Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets

TL;DR

This paper investigates approximation schemes for conformally covariant metrics on CLE$_$ gaskets, establishing tightness and properties of subsequential limits, with conjectures linking these to discrete model scaling limits.

Contribution

It proves tightness of approximation schemes for metrics on CLE$_$ gaskets and characterizes their subsequential limits, advancing understanding of conformal metrics in CLE theory.

Findings

Laws of approximation schemes are tight.

Subsequential limits form non-trivial metrics.

Limits satisfy natural conformal properties.

Abstract

We study a class of approximation schemes aimed at constructing conformally covariant metrics defined in the gasket of a conformal loop ensemble (CLE$_\kappa$) for $\kappa \in (4,8)$. This is the range of parameter values so that the loops of a CLE$_\kappa$ intersect themselves, each other, and the domain boundary. Its gasket is the closure of the union of the set of points not surrounded by a loop. The class of approximation schemes includes approximations to the geodesic metric and to the resistance metric. We show that the laws of these approximations are tight, and that every subsequential limit is a non-trivial metric on the CLE$_\kappa$ gasket satisfying a natural list of properties. Subsequent work of the second two authors will show that the limits exist and are conformally covariant both in the setting of the geodesic and resistance metrics. We conjecture that the geodesic (resp. resistance) metric describes the scaling limit of the chemical distance (resp. resistance) metric associated with discrete models that converge in the limit to CLE$_\kappa$ for $\kappa \in (4,8)$ (e.g., critical percolation for $\kappa=6$).