The uniform spanning tree (UST) is a fundamental combinatorial object, known
to be a determinantal process with correlation kernel given by transfer
impedances. However, many of its global properties remain to be understood,
especially in three dimension. We study some of the random field properties
of the spanning trees in any dimension greater than or equal to two, and
prove the conformal invariance and Gaussian white noise fluctuation of their
scaling limit. Using the burning bijection of Dhar, we also conclude the
analogous result for the zero height field of recurrent Abelian sandpiles.
Joint work with Adrien Kassel.