  We show how to compute the probabilities of various connection topologies for
uniformly random spanning trees on graphs embedded in surfaces. As an
application, we show how to compute the "intensity" of the loop-erased random
walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to
infinity passes through a given vertex or edge. For example, the probability
that it passes through (1,0) is 5/16; this confirms a conjecture from 1994
about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous
computation for the triangular lattice, honeycomb lattice and ${\mathbb Z}
\times {\mathbb R}$, for which the probabilities are 5/18, 13/36, and
$1/4-1/\pi^2$ respectively.
