Journal of Statistical Physics Vol. 74. Nos. 5/6. 1994. p. 955-979.
The height probabilities of the two-dimensional Abelian sandpile model are the fractional numbers of lattice sites having heights 1, 2, 3, 4. A combinatorial method for evaluation of these quantities is proposed. The method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice. Exact analytical expressions for all probabilities are obtained.