Self-organized criticality — group structure of the recurrent states
grains: 0 | topplings: 0
Colors: 0=black, 1=brown, 2=orange, 3=yellow
The sandpile group S(G) is abelian — toppling order doesn't matter (abelian property). The identity element e has a remarkable fractal structure. Adding e to any recurrent configuration returns it unchanged.
Firing rule: if h_i ≥ 4, lose 4, give 1 to each neighbor
Abelian Sandpile (Bak-Tang-Wiesenfeld 1987, Dhar 1990): Grains added to a grid topple when height ≥ 4, redistributing to 4 neighbors. The abelian property (Dhar) means avalanche size is independent of toppling order. Recurrent (allowed) configurations form a finite abelian group — the sandpile group — under the addition operation. The identity element of this group displays self-similar fractal structure at all scales. Avalanche sizes follow a power law s^{-τ} with τ≈1.2, a hallmark of self-organized criticality. The model connects to spanning trees (Matrix-Tree theorem), harmonic analysis on graphs, and chip-firing games.