Sandpile monomorphisms and limits

Moritz Lang; Mikhail Shkolnikov

doi:10.5802/crmath.291

Algèbre, Théorie des groupes

Sandpile monomorphisms and limits

Moritz Lang ¹ ; Mikhail Shkolnikov ²

¹ University of Applied Sciences Technikum Wien, Höchstädtplatz 6, 1200 Wien, Austria
² Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 333-341

Résumé

We introduce a tiling problem between bounded open convex polyforms $\hat{P} \subset ℝ^{2}$ with colored directed edges. If there exists a tiling of the polyform ${\hat{P}}_{2}$ by ${\hat{P}}_{1}$ , we construct a monomorphism from the sandpile group $G_{Γ_{1}} = ℤ^{Γ_{1}} / Δ (ℤ^{Γ_{1}})$ on $Γ_{1} = {\hat{P}}_{1} \cap ℤ^{2}$ to the one on $Γ_{2} = {\hat{P}}_{2} \cap ℤ^{2}$ . We provide several examples of infinite series of such tilings converging to $ℝ^{2}$ , and thus define the limit of the sandpile group on the plane.

Reçu le : 2021-01-01
Révisé le : 2021-08-25
Accepté le : 2021-10-24
Publié le : 2022-04-26

DOI : 10.5802/crmath.291

Classification : 20K30, 60K35, 47D07

Affiliations des auteurs :

Moritz Lang ¹ ; Mikhail Shkolnikov ²

¹ University of Applied Sciences Technikum Wien, Höchstädtplatz 6, 1200 Wien, Austria
² Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2022__360_G4_333_0,
     author = {Moritz Lang and Mikhail Shkolnikov},
     title = {Sandpile monomorphisms and limits},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {333--341},
     year = {2022},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     doi = {10.5802/crmath.291},
     language = {en},
}

TY  - JOUR
AU  - Moritz Lang
AU  - Mikhail Shkolnikov
TI  - Sandpile monomorphisms and limits
JO  - Comptes Rendus. Mathématique
PY  - 2022
SP  - 333
EP  - 341
VL  - 360
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.291
LA  - en
ID  - CRMATH_2022__360_G4_333_0
ER  -

%0 Journal Article
%A Moritz Lang
%A Mikhail Shkolnikov
%T Sandpile monomorphisms and limits
%J Comptes Rendus. Mathématique
%D 2022
%P 333-341
%V 360
%I Académie des sciences, Paris
%R 10.5802/crmath.291
%G en
%F CRMATH_2022__360_G4_333_0

Moritz Lang; Mikhail Shkolnikov. Sandpile monomorphisms and limits. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 333-341. doi: 10.5802/crmath.291

Bibliographie
Cité par

[1] Siva R. Athreya; Antal A. Járai Infinite volume limit for the stationary distribution of Abelian sandpile models, Commun. Math. Phys., Volume 249 (2004) no. 1, pp. 197-213 | Zbl | MR | DOI

[2] Roland Bacher; Pierre de La Harpe; Tatiana Nagnibeda The lattice of integral flows and the lattice of integral cuts on a finite graph, Bull. Soc. Math. Fr., Volume 125 (1997) no. 2, pp. 167-198 | Zbl | Numdam | MR | DOI

[3] Per Bak; Chao Tang; Kurt Wiesenfeld Self-organized criticality: an explanation of the 1/f noise, Phys. Rev. Lett., Volume 59 (1987) no. 4, p. 381

[4] Matthew Baker; Serguei Norine Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math., Volume 215 (2007) no. 2, pp. 766-788 | Zbl | MR | DOI

[5] Matthew Baker; Serguei Norine Harmonic morphisms and hyperelliptic graphs, Int. Math. Res. Not., Volume 2009 (2009) no. 15, pp. 2914-2955 | Zbl | MR

[6] Norman Biggs Algebraic potential theory on graphs, Bull. Lond. Math. Soc., Volume 29 (1997) no. 6, pp. 641-682 | Zbl | MR | DOI

[7] Michael Creutz Abelian sandpiles, Comput. Phys., Volume 5 (1991) no. 2, pp. 198-203 | DOI

[8] Deepak Dhar Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., Volume 64 (1990) no. 14, p. 1613 | Zbl | MR | DOI

[9] Deepak Dhar; Philippe Ruelle; Siddhartha Sen; Daya-Nand Verma Algebraic aspects of abelian sandpile models, J. Phys. A, Math. Gen., Volume 28 (1995) no. 4, p. 805 | MR | DOI

[10] Laura Florescu; Daniela Morar; David Perkinson; Nicholas Salter; Tianyuan Xu Sandpiles and Dominos, Electron. J. Comb., Volume 22 (2015) no. 1, pp. 1-66 | MR | Zbl

[11] Antal A. Járai; Frank Redig; Ellen Saada Approaching criticality via the zero dissipation limit in the abelian avalanche model, J. Stat. Phys., Volume 159 (2015) no. 6, pp. 1369-1407 | Zbl | MR | DOI

[12] Nikita Kalinin; Aldo Guzmán-Sáenz; Yulieth Prieto; Mikhail Shkolnikov; Vera Kalinina; Ernesto Lupercio Self-organized criticality and pattern emergence through the lens of tropical geometry, Proc. Natl. Acad. Sci., Volume 115 (2018) no. 35, p. E8135-E8142 | Zbl | MR

[13] Nikita Kalinin; Mikhail Shkolnikov Tropical curves in sandpiles, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 2, pp. 125-130 | Zbl | MR | DOI

[14] Moritz Lang; Mikhail Shkolnikov Harmonic dynamics of the abelian sandpile, Proc. Natl. Acad. Sci., Volume 116 (2019) no. 8, pp. 2821-2830 | Zbl | MR | DOI

[15] Gregory F. Lawler Random walk and the heat equation, 55, American Mathematical Society, 2010

[16] Klaus Schmidt; Evgeny Verbitskiy Abelian sandpiles and the harmonic model, Commun. Math. Phys., Volume 292 (2009) no. 3, p. 721 | Zbl | MR | DOI

Cité par Sources :

Commentaires - Politique