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Scaling limits of loop-erased random walks and uniform spanning trees

- O. Schramm
- Mathematics
- 5 April 1999

AbstractThe 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… Expand

Basic properties of SLE

- S. Rohde, O. Schramm
- Mathematics
- 5 June 2001

SLEκ 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… Expand

Recurrence of Distributional Limits of Finite Planar Graphs

- I. Benjamini, O. Schramm
- Mathematics
- 2 November 2000

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… Expand

Conformal invariance of planar loop-erased random walks and uniform spanning trees

- G. Lawler, O. Schramm, W. Werner
- Mathematics
- 20 December 2001

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ \ne \mathbb{C} \) is equal to the radial SLE2 path. In particular, the… Expand

Conformal restriction: The chordal case

- G. Lawler, O. Schramm, W. Werner
- Mathematics
- 25 September 2002

We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane Η, say) which satisfy the conformal restriction” property, i.e., K connects two fixed… Expand

Uniform Infinite Planar Triangulations

- Omer Angel, O. Schramm
- Mathematics
- 18 July 2002

The existence of the weak limit as n→∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a… Expand

Uniform spanning forests

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- Mathematics
- 1 February 2001

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)… Expand

Embeddings of Gromov Hyperbolic Spaces

- M. Bonk, O. Schramm
- Mathematics
- 1 June 2000

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the… Expand

Tug-of-war and the infinity Laplacian

- Y. Peres, O. Schramm, S. Sheffield, D. Wilson
- Mathematics
- 28 April 2006

We consider a class of zero-sum two-player stochastic games called tug-of-war and use them to prove that every bounded real-valued Lipschitz function F on a subset Y of a length space X admits a… Expand

Percolation Beyond $Z^d$, Many Questions And a Few Answers

- I. Benjamini, O. Schramm
- Mathematics
- 10 August 1996

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… Expand

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