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SLE κ is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed κ. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in(More)
Suppose that G j is a sequence of finite connected planar graphs, and in each G j a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit G of such graphs. Assume that the vertex degrees of the vertices in G j are bounded, and the bound does not depend on j. Then after passing to a subsequence, the(More)
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)
  • Itai Benjamini, Russell Lyons, Yuval Peres, Oded Schramm
  • 1998
In the introduction, Theorems 12.4 and 14.2 are incorrectly summarized. If the word " finite " is replaced by " bounded " , then the summaries become correct because in that case, the dual graph is also recurrent. Equation (4.2) is correct, but is said to give the matrix coefficient. In fact, the matrix coefficient is Y (e, f) because χ f does not have unit(More)
  • Yuval Peres, Oded Schramm, Scott Sheffield, David B Wilson
  • 2005
We prove that every bounded Lipschitz function F on a subset Y of a length space We also prove the first general uniqueness results for ∆ ∞ u = g on bounded subsets of R n (when g is uniformly continuous and bounded away from 0), and analogous results for bounded length spaces. The proofs rely on a new game-theoretic description of u. Let u ε (x) be the(More)
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)