### The Abelian sandpile model

2D `Sandpile' model[1]: Consider a 100 by 100 square grid. Each square has either 0, 1, 2, or 3 particles. A particle is added at a randomly chosen square. If as a result, the number of particles there does not exceed 3, again a square is chosen at random, and a new particle is added. If the number of particles at a square reaches 4, a so-called avalanche starts:

(i) The square with 4 particles topples, which means that it sends one particle to each of its neighbours, thereby reducing the number of particles on the square by 4, and increase the number of particles at each neighbouring square by 1. Note that when the square being toppled is on the boundary, one or more particles are removed from the grid.
(ii) Step (i) is repeated, as long as there are any squares where the number of particles is at least 4.

It is not difficult to see that the avalanche always terminates in finitely many steps. It can also be shown that the final result of the avalanche is independent of the order in which topplings were carried out[2],[3]. (This is the origin of the name `Abelian'.) Once the avalanche terminated, we add a new particle at a randomly chosen square and continue. It is straightforward to generalize to grids of higher dimension, or to a general finite graph. The above description defines a Markov chain on the space of particle configurations: a single time-step for the Markov chain consists of adding a particle at a random vertex, and stabilizing the result.

It is relatively simple to simulate the model on a computer. A fascinating feature one sees in such simulations is that after sufficiently long time (that is when the underlying Markov chain has reached stationarity), there are avalanches of all sizes: Some involve only a few squares, some span the entire grid. Let us measure the size of an avanache by how many topplings occurred during the avalanche. It is an outstanding open problem to prove that in the limit of a very large grid, the probability p(s) that an avalanche has size s decays as p(s) ~ c s-a, for some constant c and an exponent 1 < a < 2, that depends on the dimension of the grid.

References

[1] Bak, P., Tang, C. and Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364-374 (1988).

[2] Dhar, D: Theoretical studies of self-organized criticality. Phys. A 369, 29-70 (2006).

[3] Redig, F.: Mathematical aspects of Abelian sandpile models. In: Bovier, A., Dunlop, F., van Enter, A., den Hollander, F. and Dalibard, J. (eds.): Mathematical statistical physics, École d'Été de physique des houches session LXXXIII, Elsevier B. V., Amsterdam, Course 14, 657-729, (2006).