Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

TL;DR

This work proves the existence and uniqueness (up to a global constant) of a canonical geodesic metric on the gasket of CLE$_{\kappa'}$ in the non-simple regime $\kappa'\in(4,8)$, and shows this metric is conformally covariant with a universal scaling exponent. The authors develop a robust framework combining weak/strong geodesic CLE$_{\kappa'}$ metrics, internal metrics, and a GFF-imaginary-geometry coupling, and establish key quantitative controls via ball-crossing estimates and across-scale resampling. A central result is that any two such metrics are bi-Lipschitz equivalent and, moreover, share the same exponent $\alpha$, yielding a unique scale-covariant framework. They extend the construction to interior clusters in general domains and the whole plane, proving conformal covariance and providing a foundation for future connections to the continuum limits of discrete chemical-distance metrics in critical models like percolation. The work positions the CLE$_{\kappa'}$ geodesic metric as a natural continuum metric for conformally invariant random geometries and as a potential scaling limit for discrete models converging to CLE$_{\kappa'}$.

Abstract

We construct the canonical geodesic metric on the gasket of conformal loop ensembles (CLE$_κ$) in the regime $κ\in (4,8)$ where the loops intersect themselves, each other, and the domain boundary. Previous work of the authors and V. Ambrosio showed that the subsequential limits associated with certain approximation procedures for such a metric exist and are non-trivial. In this work, we show that the limit exists by proving that there is at most one geodesic metric on the CLE$_κ$ gasket which satisfies certain properties. Further, we obtain that the limit is conformally covariant. This paper is the foundation of future work which show that the metric for $κ=6$ is the continuum scaling limit of the chemical distance metric for critical percolation in two dimensions. We further conjecture that for $κ\in (4,8)$, the geodesic CLE$_κ$ metric is the scaling limit of the chemical distance metric associated with discrete models that converge to CLE$_κ$.

Figures (3)

• Figure 1.1: Left: A sample of critical percolation on the triangular lattice in the unit disk. Middle: The regions surrounded by the outermost open cluster that surrounds the origin are shown in yellow. Right: The corresponding cluster, colored according to their graph distance to a point near the center.
• Figure 5.1: Illustration of the regions between the intersecting loops in the definition of the event $E^2_{z,j}$. On the left, $D_i^*$ is shown in yellow and on the right $U_i$ is shown in purple.
• Figure 5.2: The conditional law of the pair of dotted strands is a bichordal ${\rm SLE}_{\kappa'}$.

Theorems & Definitions (71)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 1.9
• Lemma 2.1