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In this paper we rigorously construct a finite volume representation for theheight-one field of the Abelian sandpile model and the degree field of theuniform spanning tree in terms of the fermionic Gaussian free field. Thisrepresentation can be seen as the lattice representation of a free symplecticfermion field. It allows us to compute cumulants of those fields, both infinite volume and in the scaling limit, including determining the explicitnormalizing constants for fields in the corresponding logarithmic field theory.Furthermore, our results point towards universality of the height-one anddegree fields, as we prove that the scaling limits of the cumulants agree (upto constants) in the square and triangular lattice. We also recover theequivalent scaling limits for the hypercubic lattice in higher dimensions, anddiscuss how to adapt the proofs of our results to general graphs.

Fermionic Gaussian free field structure in the Abelian sandpile model and uniform spanning tree