Conformal Loop Ensembles: Construction via Loop-soups

Understand

The two-dimensional Brownian loop-soup is a Poissonian random collection of loops in a planar domain with an intensity parameter c.

• When c is not greater than 1, we show that the outer boundaries of the loop clusters are disjoint simple loops (when c>1, there is almost surely only one cluster) that satisfy certain conformal restriction axioms.
• We prove various results about loop-soups, cluster sizes, and the c=1 phase transition.
• Combining this with the results of another paper of ours on the Markovian characterization of simple conformal loop ensembles (CLE), this proves that these outer boundaries of clusters of Brownian loops are in fact SLE(k) loops for k in (8/3, 4].