Exploration trees and conformal loop ensembles

Abstract

We construct and study the conformal loop ensembles $\mathrm{CLE}(\kappa )$, defined for $8/3\le \kappa \le 8$, using branching variants of $\mathrm{SLE}(\kappa )$ called exploration trees. The $\mathrm{CLE}(\kappa )$ are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the $O(n)$ loop models