# How do I determine whether or not a system will correlate to a fractal?

So I am investigating the one dimensional Abelian Sandpile for an undergraduate research project. I am primarily investigating whether or not, there is a connection between this model and the Farey Sequence.

I've been flipping through a few papers, most far above my level, but I came across a paper "Toppling distributions in one-dimensional Abelian sandpiles," by P. Ruelle and S. Sen which states that the one-dimensional case does not exhibit criticality. If someone could explain what this means, I would be most grateful.

Finally, I wanted to read the paper by Levine, Pegden, and Smart which correlates the two-dimensional case of the Abelian Sandpile to Apollonian Circles, for further insight on the problem. However, it requires a background in PDEs. which I am currently lacking. Does anyone know if Evans' PDE book discusses this?

Take as an example the abelian sandpile model in a $$n \times n$$ square grid.
Consider random recurrent sandpile states $$\phi$$ on the grid and add to them one grain of sand at a vertex $$v$$, in other words, evolve the state $$\phi \mapsto \phi + \delta_{v}$$ where $$\delta_{v}$$ is the state that is zero at every vertex except at $$v$$ where is one. Perform this experiment a very large number $$N$$ of times and ask the following: How many avalanches $$N(s)$$ of size $$s$$ occur during the experiment?
Answer:$$Log(N(s)) = \tau Log(s) + c$$ where $$\tau \sim -1.2$$. This relation is a prototypical example of a power law, in the context of sandpiles it indicates that the two dimensional sandpile is a self-organized critical system. You could consult this paper to see the computation of some correlation functions at the $$d=1$$ case, the conclusion of the authors is that the one dimensional (BTW) sandpile is also a self-organized critical system.