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According to the paper Benjamini, Kesten, Peres, and Schramm - Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12 (Annals, 2004), the diameter of the component graph of the uniform spanning forests (USF) on $d$-dimensional transitive graph is $\lfloor(d-1)/4\rfloor$.

I wonder if the diameter of the component graph of the USF on the amenable transitive graph with superpolynomal growth is infinite. Actually, I think the answer is yes for every transitive graph with superpolynomial growth. A concrete example is that I think the component graph of WUSF on 3-regular tree has infinite diameter. But 3-regular tree is not amenable.

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  • $\begingroup$ You said that the diametre was $[n - 1]/4$, but it seemed that you meant $\lfloor(d - 1)/4\rfloor$. I edited accordingly; I hope that was correct. $\endgroup$
    – LSpice
    Commented Apr 16, 2023 at 22:51
  • $\begingroup$ Why do you think the component graph of the USF on 3-regular tree has infinite diameter? I think that the component graph of the USF consists of a single point, i.e. the tree itself. $\endgroup$
    – LeechLattice
    Commented Apr 17, 2023 at 8:52
  • $\begingroup$ @LeechLattice no! Actually, the Free USF of 3-regular tree has just one component, but the Wired USF of 3-regular tree and has infinite diameter. $\endgroup$
    – none Yuan
    Commented Apr 17, 2023 at 23:24
  • $\begingroup$ @noneYuan Do you mean "WUSF" in the sentence "I wonder if the diameter of the component graph of the USF on the amenable transitive graph with superpolynomal growth is infinite"? $\endgroup$
    – LeechLattice
    Commented Apr 18, 2023 at 1:49
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    $\begingroup$ @LeechLattice WUSF and FUSF coincide on d-dimensional transitive graph, so we just use USF. $\endgroup$
    – none Yuan
    Commented Apr 18, 2023 at 19:09

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