Sandpiles

Sandpiles are a class of models used in physics and mathematics to study how simple local interactions can produce complex global behavior. The most studied form is the Abelian sandpile model, originally introduced to illustrate self-organized criticality. It is defined on a lattice where each site holds a nonnegative integer height h_i, interpreted as the number of grains of sand at that site.

On a square lattice, a site becomes unstable when h_i reaches a threshold t, typically t = 4 for interior sites. An unstable site topples: h_i decreases by t and one grain is added to each of its t neighbors. Topplings continue until all heights are below threshold. In finite lattices with open boundaries, grains reaching the boundary are absorbed by a sink, ensuring stabilization.

A key feature of the Abelian sandpile is the Abelian property: the final stabilized configuration is independent of the order in which unstable sites topple. This enables efficient analysis. Recurrent configurations form a finite subset of all stable states, and the set of recurrent configurations has an algebraic structure known as the sandpile group. Dhar’s burning algorithm provides a practical test for recurrency.

As a paradigmatic example of self-organized criticality, the sandpile naturally evolves to a critical state without fine-tuning of parameters. Small perturbations can trigger avalanches whose sizes and durations typically obey power-law distributions, indicating scale-invariance.

Variants include stochastic and directed versions, different lattice geometries, and directed graphs; extensions explore higher dimensions and connections to other models of cascade dynamics.