Abstract
The uniform spanning forest (USF) in $\mathbb{Z}^d$ 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 $d \le 4$. We prove that any two components of the USF in $\mathbb{Z}^d$ are adjacent a.s. if $5 \le d \le 8$, but not if $d \ge 9$. More generally, let $N(x,y)$ be the minimum number of edges outside the USF in a path joining $x$ and $y$ in $\mathbb{Z}^d$. Then \[ \max\bigl\{N(x,y): x,y\in\mathbb{Z}^d\bigr\} = \bigl\lfloor (d-1)/4 \bigr\rfloor \hbox{ a.s. } \] The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.
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