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- 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)

Suppose that Gj is a sequence of finite connected planar graphs, and in each Gj a special vertex, called the root, is chosen randomlyuniformly. We introduce the notion of a distributional limit G of such graphs. Assume that the vertex degrees of the vertices in Gj are bounded, and the bound does not depend on j. Then after passing to a subsequence, the… (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)

It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random errors gives almost no prediction whether the event occurs. On the other hand, weighted majority functions are shown to be noise-stable. Several necessary… (More)

We prove that the scaling limit of loop-erased random walk in a simply connected domain D $ C 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 ∂D is a C1 simple closed curve, the… (More)

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, 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 masstransport 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 subsetH of H is identical to that of Φ(K), where Φ is a… (More)

- Oded Schramm
- 1997

Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is innnite dimensional. In Particular, Doyle's conjecture is false in this setting. MM obius invariants of circle patterns are introduced, and turn out to be… (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 6 4. In the present… (More)

Let 0 < a < b < ∞, and for each edge e of Zd let ωe = a or ωe = b, each with probability 1/2, independently. This induces a random metric distω on the vertices of Z d, called first passage percolation. We prove that for d > 1 the distance distω(0, v) from the origin to a vertex v, |v| > 2, has variance bounded by C |v|/ log |v|, where C = C(a, b, d) is a… (More)