Scaling limits of critical FK$_q$ percolation and the fuzzy Potts model: proving convergence to CLE$_{\kappa'}$ and the divide-and-color arm exponents

Miller, Sheffield, and Werner determined the continuum law of divide-and-color interfaces ("CLE percolations"): coloring the loops of a non-simple conformal loop ensemble CLE$_{\kappa'}$, $\kappa' \in (4,8)$, independently red/blue with probabilities $p$ and $1-p$ yields interfaces that are SLE$_\kappa(\rho;\kappa-6-\rho)$ curves with an explicit $p \mapsto \rho$ map, a continuum Edwards-Sokal coupling relating CLE$_{\kappa'}$ and CLE$_{16/\kappa'}$ through $q = 4\cos^2(4\pi/\kappa')$, and arctan-type arm exponents for the fuzzy Potts (divide-and-color) models. However, every discrete consequence — including the half-plane one-arm exponent of Statement 1.1 for square-lattice Bernoulli percolation and the fuzzy Potts arm exponents derived later — is conditional on the unproven convergence of critical FK$_q$ random-cluster models to CLE$_{\kappa'}$. The problem is to prove this convergence, known only for the uniform spanning tree ($q \to 0^+$), site percolation on the triangular lattice ($q=1$), and the Ising case ($q=2$): concretely, conformal invariance of critical bond percolation on the square lattice, convergence of FK$_3$ to CLE$_{24/5}$, and the general $q \in (0,4)$ statement, together with the joint convergence of the cluster coloring that would make the arm exponents unconditional.

Background

Conformal loop ensembles (CLEs) are the scaling-limit objects for loop-type interfaces in two-dimensional critical statistical-mechanics models. A CLE$_\kappa$ is a countable collection of random loops in a simply connected planar domain, characterized by conformal invariance and a Markov-type resampling property; each loop locally looks like a Schramm-Loewner evolution (SLE$_\kappa$), the conformally invariant random curve introduced by Schramm. For $\kappa \in (8/3,4]$ the CLE loops are simple and disjoint; for $\kappa' \in (4,8)$ they are non-simple, i.e. they touch themselves, each other, and the domain boundary. Following the convention of the source paper, non-simple ensembles are denoted CLE$_{\kappa'}$ with $\kappa' \in (4,8)$.

The discrete counterparts are the Fortuin-Kasteleyn (FK) random-cluster model with cluster weight $q$ at criticality on a planar lattice (a correlated percolation model generalizing Bernoulli percolation, $q=1$) and the $q$-state Potts model, which are coupled by the Edwards-Sokal construction: coloring each FK$_q$ cluster with an independent uniform color out of $q$ recovers the Potts model. If instead one colors clusters independently red/blue with probabilities $r$ and $1-r$, one obtains the fuzzy Potts model, also called the divide-and-color model (for $q=1$ it is Bernoulli percolation clusters colored independently, the setting of Statement 1.1 below). The long-standing conjecture, going back to the origins of CLE, is that the FK$_q$ interface loops (free boundary conditions) converge to CLE$_{\kappa'}$ while the boundaries of Potts spin clusters converge to CLE$_\kappa$ with $\kappa = 16/\kappa'$, where $q$ and $\kappa'$ are related by $q = 4\cos^2(4\pi/\kappa')$; as $q$ increases from $0$ to $4$, $\kappa'$ decreases from $8$ to $4$. The precise formulation — critical FK$_q$ with free boundary conditions on the square lattice $\mathbb{Z}^2$ converging to the nested CLE$_{\kappa'}$ — goes back to Sheffield's construction of CLEs via exploration trees and is stated as Conjecture 2.8 (the "FK conformal invariance conjecture") by Kohler-Schindler and Lehmkuehler.

Miller, Sheffield, and Werner's "CLE percolations" (2017) described the continuum divide-and-color interfaces — the law of the red/blue interface when loops of a CLE$_{\kappa'}$ are colored independently — up to one missing parameter $\rho$. Miller, Sheffield, and Werner (2021), "Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces", completed the identification: it explores a CLE$_{\kappa'}$ drawn on an independent Liouville quantum gravity (LQG) surface — the random planar geometry built from the Gaussian free field with parameter $\gamma$, here $\gamma^2 = 16/\kappa'$, where the boundary surfaces cut out by the exploration are quantum disks and their "forested" variants — and shows the exploration is a Poissonian growth-fragmentation process (a Markovian tree encoding how explored regions split and shrink, as also arises in peeling of random planar maps). Consequences: (i) Theorem 1.2, the coloring parameter matches the cluster weight via $q(\kappa') = 4\cos^2(4\pi/\kappa')$, giving a continuum analog of the Edwards-Sokal coupling between CLE$_{\kappa'}$ and CLE$_{16/\kappa'}$; (ii) Theorem 1.3, the explicit $p \mapsto \rho$ map identifying the interface as an SLE$_\kappa(\rho;\kappa-6-\rho)$ with $\kappa = 16/\kappa'$; (iii) arctan-type half-plane arm exponents for fuzzy Potts models, in particular Statement 1.1: if critical Bernoulli percolation is conformally invariant in the scaling limit, then in the divide-and-color model on $\mathbb{Z} \times \mathbb{N}$ (critical bond percolation clusters colored independently red with probability $p$, blue with probability $1-p$), the probability $\mathbb{P}[E_R(p)]$ of the event $E_R(p)$ that a path of red sites joins the origin to the semi-circle of radius $R$ around the origin satisfies $\mathbb{P}[E_R(p)] = R^{-a(p)+o(1)}$ with

