Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12

Details Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12 Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12

2001-07-19

arxiv.org/abs/math/0107140v3

Annals Math.160:465-491,2004

Mathematics

Probability

Current version: added some comments regarding related problems and implications, and made some corrections

Scientific paper

10.4007/=

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.

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