Random metrics on the CLE carpet

共形环系综(CLE)是定义在单连通区域中的随机环集合。自从它们被引入以来，CLE、Schramm-Loewner演化(SLE)曲线和高斯自由场(GFF)之间的联系已经被证明和分类。在这个项目中，我们考虑定义在不被CLE环包围的点集(CLE地毯)上的自然随机度量，并研究它们的性质。第一个目标是建立CLE、SLE和GFF之间的关系，以表明自然CLE度量中的测地线相对于SLE是奇异的。这证明了2014年的一个基于经验结果的猜测是错误的。

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.