Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12, …

Abstract

The uniform spanning forest (USF) in ℤd is the weak limit of random, uniformly chosen, spanning trees in [−n, n]d. 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. Then $$\max \{ N(x,y):x,y{ \in \mathbb{Z}^d}\} = \left\lfloor {(d - 1)/4} \right\rfloor {\textrm{a.s}}.$$ The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.Key words and phrasesStochastic dimensionUniform spanning forest