Existence and uniqueness of the conformally covariant geodesic metric on simple conformal loop ensemble carpets

TL;DR

This paper establishes the existence and uniqueness of a canonical conformally covariant geodesic metric on CLE$_rac{8}{3}, 4)$ carpets, constructed as a scaling limit of Minkowski first passage percolation, and characterizes its properties.

Contribution

It proves the uniqueness of the CLE$_rac{8}{3}, 4)$ carpet metric as a scaling limit of MFPP, providing a canonical geometric structure for these fractal sets.

Findings

The geodesic metric on CLE$_rac{8}{3}, 4)$ carpets is unique and characterized by specific axioms.

The metric can be explicitly constructed as a limit of Minkowski first passage percolation.

Conjecture that this metric describes the scaling limit of chemical distances in discrete loop models.

Abstract

We prove that for each $\kappa \in (8/3, 4)$ there exists a geodesic metric on the carpet of a CLE$_\kappa$ which is canonical in the sense that it is characterized by a certain list of axioms. Our metric can be constructed explicitly as the scaling limit of Minkowski first passage percolation (MFPP), i.e., the metric obtained by taking the infimum of the Lebesgue measure of the $\varepsilon$-neighborhood of all paths connecting each pair of points. Earlier work by the first co-author showed that MFPP admits nontrivial subsequential limits. The present paper shows that this subsequential limit is unique and is characterized by our list of axioms. We conjecture that our metric describes the scaling limit of the chemical distance metric for discrete loop models that converge to CLE$_\kappa$ for $\kappa \in (8/3, 4)$ in the scaling limit, e.g., the critical Ising model for $\kappa=3$. Our argument is inspired by recent works of Gwynne and Miller and Ding and Gwynne on the uniqueness of Liouville quantum gravity metrics.