Tropical curves in sandpiles

Nikita Kalinin; Mikhail Shkolnikov

doi:10.1016/j.crma.2015.11.003

Combinatorics/Mathematical physics

Tropical curves in sandpiles

Présenté par : the Editorial Board

Nikita Kalinin ¹ ; Mikhail Shkolnikov ¹

¹ Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 125-130.

Résumé
Abstract

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 $μ \equiv 3$ . Le résultat $ψ^{\circ}$ de la relaxation est presque partout égal à μ. On appelle lieu de déviation l'ensemble des points où $ψ^{\circ} \neq μ$ . 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 $μ \equiv 3$ is obtained by adding extra grains at several points. It appears that the result $ψ^{\circ}$ of the relaxation of ψ coincides with μ almost everywhere; the set where $ψ^{\circ} \neq μ$ 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 : 2015-08-18
Accepté le : 2015-11-06
Publié le : 2016-01-26

DOI : 10.1016/j.crma.2015.11.003

Mots-clés : Combinatorics, Mathematical physics

Affiliations des auteurs :

Nikita Kalinin ¹ ; Mikhail Shkolnikov ¹

¹ Université de Genève, Section de mathématiques, route de Drize 7, villa Battelle, 1227 Carouge, Switzerland

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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/

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Nikita Kalinin Sandpile Solitons in Higher Dimensions, Arnold Mathematical Journal, Volume 9 (2023) no. 3, p. 435 | DOI:10.1007/s40598-023-00224-7
Arkadiy Artemovich Aliev; Nikita Sergeevich Kalinin Convergence of a sandpile on a triangular lattice under rescaling, Sbornik: Mathematics, Volume 214 (2023) no. 12, p. 1651 | DOI:10.4213/sm9789e
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Nikita Kalinin; Mikhail Shkolnikov Sandpile Solitons via Smoothing of Superharmonic Functions, Communications in Mathematical Physics, Volume 378 (2020) no. 3, p. 1649 | DOI:10.1007/s00220-020-03828-8
Nikita Kalinin Pattern Formation and Tropical Geometry, Frontiers in Physics, Volume 8 (2020) | DOI:10.3389/fphy.2020.581126
Robert Hough; Hyojeong Son The spectrum of the abelian sandpile model, Mathematics of Computation, Volume 90 (2020) no. 327, p. 441 | DOI:10.1090/mcom/3565
Grigory Mikhalkin Examples of tropical-to-Lagrangian correspondence, European Journal of Mathematics, Volume 5 (2019) no. 3, p. 1033 | DOI:10.1007/s40879-019-00319-6
Moritz Lang; Mikhail Shkolnikov Harmonic dynamics of the abelian sandpile, Proceedings of the National Academy of Sciences, Volume 116 (2019) no. 8, p. 2821 | DOI:10.1073/pnas.1812015116
N. Kalinin; A. Guzmán-Sáenz; Y. Prieto; M. Shkolnikov; V. Kalinina; E. Lupercio Self-organized criticality and pattern emergence through the lens of tropical geometry, Proceedings of the National Academy of Sciences, Volume 115 (2018) no. 35 | DOI:10.1073/pnas.1805847115
Nikita Kalinin; Mikhail Shkolnikov The Number $π$ π and a Summation by $S L (2, Z)$ S L ( 2 , Z ), Arnold Mathematical Journal, Volume 3 (2017) no. 4, p. 511 | DOI:10.1007/s40598-017-0075-9

Cité par 11 documents. Sources : Crossref

^☆ 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.

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