The uniform spanning forest (USF) in ℤd 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 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
The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.