On The Avalanche-finiteness Of Abelian Sandpiles

Abstract

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 $\mbox{O} \left( N^3 \right)$ arithmetic computations where $N$ is the total number of sites of the system. Key words: Abelian sandpile model; avalanche-finiteness; self-organized criticality