  We prove that the wired uniform spanning forest exhibits mean-field behaviour
on a very large class of graphs, including every transitive graph of at least
quintic volume growth and every bounded degree nonamenable graph. Several of
our results are new even in the case of $\mathbb{Z}^d$, $d\geq 5$. In
particular, we prove that every tree in the forest has spectral dimension $4/3$
and walk dimension $3$ almost surely, and that the critical exponents governing
the intrinsic diameter and volume of the past of a vertex in the forest are $1$
and $1/2$ respectively. (The past of a vertex in the uniform spanning forest is
the finite component that is disconnected from infinity when that vertex is
deleted from the forest.) We obtain as a corollary that the critical exponent
governing the extrinsic diameter of the past is $2$ on any transitive graph of
at least five dimensional polynomial growth, and is $1$ on any bounded degree
nonamenable graph. We deduce that the critical exponents describing the
diameter and total number of topplings in an avalanche in the Abelian sandpile
model are $2$ and $1/2$ respectively for any transitive graph with polynomial
growth of dimension at least five, and are $1$ and $1/2$ respectively for any
bounded degree nonamenable graph.
  In the case of $\mathbb{Z}^d$, $d\geq 5$, some of our results regarding
critical exponents recover earlier results of Bhupatiraju, Hanson, and J\'arai
(2017). In this case, we improve upon their results by showing that the tail
probabilities in question are described by the appropriate power laws to within
constant-order multiplicative errors, rather than the polylogarithmic-order
multiplicative errors present in that work.
