# 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…

## 36 Citations

Deterministic Abelian Sandpile and square-triangle tilings

- Mathematics
- 2015

The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a…

Harmonic dynamics of the abelian sandpile

- MathematicsProceedings of the National Academy of Sciences
- 2019

It is demonstrated that the self-similar fractal structures arising in the abelian sandpile show smooth dynamics under harmonic fields, similar to sand dunes which travel, transform, and merge, depending on the wind, and that the existence of several scaling limits for infinite domains is conjecture.

Sandpiles and Dominos

- MathematicsElectron. J. Comb.
- 2015

A new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of a $2m \times 2n$ rectangular checkerboard and a new way of counting the many-sided domino Tilings on a Mobius strip are provided.

Tropical curves in sandpile models

- Mathematics
- 2015

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to…

Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

- Mathematics
- 2018

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner…

The spectrum of the abelian sandpile model

- MathematicsMath. Comput.
- 2021

A general method of determining the spectral factor either computationally or asymptotically and performs the determination in specific examples is given.

The Apollonian structure of integer superharmonic matrices

- Mathematics
- 2013

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…

Integer superharmonic matrices on the $F$-lattice

- Mathematics
- 2021

We prove that the set of quadratic growths achievable by integer superharmonic functions on the F -lattice, a periodic subgraph of the square lattice with oriented edges, has the structure of an…

The Divisible Sandpile at Critical Density

- Mathematics
- 2015

The divisible sandpile starts with i.i.d. random variables (“masses”) at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make…

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