N-dimensional Abelian sandpile model with nearest-neighbor toppling

Abstract

We have derived an exact expression for the determinant of the toppling matrix of the Abelian sandpile model in any finite dimension with open boundary conditions. The result can also apply to the semiopen boundary cases in certain limiting cases. An analytic result can also be obtained in the thermodynamic limit where the grid size tends to infinite: namely, the total number of system configurations in the self-organized critical state follows a power law as the grid size increases. Therefore, under a uniform but random particle addition, 1/f2 instead of 1/f scaling is observed. The relation with the site percolation problem is also discussed. © 1993 The American Physical Society.