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Article Contents

# The sandpile identity element on an ellipse

• * Corresponding author: Andrew Melchionna

The first author is supported by NSF grant DMS-1455272

• We consider certain elliptical subsets of the square lattice. The recurrent representative of the identity element of the sandpile group on this graph consists predominantly of a biperiodic pattern, along with some noise. We show that as the lattice spacing tends to 0, the fraction of the area taken up by the pattern in the identity element tends to 1.

Mathematics Subject Classification: 35Q70, 31A05, 05C57.

 Citation:

• Figure 1.  Figures 1a) and 1b) correspond to the identity elements for the ellipses $E_{A,k}$, with $A = \begin{pmatrix}\frac{10}{9} & \frac{1}{3} \\[0.15cm] \frac{1}{3} & 1\end{pmatrix}$ and $\begin{pmatrix}\frac{4}{3} & \frac{1}{2} \\[0.15cm] \frac{1}{2} & \frac{3}{4}\end{pmatrix},$ respectively, with $k = 18^2$ in both. Figures 1c) and 1d) show the patterns $p_A$ corresponding to the ellipses in 1a) and 1b) respectively. A black square represents a vertex with 3 grains of sand; dark patterned (parallel lines), 2; light patterned (cross), 1; white, 0.

Figure 2.  The identity elements for the ellipses given by $E_{A,k}$, with $A = \begin{pmatrix}\frac{5}{4} & \frac{1}{2} \\[0.15cm] \frac{1}{2} & 1\end{pmatrix}$ and various $k$. A black square represents a vertex with 3 grains of sand; dark patterned (parallel lines), 2; light patterned (cross), 1; white, 0.

Figure 7.  A demonstration of r-goodness. The white point $x_1$ is not r-good, while the light gray point $x_2$ is $r$-good

Figure 3.  The identity element for the graph $E^p_{A,k},$ with} $A = \begin{pmatrix}\frac{10}{9} & \frac{1}{3} \\[0.15cm] \frac{1}{3} & 1\end{pmatrix}$, $k = 18^2,$ and p = (.47,.5)

Figure 4.  [5] A portion of the Apollonian circle packing between the lines $\{x = 0\}$ and $\{x = 2\}$

Figure 5.  [5] A portion of the boundary of the set $\Theta$

Figure 6.  Identity elements of $E_{A_i,k}$, $A_i \notin \Gamma^+,$ $k = 18^2$ $A_1 \in \Gamma \backslash \partial \Gamma,$ $A_2 \notin \Gamma$

Figure 8.  $x \in \partial E$ and the corresponding $y_x \in \partial \tilde{F}_1$

Figure 9.  A schematic for Lemma 4.5. $x \in B - A$

Figure 10.  The identity element of $B_{1152}(0) \cap \mathbb{Z}^2$

•  [1] S. Corry and D. Perkinson, Divisors and sandpiles, American Mathematical Soc., (2018), 33–75. doi: 10.1090/mbk/114. [2] A. Fey, L. Levine and Y. Peres, Growth rates and explosions in sandpiles, J. Stat. Phys., 138 (2010), 143-159.  doi: 10.1007/s10955-009-9899-6. [3] A. A. Jáarai, Sandpile models, Probab. Surv., 15 (2018), 243–306. preprint, arXiv: 1401.0354. doi: 10.1214/14-PS228. [4] L. Levine, W. Pegden and C. K. Smart, Apollonian structure in the abelian sandpile, Geometric and Functional Analysis, 26 (2016), 306-336.  doi: 10.1007/s00039-016-0358-7. [5] L. Levine, W. Pegden and C. K. Smart, The apollonian structure of integer superharmonic matrices, Annals of Mathematics, 186 (2017), 1-67.  doi: 10.4007/annals.2017.186.1.1. [6] W. Pegden and C. K. Smart, Stability of patterns in the abelian sandpile, Annales Henri Poincaré, 21 (2020), 1383–1399. doi: 10.1007/s00023-020-00898-1. [7] G. Stefani, On the monotonicity of perimeter of convex bodies, J. Convex Anal., 25 (2018), 93–102. preprint, arXiv: 1612.00295.

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