Random metrics on the CLE carpet

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Conformal loop ensembles (CLE) are random collections of loops defined in simply connected domains. They exhibit a fractal structure and arise as conjectured and proved scaling limits of a number of lattice models from Statistical Physics.Since their introduction, connections between CLE, Schramm-Loewner evolution (SLE) curves, and the Gaussian Free Field (GFF) have been shown and categorized. In this project we consider natural random metrics defined on the set of points not surrounded by CLE loops (the CLE carpet) and investigate their properties. A first goal is to build on the relationship between CLE, SLE, and the GFF to show that geodesics in a natural CLE metric are singular with respect to SLE. This disproves a conjecture from 2014 that was based on empirical results.