PR ] 1 3 Fe b 20 03 Geometry of the Uniform Spanning Forest : Transitions in Dimensions

The uniform spanning forest (USF) in Z is the weak limit of random, uniformly chosen, spanning trees in [−n, n]. Pemantle (1991) 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 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. 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. 1991 Mathematics Subject Classification. Primary 60K35. Secondary 60J15.