Why is the identity element of the sandpile group self-similar?

Question

I've been reading about the Abelian Sandpile Model and noticed the identity element of the sandpile group on the square has self-similar components.

The sandpile group of the 198x198 square of height 4 above is a finite abelian group. However, the sandpile corresponding to elements of this group can be fractal. Where is the complexity coming from?

Also, how does can you find the identity element (say, with a computer)?

Answer 1

Since the time this question was asked, there have been some important breakthroughs on the convergence of the scaling limit of the sandpile model (and its identity element), which may explain some of the self-similarity you note.

See especially the papers: Pegden and Smart - Convergence of the Abelian sandpile, Levine, Pegden, and Smart - Apollonian structure in the Abelian sandpile, and Levine, Pegden, and Smart - The Apollonian structure of integer superharmonic matrices.

Answer 2

As per your second question, the following algorithm allows one to compute the identity element.

Let $c$ denote the maximal stable configuration; i.e. $c = \sum_{v\in V}(d(v)-1) v$ This is always recurrent. Let $a^{\circ}$ denote the stabilization of a configuration $a$. Then this will give you the identity $e$:

$e =(2c - (2c)^\circ)^\circ$

If you are interested, check out this applet for doing a lot of this stuff (and it's pretty, too): http://people.reed.edu/~davidp/sand/program/program.html

Answer 3

Another way to compute the identity is to do it iteratively: start with an empty grid (all cells contain 0 chips), then iterate the following:

• perform a reverse firing from the edges: add 2 chips to each corner cell and one chip to each edge cell
• if the resulting configuration is not stable, i.e. some cells contain at least 4 chips, then fire these cells until you obtain a stable configuration (some chips will disappear from the grid at corner and edge cells).

Once you obtain twice in a row the same stable configuration, you have the identity. During my PhD I have made a video of the stable configurations that you obtain at each step of this algorithm:

          

(many initial steps are missing to speed up the video).

You can see how the self-similar patterns evolve at each step and the formation of the center square.