On the avalanche-finiteness of Abelian Sandpiles

We prove a necessary and sufficient condition for an Abelian Sandpile Model (ASM) to be avalanche-finite, namely: all unstable states of the system can be brought back to stability in finite number of topplings. The method is also computationally feasible since it involves no greater than O(N3) arithmetic computations where N is the total number of sites of the system