CLE PERCOLATIONS

Abstract

Conformal loop ensembles (CLEs) 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 loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket. In the present paper, we derive a direct relationship between the CLEs with simple loops (CLE κ for κ ∈ ( 8 / 3 , 4 ) , whose loops are Schramm’s SLE κ -type curves) and the corresponding CLEs with nonsimple loops (CLE κ′ with κ′ := 16 /κ ∈ ( 4 , 6 ) , whose loops are SLE κ′ -type curves). This correspondence 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 noninteger q . Like its discrete analog, our continuum correspondence has two directions. First, we show that for each κ ∈ ( 8 / 3 , 4 ) , one can construct a variant of CLE κ as follows: start with an instance of CLE κ ′ , then use a biased coin to independently color each CLE κ′ loop in 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 Bernoulli percolation within CLE κ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by SLE₆ and CLE 6 . These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized SLEκ (ρ) curves for ρ < − 2, such as their decomposition into collections of SLE κ-type ‘loops’ hanging off of SLE κ′-type ‘trunks’, and vice versa (exchanging κ and κ′ ). We also 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 κ /CLE κ′ correspondence to a BCLE κ /BCLE κ′ correspondence that makes sense for the wider range κ ∈ ( 2 , 4 ] and κ′ ∈[ 4 , 8 )