I try to understand the Geometry of the Uniform Spanning Forest in higher Dimensions and looking especially at the Work from ITAI BENJAMINI, HARRY KESTEN, YUVAL PERES, AND ODED SCHRAMM "Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12" https://arxiv.org/abs/math/0107140
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_{x,y \in \mathbb{Z}^{d}} N(x,y) = \lfloor (d- 1)/4 \rfloor a.s. $
The challening part of the proof is proofing the upper bound. Short Version: They establish a random Relation $U^{m+1}:=\{ (x,y) \in \mathbb{Z}^{d} | N(x,y) \leq m \}$, proofing that $U^{m+1} $ stochastically dominatess the compostion of (m+1) independent Copies of $U^{1} $. Some very beautiful arguments leeds to $P((x,y) \in (m+1)-independent \; Copies \; of \; U^{1} ) > 0 \; \forall x,y \in \mathbb{Z}^{d} $. Afterwards some non-obvious "Tail-Trivialty" Arguments finish the Proof.
Basic to the non-obvious "Tail-Trivialty" Arguments, is proofing that the Relation $U^{1}$ has trivial left, right and remote Tails. Definition: Let $\lambda \subset \mathbb{Z}^{d} \times \mathbb{Z}^{d}$. Let $\mathcal{F}_{\lambda}$ be the $\sigma$-Field generated by the events $\{ (x,y) \in U^{1} | (x,y) \in \lambda \}$. Fix some $v \in \mathbb{Z}^{d}$ and let the corresponding left tail field $\mathcal{F}_{L}(v)$ for the Relation $U^{1}$ be the intersection of all $\mathcal{F}_{\{v \} \times K }$, where K ranges over all subsets of $\mathbb{Z}^{d}$, such that $\mathbb{Z}^{d} \setminus K $ finite . $\mathcal{F}_{L}(v):=\underset{\substack{ K\subset \mathbb{Z}^{d} \\ \mathbb{Z}^{d} \setminus K \; finite}} {\bigcap} \mathcal{F}_{\{v\} \times K}$ (analogue define the right tail field)
The Remote tail is definied as: $\mathcal{F}_{Rem}:=\underset{\substack{ K_{1}, K_{2} \subset \mathbb{Z}^{d} \\ \mathbb{Z}^{d} \setminus K_{1} \; finite \\ \mathbb{Z}^{d} \setminus K_{2} \; finite }} {\bigcap} \mathcal{F}_{K_{1}\times K_{2}}.$
To establish the trivialty of these Fields, they proofing that:
(1): For any elementary Cylinder S and for any $A \in \mathcal{F}_{L}(v)$: A and S are independent -- but i am not sure why this is a suffiecient condition for the trivialty of $\mathcal{F}_{L}(v)$.
My guess: If (1) is true, then any Cylinder Event (finite union of (disjoint) elementary Cylinder) are independent of any $A \in \mathcal{F}_{L}(v)$. Cylinder Events generating some Borel $\sigma$-Field $\mathcal{F}$, on which the USF is also defined on. If $\mathcal{F}_{L}(v) \subset \mathcal{F} $ and $\mathcal{F}_{L}(v)$ independent of $\mathcal{F}$, we could conclude that $\mathcal{F}_{L}(v)$ is trivial.
-- Is that the right way?
(2) They find some $\Gamma_{R}$ which determines the Remote Tail for every R and proofing the following : $ \lim_{R \rightarrow \infty}E[ \; | P (S| \Gamma_{R}) - P (S) | \;] = 0$. Its the same as: $ P (S| \Gamma_{R}) \underset{R \rightarrow \infty}{\rightarrow} P(S) \; in \; L^{1}$ --- but i dont know why its suffiecient for the trivialty of the Remote Tail.
I appreciate every Comment, be it references to literature, help with my questions/uncertainties or some basic Explanations to mistakes i have made. I would like to apologize for any (grammatical) errors i have made in this text.
In [1] the relevant events are denoted by $\Upsilon_R$ rather than $\Gamma_R$. As explained in page 482, the formula established there implies that the remote tail is independent of every cylinder event $S$. But since the cylinder events generate the full $\sigma$-algebra (by the Dynkin $\pi$-$\lambda$ theorem) it follows that every event in the remote tail is independent of itself, so it must have probability $0$ or 1.
[1] https://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p03.pdf