Abelian Sandpile Model

The abelian sandpile model is a discrete dynamical model in which vertices of a graph carry nonnegative integer numbers of grains. A vertex with at least its vertex degree grains topples by sending one grain along each incident edge. Grains reaching a designated sink are removed. The abelian property means that, when stabilization is possible, the stabilized configuration and the number of topplings at each vertex do not depend on the legal toppling order (Bou-Rabee 2026).

The dollar game instead allows negative balances and borrowing and preserves the total balance, with no money removed at a sink (Baker and Norine 2007).

On , each site has neighbors. Adding a grain to a stable configuration and stabilizing produces an avalanche. The stationary state is the infinite-volume limit of the stationary distributions in finite boxes with absorbing boundaries. Bou-Rabee (2026) proved that, in the stationary state on , adding one grain at the origin produces an almost surely finite avalanche. If is its radius in the maximum norm, then

for some constants and and every . The expected number of topplings at the origin is nevertheless infinite (Bou-Rabee 2026). Bou-Rabee (2026) reports that GPT models contributed mathematical ideas, computations, and proof attempts, and supplies a Lean formalization.