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 κ∈(8/3,4] — and that the loops are forms of SLEκ. 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 (SLEκ) curves for κ in (8/3,4],
(2) the outer-cluster-boundary ensembles of Brownian loop-soups for c∈(0,1],
(3) the (only) random loop ensembles satisfying the conformal restriction axioms.

KEYWORDS

SHARE & LIKE

COMMENTS

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Read MoreAbstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Read MoreAbstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Read MoreAbstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Read MoreAbstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Read MoreAbstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Read MoreAbstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Read MoreAbstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Read MoreAbstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Read MoreAbstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt

Read More