Conformal loop ensembles: the Markovian characterization and the loop-soup construction

Abstract

For random collections of self-avoiding loops in two-dimensional domains, we dene a simple and natural conformal restriction property that is conjecturally satised by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears at rst glance to be weaker. Our rst main result is that there exists exactly a one-dimensional family of random loop collections with this property | one for each 2 (8=3; 4] | and that the loops are forms of SLE . The proof proceeds in two steps. First, uniqueness is established by showing that every such loop ensemble can be generated by an \exploration process based on SLE. Second, existence is obtained using the two-dimensional Brownian loopsoup, which is a Poissonian random collection of loops in a planar domain. When the intensity parameterc of the loop-soup is less 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 the conformal restriction axioms. We prove various results about loop-soups, cluster sizes, and the c = 1 phase transition. Taken together, our results imply that the following families are equiv