= Abelian sandpile model
= Sandpile
{synonym}
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>.
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