Apollonian structure in the Abelian sandpile

@article{Levine2012ApollonianSI,
  title={Apollonian structure in the Abelian sandpile},
  author={Lionel Levine and Wesley Pegden and Charles K. Smart},
  journal={Geometric and Functional Analysis},
  year={2012},
  volume={26},
  pages={306-336}
}
The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling… 
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References

SHOWING 1-10 OF 49 REFERENCES
Convergence of the Abelian sandpile
The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple},
Explicit characterization of the identity configuration in an Abelian sandpile model
Since the work of Creutz, identifying the group identities for the Abelian sandpile model (ASM) on a given lattice is a puzzling issue: on rectangular portions of complex quasi-self-similar
Patterns formed by addition of grains to only one site of an abelian sandpile
Results and conjectures on the Sandpile Identity on a lattice
TLDR
This paper studies the identity of the Abelian Sandpile Model on a rectangular lattice with the burning algorithm, which is extended to an infinite lattice, which allows it to be proved that the first steps of the algorithm on a finite lattice are the same whatever its size.
Mathematical aspects of the abelian sandpile model
In 1988, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been studied intensively, both in the physics
The Apollonian structure of integer superharmonic matrices
We prove that the set of quadratic growths attainable by integer-valued superharmonic functions on the lattice $\mathbb{Z}^2$ has the structure of an Apollonian circle packing. This completely
Conservation laws for strings in the Abelian Sandpile Model
The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures
The Abelian Sandpile Model
It has been more than 20 years since Bak, Tang and Wiesenfeld’s landmark papers on self-organized criticality (SOC) appeared [1]. The concept of self-organized criticality has been invoked to
...
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5
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