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We prove that the scaling limit of loop-erased random walk in a simply connected domain D C is equal to the radial SLE 2 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 ∂D is a C 1 simple closed curve, the… (More)
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… (More)
We show how to combine Kesten's scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov's proof of Cardy's formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.
This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a half-plane. For instance, suppose that B and B ′ are two independent planar Brownian motions started from distinct points in a half-plane H. Then as t → ∞, P B[0, t] ∩ B ′ [0, t] = ∅ and B[0, t] ∪ B ′ [0, t] ⊂ H = t −5/3+o(1). The… (More)
The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than R is proved to decay like R −5/48 as R → ∞.
We show that there exists (up to multiplicative constants) a unique and natural measure on simple loops on Riemann surfaces, such that the measure is conformally invariant and also invariant under restriction (i.e. the measure on a Riemann surface S ′ that is contained in another Riemann surface S, is just the measure on S restricted to those loops that… (More)
We derive the exact value of intersection exponents between planar Brow-nian motions or random walks, confirming predictions from theoretical physics by Duplantier and Kwon. Let B and B ′ be independent Brownian motions (or simple random walks) in the plane, started from distinct points. We prove that the probability that the paths B[0, t] and B ′ [0, t] do… (More)
A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the scaling limit of the uniform measures on these objects. The conjectures are based on recent results on the stochastic… (More)
This is the preliminary version of the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007.