Exact integration of height probabilities in the Abelian Sandpile Model

The height probabilities for the recurrent configurations in the Abelian Sandpile Model on the square lattice have analytic expressions, in terms of multidimensional quadratures. At first, these quantities have been evaluated numerically with high accuracy, and conjectured to be certain cubic rational-coefficient polynomials in 1/pi. Later their values have been determined by different methods. We revert to the direct derivation of these probabilities, by computing analytically the corresponding integrals. Yet another time, we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height.