Comptes Rendus
Combinatorics/Mathematical physics
Tropical curves in sandpiles
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 125-130.

Nous considérons le modèle du tas de sable sur l'ensemble des points entiers d'un polygone entier. En ajoutant des grains de sable en certains points, on obtient une perturbation mineure de la configuration stable maximale μ3. Le résultat ψ de la relaxation est presque partout égal à μ. On appelle lieu de déviation l'ensemble des points où ψμ. La limite au sens de la distance de Hausdorff du lieu de déviation est une courbe tropicale spéciale, qui passe par les points de perturbation.

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation ψ of the maximal stable state μ3 is obtained by adding extra grains at several points. It appears that the result ψ of the relaxation of ψ coincides with μ almost everywhere; the set where ψμ is called the deviation locus. The scaling limit of the deviation locus turns out to be a distinguished tropical curve passing through the perturbation points.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.11.003
Mots clés : Combinatorics, Mathematical physics

Nikita Kalinin 1 ; Mikhail Shkolnikov 1

1 Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland
@article{CRMATH_2016__354_2_125_0,
     author = {Nikita Kalinin and Mikhail Shkolnikov},
     title = {Tropical curves in sandpiles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {125--130},
     publisher = {Elsevier},
     volume = {354},
     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.11.003},
     language = {en},
}
TY  - JOUR
AU  - Nikita Kalinin
AU  - Mikhail Shkolnikov
TI  - Tropical curves in sandpiles
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 125
EP  - 130
VL  - 354
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2015.11.003
LA  - en
ID  - CRMATH_2016__354_2_125_0
ER  - 
%0 Journal Article
%A Nikita Kalinin
%A Mikhail Shkolnikov
%T Tropical curves in sandpiles
%J Comptes Rendus. Mathématique
%D 2016
%P 125-130
%V 354
%N 2
%I Elsevier
%R 10.1016/j.crma.2015.11.003
%G en
%F CRMATH_2016__354_2_125_0
Nikita Kalinin; Mikhail Shkolnikov. Tropical curves in sandpiles. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 125-130. doi : 10.1016/j.crma.2015.11.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.11.003/

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

[2] E. Brugallé; I. Itenberg; G. Mikhalkin; K. Shaw Brief introduction to tropical geometry, Proceedings of 21st Gökova Geometry-Topology Conference, 2015

[3] S. Caracciolo; G. Paoletti; A. Sportiello Conservation laws for strings in the Abelian sandpile model, Europhys. Lett., Volume 90 (2010) no. 6, p. 60003

[4] S. Caracciolo; G. Paoletti; A. Sportiello Multiple and inverse topplings in the Abelian sandpile model, Eur. Phys. J. Spec. Top., Volume 212 (2012) no. 1, pp. 23-44

[5] D. Dhar Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., Volume 64 (1990) no. 14, pp. 1613-1616

[6] A. Fey; L. Levine; Y. Peres Growth rates and explosions in sandpiles, J. Stat. Phys., Volume 138 (2010) no. 1–3, pp. 143-159

[7] N. Kalinin; M. Shkolnikov Tropical curves in sandpile models, 2015 (in preparation) | arXiv

[8] Y. Le Borgne; D. Rossin On the identity of the sandpile group, Discrete Math., Volume 256 (2002) no. 3, pp. 775-790 LaCIM 2000 Conference on Combinatorics, Computer Science and Applications (Montreal, QC)

[9] L. Levine; W. Pegden; C.K. Smart Apollonian structure in the Abelian sandpile, Geom. Funct. Anal. (2012) (in press) | arXiv

[10] L. Levine; J. Propp What is a sandpile?, Not. Amer. Math. Soc. (2010)

[11] G. Mikhalkin Enumerative tropical algebraic geometry in R2, J. Amer. Math. Soc., Volume 18 (2005) no. 2, pp. 313-377

[12] G. Paoletti Deterministic Abelian sandpile models and patterns, Springer Theses, Springer, Cham, 2014 (PhD Thesis, University of Pisa, Pisa, 2012)

[13] W. Pegden; C.K. Smart Convergence of the Abelian sandpile, Duke Math. J., Volume 162 (2013) no. 4, pp. 627-642

[14] T. Sadhu; D. Dhar Pattern formation in fast-growing sandpiles, Phys. Rev. E, Volume 85 (2012) no. 2

[15] T.Y. Yu The number of vertices of a tropical curve is bounded by its area, Enseign. Math., Volume 60 (2014) no. 3–4, pp. 257-271

Cité par Sources :

Research is supported in part the grant 159240 of the Swiss National Science Foundation as well as by the National Center of Competence in Research SwissMAP of the Swiss National Science Foundation.

Commentaires - Politique