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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 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)
Testing a property P of graphs in the bounded degree model deals with the following problem: given a graph G of bounded degree d we should distinguish (with probability 0.9, say) between the case that G satisfies P and the case that one should add/remove at least ε d n edges of G to make it satisfy P. In sharp contrast to property testing of dense(More)