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Scaling limits of loop-erased random walks and uniform spanning trees
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…
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Basic properties of SLE
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…
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Recurrence of Distributional Limits of Finite Planar Graphs
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…
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Conformal invariance of planar loop-erased random walks and uniform spanning trees
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…
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Conformal restriction: The chordal case
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…
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Uniform Infinite Planar Triangulations
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…
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Uniform spanning forests
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)…
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Embeddings of Gromov Hyperbolic Spaces
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…
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Tug-of-war and the infinity Laplacian
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…
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Percolation Beyond $Z^d$, Many Questions And a Few Answers
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…
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