$a(p) = \frac{1}{6}\left(2-\frac{3}{\pi}\arctan\frac{p\sqrt{3}}{2-p}\right)\left(1-\frac{3}{\pi}\arctan\frac{p\sqrt{3}}{2-p}\right).$

Every discrete consequence of this program rests on one unproven hypothesis: the discrete-to-continuum convergence of critical FK$_q$ models to CLE$_{\kappa'}$. This has been proved only in the integrable cases: the uniform spanning tree ($q \to 0^+$), site percolation on the triangular lattice ($q=1$; Smirnov's conformal invariance and the Camia-Newman full scaling limit), and the Ising case $q=2$ (Smirnov's holomorphic fermions, the Kemppainen-Smirnov full scaling limit as a branching SLE, and Benoist-Hongler's identification of Ising interfaces with CLE$_3$). It is open for Bernoulli bond percolation on the square lattice, for FK$_3$ (expected to converge to CLE$_{24/5}$, with the coupled 3-state Potts clusters converging to CLE$_{10/3}$), and for all non-integer $q \in (0,4)$. The strongest progress toward the square-lattice cases is the large-scale rotational invariance of the critical random-cluster model with $q \in [1,4]$ on the square lattice proved by Duminil-Copin, Kozlowski, Krachun, Manolescu, and Oulamara, which does not imply conformal invariance. Later works in the line — the fuzzy Potts arm exponents of Kohler-Schindler and Lehmkuehler and the bulk one-arm exponent of Liu, Sun, Yu, and Zhuang — still take exactly this convergence as an assumption.

Problem Statement

Prove the discrete-to-continuum convergence conjecture underlying the CLE-percolation program of Miller-Sheffield-Werner in the non-simple regime, and thereby make its divide-and-color exponent formulas unconditional.

(Core conjecture) Let $q \in (0,4)$ and $\kappa' \in (4,8)$ satisfy $q = 4\cos^2(4\pi/\kappa')$ (equivalently $\kappa' = 4\pi/\arccos(-\sqrt{q}/2)$). In the precise formulation of the conjecture used in this line of work (Conjecture 2.8 of Kohler-Schindler and Lehmkuehler, the "FK conformal invariance conjecture", tracing back to Sheffield's exploration-trees paper): for a Jordan domain $D$ and discrete domains $D_n$ of mesh $\delta_n \to 0$ on the square lattice $\mathbb{Z}^2$ with $D_n \to D$, the interface loop ensemble of the critical FK$_q$ model on $D_n$ with free boundary conditions converges in law, as $n \to \infty$, to the nested CLE$_{\kappa'}$ in $D$. Prove this convergence. The boundary case $q = 4$ ($\kappa' = 4$) lies outside the non-simple regime $\kappa' \in (4,8)$ of the source paper and is not part of the problem as stated here (Kohler-Schindler and Lehmkuehler state their conjecture for $q \in (0,4]$). The conjecture is open already for: (i) $q=1$, critical Bernoulli bond percolation on the square lattice, where the conjecture is exactly the conformal invariance of the critical scaling limit that Statement 1.1 of the source paper hypothesizes — a Cardy-type crossing formula, convergence of exploration paths to SLE$_6$, and convergence of the loop ensemble to CLE$_6$; (ii) $q=3$, where FK$_3$ should converge to CLE$_{24/5}$ with the coupled 3-state Potts clusters converging to CLE$_{10/3}$; and (iii) all non-integer $q \in (0,4)$.

(Joint coloring convergence) Prove, jointly with the loop-ensemble convergence, that coloring the FK$_q$ clusters independently red/blue with probabilities $r$ and $1-r$ (the fuzzy Potts, or divide-and-color, model) converges to the continuum divide-and-color construction of CLE percolations — the coupled pair (CLE$_{\kappa'}$, CLE$_{16/\kappa'}$) with the Miller-Sheffield-Werner $p \mapsto \rho$ map — so that the fuzzy Potts arm exponents hold unconditionally for the discrete models. This includes Statement 1.1 of the source paper for $q=1$: for critical bond percolation on $\mathbb{Z} \times \mathbb{N}$ with clusters colored independently red with probability $p$ and blue with probability $1-p$ (the divide-and-color model), if $E_R(p)$ denotes the event that there exists a path of red sites joining the origin to the semi-circle of radius $R$ around the origin, then

$\mathbb{P}[E_R(p)] = R^{-a(p)+o(1)} \quad \text{as } R \to \infty, \quad a(p) = \frac{1}{6}\left(2-\frac{3}{\pi}\arctan\frac{p\sqrt{3}}{2-p}\right)\left(1-\frac{3}{\pi}\arctan\frac{p\sqrt{3}}{2-p}\right),$

together with the FK$_q$ generalizations established for $q \in [1,4)$ (fuzzy Potts half-plane and bulk arm exponents, including the bichromatic one-arm exponent $4/135$ for the critical 3-state Potts model).

The complete problem is the general $q \in (0,4)$ statement on the square lattice $\mathbb{Z}^2$.

References

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