Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12,… | Annals of Mathematics

Abstract The uniform spanning forest (USF) in ℤd 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 ℤd 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 ℤd. Then max{N(x,y):x,y∈ℤd}=⌊(d−1)/4⌋ a.s. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.

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数学年刊(Annals of Mathematics)

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