In 2013, Werner and Wu introduced conformally invariant explorations of the conformal loop ensemble (CLE): an asymmetric family, obtained by signing the excursions of the Bessel-type process that drives the exploration tree with independent biased coins, and a uniform exploration of the CLE with critical parameter κ=4\kappa=4. Their paper closed with three determinacy questions - whether these explorations are deterministic functions of the underlying CLE, which for the uniform exploration would yield a conformally invariant distance between the loops of a CLE4_4 - and with a technical challenge: the natural discrete, step-size-ϵ\epsilon counterparts of the asymmetric and uniform explorations were asserted, but not proved, to converge in distribution to the continuous constructions. Later literature settled the determinacy questions: for κ∈(8/3,4)\kappa\in(8/3,4) the exploration tree is not a deterministic function of the labeled CLEκ_\kappa and its root, while at κ=4\kappa=4 the explorations are deterministic functions of the labeled CLE4_4, the uniform exploration is almost surely determined by the CLE4_4 itself, and the conjectured conformally invariant metric on the loops of the CLE4_4 has been constructed and shown unique. The piece that remains open, and that this problem records, is the original discrete challenge: prove or disprove that the asymmetric discrete exploration (biased coin choice compensated by a vanishing deterministic rotation) and the harmonic-measure uniform discrete exploration converge in distribution, as ϵ→0\epsilon\to 0, to their continuous counterparts.

Background

The Schramm-Loewner evolution SLEκSLE_\kappa is the one-parameter family of random planar curves, built from Loewner's conformal-mapping evolution of a domain driven by a Brownian motion with speed parameter κ>0\kappa>0, that describes the scaling limits of interfaces in two-dimensional critical statistical-mechanics models. The conformal loop ensemble CLEκCLE_\kappa is the corresponding canonical random collection of nested, non-crossing loops in a simply connected planar domain; for κ∈(8/3,4]\kappa\in(8/3,4] the loops are simple and the ensemble is the (conjectured or proved, depending on the model) scaling limit of the full family of cluster interfaces - for instance the loops of the CLE4CLE_4 are the level lines of the continuum Gaussian free field (the scale-invariant random generalized function on the domain), i.e. the curves on which that field vanishes with a fixed sign on each side.

The most effective way to analyze a CLE is to explore it. Starting from a boundary point, one grows a curve that traces each loop it meets and then continues along one side of it, so that the explored branches form a tree of SLEκ(κ−6)SLE_\kappa(\kappa-6)-type paths (Loewner chains whose driving Brownian motion receives an additional drift term from a force point that follows the curve). Sheffield introduced these exploration trees and, together with Werner, used a discrete version of the symmetric exploration to give the Markovian characterization of CLE. After normalization, the driving function of the trunk of the exploration (the exploration path with the loop-tracings removed) is a Bessel process of dimension δ=3−8/κ∈(0,1]\delta=3-8/\kappa\in(0,1], a self-similar diffusion on the nonnegative reals, and each excursion of this process away from zero corresponds to the tracing of one loop. Werner and Wu studied the explorations obtained by biasing the side on which the exploration bypasses each discovered loop: one glues the excursions back into a driving function after multiplying each by an independent sign ϵ(e)∈{−1,+1}\epsilon(e)\in\{-1,+1\} chosen with probability (1+β)/2(1+\beta)/2 for +1+1, and the bias parameter β∈[−1,1]\beta\in[-1,1] produces a one-parameter family of asymmetric explorations, with β=0\beta=0 the symmetric tree and β=1\beta=1 the totally asymmetric one. Since each excursion corresponds to one discovered loop, the signs can equivalently be indexed by the loops of the CLE. At κ=4\kappa=4 the side-swapping must be symmetric, and asymmetry is instead carried by a real drift parameter μ\mu of the driving process; the μ→∞\mu\to\infty limit is a genuinely new object, the uniform exploration, which Werner and Wu also construct directly as a Poisson point process of discovered loops (a random countable family of loops, locally finite and with independent increments in the exploration time) with an intensity invariant under the Moebius self-maps of the disk, and which is conformally invariant without any time change.

