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

Question

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?

Answer 1

It is a general fact that the correlation functions of a dynamical system at criticality are power laws of the parameters that they depend on. The key point is to notice that (at least) in the context of sandpiles, having correlation functions that behave as power laws is equivalent to being a "dynamical system at criticality".

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.