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.
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Benjamini et al. (2004) studied this question.
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