On conformally invariant CLE explorations

TL;DR

This paper introduces conformally invariant dynamic methods to construct Conformal Loop Ensembles (CLE), providing a new way to measure distances within CLE(4) loops while maintaining conformal invariance.

Contribution

It presents novel conformally invariant exploration techniques for CLE and a method to measure distances within CLE(4) loops, advancing understanding of their geometric properties.

Findings

Developed conformally invariant CLE exploration methods

Established a way to measure distances within CLE(4) loops

Enhanced understanding of CLE geometric structure

Abstract

We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles with simple loops introduced in earlier papers by Sheffield, and by Sheffield and Werner. One outcome is a conformally invariant way to measure a distance of a CLE(4) loop to the boundary "within" the CLE(4), when one identifies all points of each loop.