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

Abstract

For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied 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 first glance to be weaker.

Our first main result is that there exists exactly a one-dimensional family of random loop collections with this property — one for each $\kappa \in (8/3,4]$ — and that the loops are forms of ${\rm SLE}_\kappa$. 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 loop soup, which is a Poissonian random collection of loops in a planar domain. When the intensity parameter $c$ 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 equivalent:

(1) the random loop ensembles traced by branching Schramm-Loewner Evolution (${\rm SLE}_\kappa$) curves for $\kappa$ in $(8/3, 4]$,
(2) the outer-cluster-boundary ensembles of Brownian loop-soups for $c \in (0, 1]$,
(3) the (only) random loop ensembles satisfying the conformal restriction axioms.

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Authors

Scott Sheffield

Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Ave.
Cambridge, MA 02139

Wendelin Werner

Laboratoire de Mathématiques
Université Paris-Sud
91405 Orsay
France
and
Ecole Normale Supérieure
75230 Paris
France