### 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).