Geometry Of The Uniform Spanning Forest: Transitions In Dimensions 4, 8, 12, ...

The uniform spanning forest (USF) in Z(d) is the weak limit of random, uniformly chosen, spanning trees in [-n, n]. 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. Thenmax{N(x,y): x, y epsilon Z(d)} = [(d - 1)/4] a.s.The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.