  When a simply connected domain $D\subset{\mathbb{R}}^d$ ($d\ge 2$) is
approximated in a "good" way by embedded connected weighted graphs, we prove
that the transfer current matrix (defined on the edges of the graph viewed as
an electrical network) converges, up to a local weight factor, to the
differential of Green's function on $D$.
  This observation implies that properly rescaled correlations of the spanning
tree model and correlations of minimal subconfigurations in the abelian
sandpile model have a universal and conformally covariant limit.
  We further show that, on a periodic approximation of the domain, all pattern
fields of the spanning tree model, as well as the minimal-pattern (e.g.
zero-height) fields of the sandpile, converge weakly in distribution to
Gaussian white noise.
