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

TL;DR

This paper establishes the existence, uniqueness, and conformal covariance of a canonical geodesic metric on CLE$_$ gaskets for  in (4,8), laying groundwork for linking discrete models to continuum limits.

Contribution

It proves the unique existence and conformal covariance of the geodesic metric on CLE$_$ gaskets, advancing understanding of their geometric structure.

Findings

Existence of a unique conformally covariant geodesic metric on CLE$_$ gaskets.

The metric is the continuum scaling limit of the chemical distance in critical percolation for =6.

Foundation for future work linking discrete models to CLE$_$ geometries.

Abstract

We construct the canonical geodesic metric on the gasket of conformal loop ensembles (CLE$_\kappa$) in the regime $\kappa \in (4,8)$ 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$_\kappa$ 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 $\kappa=6$ is the continuum scaling limit of the chemical distance metric for critical percolation in two dimensions. We further conjecture that for $\kappa \in (4,8)$, the geodesic CLE$_\kappa$ metric is the scaling limit of the chemical distance metric associated with discrete models that converge to CLE$_\kappa$.