We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H, say) which satisfy the "conformal restriction" property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ(K), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 .

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that aD is a C 1 -simp1e closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc A c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

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 loopsoup, 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.