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?