Abelian Sandpile Model

The Abelian Sandpile Model (Bak-Tang-Wiesenfeld 1987; Dhar 1990) is the canonical model of self-organized criticality. Grains stack on a lattice; when a site reaches 4 grains it topples: loses 4, each neighbor gains 1.

z(x,y) ≥ 4 → z(x,y) -= 4; z(neighbors) += 1

The model is abelian: the final stable state is independent of the order topplings occur. Recurrent configurations form a group under addition-and-topple. The identity element of this group has a striking fractal structure with exact self-similarity. Avalanche sizes follow a power law P(s) ~ s^-τ with τ ≈ 5/4 — the system tunes itself to criticality without external parameter tuning.