CLE percolations

Abstract

Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and conformally invariant analog of the Sierpinski gasket or carpet. In the present paper, we derive a direct relationship between each CLE consisting of simple disjoint loops (CLE($κ$) with $κ$ between 8/3 and 4) and the corresponding CLE($κ'$) where $κ':=16/κ$, a CLE consisting of non-disjoint loops. This is the continuum analog of the Edwards-Sokal coupling (between the q-state Potts model and the associated FK random cluster model) and its generalization to non-integer q. Like its discrete analog, our continuum correspondence has two directions. First, we show that one can construct (variants of) CLE($κ$) as follows: sample a CLE($κ'$), then use a biased coin to independently color each loop one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret CLE($κ'$) loops as interfaces of a continuum analog of critical percolation within a CLE($κ)$ carpet. This is the first description of continuous percolation interfaces in fractal domains. These constructions provide new interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain results about generalized SLE$(κ;ρ)$ curves, and define a continuous family of natural CLE variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize CLEs, and that should be scaling limits of critical models with special boundary conditions. We extend the CLE correspondence to a BCLE correspondence that makes sense for all $κ$ between 2 and 4.