi_B'$ are then the outer boundaries and $\Xi_B"$ the inner boundaries of the blue percolation clusters touching the divide and color interfaces from the inside; similarly, $\Xi_R'$ are the outer boundaries and $\Xi_R"$ are the inner boundaries of the percolation clusters touching the outside of the divide and color interfaces (and are hence contained in $\Xi_R:=\Xi_B^*$). The situation is symmetric when 'red' is interchanged with 'blue'. In each of the areas shaded in red (resp. blue), we now iterate with red (resp. blue) boundary conditions on the outside.
• Figure 4: Top left. This figure illustrates Proposition \ref{['prop:msw-interface-approx']}; the green curve is $\gamma^{a,b}$, the area shaded in green is $\gamma^{a,b}([s,t])+B_\epsilon(0)$ and the dashed blue curve is $\widetilde{\gamma}$. Top right. This graphic explains Corollary \ref{['cor:msw-loop-approx']}. The green loop is $\eta$, the shaded are in green is $\eta(\partial\mathbb{D})+B_\epsilon(0)$ and the dashed blue loop is $\widetilde{\eta}$. The bottom row illustrates the argument which can be used to derive Corollary \ref{['cor:msw-loop-approx']} from Proposition \ref{['prop:msw-interface-approx']}. Bottom left. The green curve is $\gamma^{a,b}$ restricted to $[s,t]$ and the dashed blue curve is the approximating curve appearing in the proposition (see also the top left part of this figure). Crucially, the restriction of $\gamma^{a,b}$ restricted to $[s,t]$ yields a segment of the loop $\eta$. Bottom right. The green curve is $\gamma^{b,a}$ restricted to $[s',t']$ and the dashed blue curve is again the approximating curve as in the proposition. Again this curve segment forms part of $\eta$. The key is that the approximating curves appearing in the two bottom figures intersect which readily implies the corollary.
• Figure 5: Left. This figure illustrates Lemma \ref{['lem:imaginary-map-in']}. The relevant imaginary geometry boundary conditions have been drawn in for the reader's convenience where $\lambda = \pi/\sqrt{\kappa}$. Right. Lemma \ref{['lem:sle-excursion']} is illustrated: The conditional law of the blue curve given the red curves is an $\mathop{\mathrm{SLE}}\nolimits_\kappa(0,\rho')$ in the complementary domain (up to time reparametrization). Both figures are drawn in the upper halfplane rather than the unit disk for illustrational purposes.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Corollary 2.1
• Corollary 2.2
• Theorem 2.3: duminil2021planar
• Definition 2.4
• Corollary 2.5: duminil2021planar
• Theorem 2.6: duminil2021planar