Key points are not available for this paper at this time.
The uniform spanning forest (USF) in Z d is the weak limit of random, uniformly chosen, spanning trees in -n, n d . Pemantle 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 max N (x, y) : x, y Z d = (d -1)/4 a.s.
Benjamini et al. (Wed,) studied this question.