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We compute the lattice 1-site probabilities, on the upper half-plane, of thefour height variables in the two-dimensional Abelian sandpile model. We findtheir exact scaling form when the insertion point is far from the boundary, andwhen the boundary is either open or closed. Comparing with the predictions of alogarithmic conformal theory with central charge c=-2, we find a fullcompatibility with the following field assignments: the heights 2, 3 and 4behave like (an unusual realization of) the logarithmic partner of a primaryfield with scaling dimension 2, the primary field itself being associated withthe height 1 variable. Finite size corrections are also computed andsuccessfully compared with numerical simulations. Relying on these fieldassignments, we formulate a conjecture for the scaling form of the lattice2-point correlations of the height variables on the plane, which remain as yetunknown. The way conformal invariance is realized in this system points to alocal field theory with c=-2 which is different from the triplet theory.Comment: 68 pages, 17 figures; v2: published version (minor corrections, one  comment added

Height variables in the Abelian sandpile model: scaling fields and correlations