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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)

- Oded Schramm
- 1999

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish… (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)

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- 1999

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodu-larity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new mass-transport technique that has… (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 study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF) boundary conditions. Pemantle (1991) proved that the free and wired spanning forests coincide in Z d and that they give a single tree iff d 4. In the present… (More)

A comprehensive study of percolation in a more general context than the usual Z d setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results concerning uniqueness of innnite clusters and inequalities for the critical value p c are given, and a simple planar example exhibiting uniqueness and… (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)