  We compute the precise logarithmic corrections to mean-field scaling for
various quantities describing the uniform spanning tree of the four-dimensional
hypercubic lattice $\mathbb{Z}^4$. We are particularly interested in the
distribution of the past of the origin, that is, the finite piece of the tree
that is separated from infinity by the origin. We prove that the probability
that the past contains a path of length $n$ is of order $(\log n)^{1/3}n^{-1}$,
that the probability that the past contains at least $n$ vertices is of order
$(\log n)^{1/6} n^{-1/2}$, and that the probability that the past reaches the
boundary of the box $[-n,n]^4$ is of order $(\log n)^{2/3+o(1)}n^{-2}$. An
important part of our proof is to prove concentration estimates for the
capacity of the four-dimensional loop-erased random walk which may be of
independent interest.
  Our results imply that the Abelian sandpile model also exhibits non-trivial
polylogarithmic corrections to mean-field scaling in four dimensions, although
it remains open to compute the precise order of these corrections.
