Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction

TL;DR

This paper characterizes a unique family of conformally invariant loop ensembles in two dimensions, linking SLE curves, Brownian loop-soups, and conformal restriction properties, and explores their properties and phase transitions.

Contribution

It establishes the uniqueness and existence of conformal restriction loop ensembles related to SLE(k) and Brownian loop-soups, unifying different models under a common framework.

Findings

Existence of a one-parameter family of loop ensembles satisfying conformal restriction.

Equivalence of SLE(k) loop ensembles, Brownian loop-soup boundaries, and conformal restriction axioms.

Identification of phase transition at c=1 in loop-soup clusters.

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 k in (8/3,4]---and that the loops are forms of SLE(k). 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 certain branching SLE(k) curves for k 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.