Abelian sandpile model

The abelian sandpile model is a variant of the BTW sandpile model such that the order of the perturbations (sand grains) does not affect the final outcome.

Model definition

The model can be defined in a network through a matrix Δ. Each site i has a critical value K_i and a height value H_i assigned to it. At each time step a site is chosen at random, and its height value is increased. Once a site's height reaches the critical value the grains of sand spread to the rest of the network in a non-conservative way (there are always grains of sand that are lost) defined by Δ, possibly causing a chain reaction (avalanches). More precisely, while K_i < H_i for every i a grain of sand is randomly placed. If not:

$H_j \leftarrow H_j + \Delta_{i,j}\, \forall\, j\, ,\forall\, i\, | \, K_i \geq H_i .$

As the relaxations are non-conservative, we must have

$\sum_j \Delta_{i,j} < 0 \,\forall \,j.$

This implies that infinite chain reactions are impossible, and as the amount of sand spread during the relaxation of site i does not depend on H_i, the model is trivially abelian (i.e. commutative under permutations).