Definition
The iteration rules for the model on the square lattice can be defined as follows:
Begin with some nonnegative configuration which is finite, in the sense that
- .
Any site with
is unstable and can topple, sending one of its chips to each of its 4 neighbors:
The process is guaranteed to terminate given that the initial configuration was finite. Moreover, although there will often be many possible choices for the order in which to topple vertices, the final configuration does not depend on the chosen order; this is one sense in which the sandpile is Abelian. The number of times each vertex topples in this process is also independent of the choice of toppling order.
On an arbitrary graph with a sink, the rules are that any non-sink vertex with
is unstable; toppling again sends one of its chips to each of its neighbors:
and, for each :
Read more about this topic: Abelian Sandpile Model
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