August  2022, 42(8): 3709-3732. doi: 10.3934/dcds.2022029

The sandpile identity element on an ellipse

Cornell University Department of Mathematics, 301 Tower Rd., Ithaca, NY 14853, USA

* Corresponding author: Andrew Melchionna

Received  October 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

Fund Project: 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.

Citation: Andrew Melchionna. The sandpile identity element on an ellipse. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3709-3732. doi: 10.3934/dcds.2022029
References:
[1]

S. Corry and D. Perkinson, Divisors and sandpiles, American Mathematical Soc., (2018), 33–75. doi: 10.1090/mbk/114.

[2]

A. FeyL. 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. LevineW. 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. LevineW. 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.

show all references

References:
[1]

S. Corry and D. Perkinson, Divisors and sandpiles, American Mathematical Soc., (2018), 33–75. doi: 10.1090/mbk/114.

[2]

A. FeyL. 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. LevineW. 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. LevineW. 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.

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]

Xinxin Tan, Shujuan Li, Sisi Liu, Zhiwei Zhao, Lisa Huang, Jiatai Gang. Dynamic simulation of a SEIQR-V epidemic model based on cellular automata. Numerical Algebra, Control and Optimization, 2015, 5 (4) : 327-337. doi: 10.3934/naco.2015.5.327

[2]

T.K. Subrahmonian Moothathu. Homogeneity of surjective cellular automata. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 195-202. doi: 10.3934/dcds.2005.13.195

[3]

Achilles Beros, Monique Chyba, Oleksandr Markovichenko. Controlled cellular automata. Networks and Heterogeneous Media, 2019, 14 (1) : 1-22. doi: 10.3934/nhm.2019001

[4]

Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723

[5]

Achilles Beros, Monique Chyba, Kari Noe. Co-evolving cellular automata for morphogenesis. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2053-2071. doi: 10.3934/dcdsb.2019084

[6]

Hisashi Morioka. Inverse boundary value problems for discrete Schrödinger operators on the multi-dimensional square lattice. Inverse Problems and Imaging, 2011, 5 (3) : 715-730. doi: 10.3934/ipi.2011.5.715

[7]

Bernard Host, Alejandro Maass, Servet Martínez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1423-1446. doi: 10.3934/dcds.2003.9.1423

[8]

Marcelo Sobottka. Right-permutative cellular automata on topological Markov chains. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1095-1109. doi: 10.3934/dcds.2008.20.1095

[9]

Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51

[10]

Laurence Halpern, Jeffrey Rauch. Hyperbolic boundary value problems with trihedral corners. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4403-4450. doi: 10.3934/dcds.2016.36.4403

[11]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[12]

J. R. L. Webb. Uniqueness of the principal eigenvalue in nonlocal boundary value problems. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 177-186. doi: 10.3934/dcdss.2008.1.177

[13]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems and Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1

[14]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377

[15]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283

[16]

Hugo Beirão da Veiga. Elliptic boundary value problems in spaces of continuous functions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 43-52. doi: 10.3934/dcdss.2016.9.43

[17]

Marek Macák, Róbert Čunderlík, Karol Mikula, Zuzana Minarechová. Computational optimization in solving the geodetic boundary value problems. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 987-999. doi: 10.3934/dcdss.2020381

[18]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851

[19]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83

[20]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (155)
  • HTML views (121)
  • Cited by (0)

Other articles
by authors

[Back to Top]