 Abelian sandpile model

The finite Abelian sandpile model places a nonnegative number of chips at each nonsink vertex of a finite connected graph with a distinguished sink. An unstable vertex, holding at least its degree in chips, fires by sending one chip along every incident edge; chips reaching the sink disappear.

Repeated legal firings stabilize the configuration. The stable result and the number of firings at each vertex are independent of the firing order, which explains the word Abelian. Adding a chip and stabilizing therefore defines commuting operations. The sink is essential for this finite dissipative formulation: without it, chips can circulate indefinitely. Recurrent stable configurations carry the associated sandpile group, linking a local chip-firing game to the graph Laplacian.