# Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12 ,...

@article{Benjamini2001GeometryOT,
title={Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12 ,...},
author={Itai Benjamini and Harry Kesten and Yuval Peres and Oded Schramm},
journal={Annals of Mathematics},
year={2001},
volume={160},
pages={465-491}
}
• Published 19 July 2001
• Mathematics
• Annals of Mathematics
The uniform spanning forest (USF) in ℤ d is the weak limit of random, uniformly chosen, spanning trees in [−n, n] d . Pemantle [11] proved that the USF consists a.s. of a single tree if and only if d ≤ 4. We prove that any two components of the USF in ℤ d are adjacent a.s. if 5 ≤ d ≤ 8, but not if d ≥ 9. More generally, let N(x, y) be the minimum number of edges outside the USF in a path joining x and y in ℤ d . Then \max \{ N(x,y):x,y{ \in \mathbb{Z}^d}\} = \left\lfloor {(d - 1)/4} \right…

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## References

SHOWING 1-10 OF 23 REFERENCES
Choosing a Spanning Tree for the Integer Lattice Uniformly
Consider the nearest neighbor graph for the integer lattice Zd in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs
Generating random spanning trees more quickly than the cover time
This paper gives a new algorithm for generating random spanning trees of an undirected graph that is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities.
STEIN, Spin-glass model with dimension-dependent ground state
• Phys. Rev. Lett
• 1994
Uniform spanning forests
• Mathematics
• 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)
Principles Of Random Walk
This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of
Some Random Series of Functions
1. A few tools from probability theory 2. Random series in a Banach space 3. Random series in a Hilbert space 4. Random Taylor series 5. Random Fourier series 6. A bound for random trigonometric
Principles of random walks