Cle Percolations

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 (CLEₗ₁₃₇₀₅ for x1D705 (8/3, 4), whose loops are Schramm’s SLEₗ₁₃₇₀₅ -type curves) and the corresponding CLEs with nonsimple loops (CLEₗ₁₃₇₀₅^ with x1D705^: =16/x1D705 (4, 6), whose loops are SLEₗ₁₃₇₀₅^ -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 x1D705 (8/3, 4), one can construct a variant of CLEₗ₁₃₇₀₅ as follows: start with an instance of CLEₗ₁₃₇₀₅^, then use a biased coin to independently color each CLEₗ₁₃₇₀₅^ 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 CLEₗ₁₃₇₀₅^ loops as interfaces of a continuum analog of critical Bernoulli percolation within CLEₗ₁₃₇₀₅ 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 CLE₆. 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 SLEₗ₁₃₇₀₅ (x1D70C) curves for x1D70C<-2, such as their decomposition into collections of SLEₗ₁₃₇₀₅ -type ‘loops’ hanging off of SLEₗ₁₃₇₀₅^ -type ‘trunks’, and vice versa (exchanging x1D705 and x1D705^). 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 CLE s, and that should be scaling limits of critical models with special boundary conditions. We extend the CLEₗ₁₃₇₀₅ / CLEₗ₁₃₇₀₅^ correspondence to a BCLE_{[S