Section 5 of Werner and Wu records what their methods leave open: (i) whether the asymmetric exploration with parameter β\beta is a deterministic function of the CLE, the starting point, and the coin variables; (ii) whether the totally asymmetric exploration is a deterministic function of the CLE and the starting point alone; and (iii) whether the uniform exploration is a deterministic function of the CLE4CLE_4, a positive answer giving a conformally invariant distance between the loops of a CLE4CLE_4. These determinacy questions have since been resolved - negatively for κ∈(8/3,4)\kappa\in(8/3,4) and positively at κ=4\kappa=4 - and the conformally invariant metric on the loops of the CLE4CLE_4 has been constructed; the lineage is detailed in references 4-10 below.

The same section also defines discrete counterparts of the asymmetric and uniform explorations, in direct analogy with the symmetric discrete exploration that powers the Markovian characterization. Fix a step size ϵ>0\epsilon>0 and iterate: in the unit disk, explore the small boundary set D(1,ϵ)D(1,\epsilon), discover all loops that meet it, map the connected component containing the origin conformally back onto the disk by a map fixing the origin, and choose which boundary point is sent to 11 according to a rule that uses no information about the loops not yet discovered; the number of steps until the loop surrounding the origin is discovered is geometric (a memoryless waiting-time distribution), so the iteration terminates almost surely. The symmetric rule tosses a fair coin between the two intersection points of the explored set with the unit circle; Sheffield and Werner proved that the resulting discrete exploration converges in distribution, as ϵ→0\epsilon\to 0, to the continuous symmetric exploration, and this is a key step in the Markovian characterization of CLE. The asymmetric rule tosses a coin with probabilities (1+β)/2(1+\beta)/2 and (1−β)/2(1-\beta)/2 for the two points, and compensates the bias thereby created by post-composing each conformal map with a deterministic rotation of angle θ(ϵ)\theta(\epsilon) tending to 00 with ϵ\epsilon - a rotation that depends on the chosen base point, which is why the authors call this discrete approximation "a little harder to master" than the symmetric case. The uniform rule chooses the boundary point according to the harmonic measure seen from the origin - the exit distribution of a Brownian motion started at the origin, which is the conformally invariant notion of a uniform choice seen from the target point - equivalently, by composing with an independent uniformly distributed rotation. Werner and Wu leave it "to the interested (and motivated) reader" to check that these discrete explorations converge in distribution to the continuous asymmetric and uniform constructions; no proof of these two convergence statements has appeared, and establishing them is the open problem recorded here.

Problem Statement

Let Γ\Gamma be a CLEκCLE_\kappa in the unit disk U\mathbb{U} with κ∈(8/3,4)\kappa\in(8/3,4), and let ϵ>0\epsilon>0. Define the asymmetric discrete exploration of Γ\Gamma with parameter β∈[−1,1]\beta\in[-1,1] and step size ϵ\epsilon as in Section 5.2 of Werner and Wu (2013): iteratively explore the set D(1,ϵ)D(1,\epsilon) (the image of {z∈H:∣z∣<ϵ}\{z\in\mathbb{H}:|z|<\epsilon\} under the conformal map z↦(i−z)/(i+z)z\mapsto(i-z)/(i+z) from the upper half-plane onto the disk that sends ii to 00 and 00 to 11), discover the loops of the current ensemble that intersect it, map the connected component containing the origin conformally back onto U\mathbb{U} by a map that fixes the origin, and choose the boundary point sent to 11 by tossing a coin with probabilities (1+β)/2(1+\beta)/2 and (1−β)/2(1-\beta)/2 for the two intersection points of ∂D(1,ϵ)\partial D(1,\epsilon) with the unit circle, post-composing the map with a deterministic rotation of angle θ(ϵ)\theta(\epsilon), where θ(ϵ)→0\theta(\epsilon)\to 0 as ϵ→0\epsilon\to 0 and θ(ϵ)\theta(\epsilon) is a function only of ϵ\epsilon and the chosen base point; we call such a rotation admissible; at every step the choice rule must remain conditionally independent of the loops not yet discovered. Define the uniform discrete exploration by the same iteration, with the point sent to 11 chosen according to the harmonic measure on the boundary seen from the origin, equivalently by composing with an independent uniformly distributed rotation.

Prove, or disprove, the two statements that Werner and Wu left to "the interested (and motivated) reader":

(1) For every κ∈(8/3,4)\kappa\in(8/3,4) and every β∈[−1,1]\beta\in[-1,1], there is an admissible rotation θ(ϵ)→0\theta(\epsilon)\to 0 such that the asymmetric discrete exploration with parameter β\beta converges in distribution, as ϵ→0\epsilon\to 0, to the continuous asymmetric exploration of the CLEκCLE_\kappa with parameter β\beta - the exploration whose driving function is obtained by gluing the excursions of the Bessel process of dimension 3−8/κ3-8/\kappa back together with independent signs of bias β\beta.

