Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12
Probability
2009-04-28 v3 Mathematical Physics
math.MP
Abstract
The uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle 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 a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.
Cite
@article{arxiv.math/0107140,
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= {arXiv preprint arXiv:math/0107140},
year = {2009}
}
Comments
Current version: added some comments regarding related problems and implications, and made some corrections