(2) The uniform discrete exploration of a CLE4CLE_4 in U\mathbb{U} converges in distribution, as ϵ→0\epsilon\to 0, to the continuous uniform exploration of the CLE4CLE_4 - the exploration defined by the Poisson point process of discovered loops with Moebius-invariant intensity, equivalently the limit of the drift-μ\mu CLE4CLE_4 explorations as μ→∞\mu\to\infty.

Convergence is to be understood for the full exploration - the collection of discovered loops together with the exploration path - in any explicitly stated standard topology that determines it, for instance locally uniform convergence of the induced conformal maps (the Caratheodory topology), or convergence of the encoded exploration paths.

The determinacy questions of Section 5.1 of Werner and Wu have been resolved in later work (see references 4-10), and the discrete-to-continuum convergence of the symmetric procedure was proved by Sheffield and Werner in their Markovian characterization of CLE; the asymmetric and uniform discrete procedures are the remaining cases.

References

  1. Wendelin Werner and Hao Wu. On conformally invariant CLE explorations. Communications in Mathematical Physics 320 (2013), no. 3, 637-661. DOI: 10.1007/s00220-013-1719-9. https://doi.org/10.1007/s00220-013-1719-9 (author preprint: arXiv:1112.1211, https://arxiv.org/abs/1112.1211)
  2. Scott Sheffield. Exploration trees and conformal loop ensembles. Duke Mathematical Journal 147 (2009), no. 1, 79-129. DOI: 10.1215/00127094-2009-007. https://doi.org/10.1215/00127094-2009-007
  3. Scott Sheffield and Wendelin Werner. Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Annals of Mathematics 176 (2012), no. 3, 1827-1917. DOI: 10.4007/annals.2012.176.3.8. https://doi.org/10.4007/annals.2012.176.3.8
  4. Jason Miller, Scott Sheffield, and Wendelin Werner. CLE percolations. Forum of Mathematics, Pi 5 (2017), paper no. e4. DOI: 10.1017/fmp.2017.5. https://doi.org/10.1017/fmp.2017.5 (author preprint: arXiv:1602.03884, https://arxiv.org/abs/1602.03884)
  5. Jason Miller, Scott Sheffield, and Wendelin Werner. Non-simple SLE curves are not determined by their range. Journal of the European Mathematical Society 22 (2020), no. 3, 669-716. DOI: 10.4171/jems/930. https://doi.org/10.4171/jems/930 (author preprint: arXiv:1609.04799, https://arxiv.org/abs/1609.04799)
  6. Matthis Lehmkuehler. The trunks of CLE(4) explorations. The Annals of Applied Probability 33 (2023), no. 5, 3387-3417. DOI: 10.1214/22-AAP1895. https://doi.org/10.1214/22-AAP1895 (author preprint: arXiv:2107.05310, https://arxiv.org/abs/2107.05310)
  7. Emmanuel Kammerer. Distances on the CLE4, critical Liouville quantum gravity and 3/2-stable maps. Probability and Mathematical Physics 6 (2025), no. 3, 1111-1180. DOI: 10.2140/pmp.2025.6.1111. https://doi.org/10.2140/pmp.2025.6.1111 (author preprint: arXiv:2311.08571, https://arxiv.org/abs/2311.08571)
  8. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, and Yi Tian. The conformally invariant metric on CLE4 I: subsequential limits of the non-simple CLE graph metric. Preprint, September 2026. arXiv:2609.04138. https://arxiv.org/abs/2609.04138
  9. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, and Yi Tian. The conformally invariant metric on CLE4 II: existence of geodesics. Preprint, September 2026. arXiv:2609.04139. https://arxiv.org/abs/2609.04139
  10. Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, and Yi Tian. The conformally invariant metric on CLE4 III: uniqueness. Preprint, September 2026. arXiv:2609.04140. https://arxiv.org/abs/2609.04140
  11. Wendelin Werner and Hao Wu. From CLE(κ) to SLE(κ,ρ)'s. Electronic Journal of Probability 18 (2013), no. 36, 1-20. arXiv:1210.3264. https://arxiv.org/abs/1210.3264