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 . Le résultat de la relaxation est presque partout égal à μ. On appelle lieu de déviation l'ensemble des points où . 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 is obtained by adding extra grains at several points. It appears that the result of the relaxation of ψ coincides with μ almost everywhere; the set where 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.
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@article{CRMATH_2016__354_2_125_0,
author = {Kalinin, Nikita and Shkolnikov, Mikhail},
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},
url = {http://www.numdam.org/articles/10.1016/j.crma.2015.11.003/}
}
TY - JOUR AU - Kalinin, Nikita AU - Shkolnikov, Mikhail TI - Tropical curves in sandpiles JO - Comptes Rendus. Mathématique PY - 2016 SP - 125 EP - 130 VL - 354 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2015.11.003/ DO - 10.1016/j.crma.2015.11.003 LA - en ID - CRMATH_2016__354_2_125_0 ER -
Kalinin, Nikita; Shkolnikov, Mikhail. Tropical curves in sandpiles. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 125-130. doi : 10.1016/j.crma.2015.11.003. http://www.numdam.org/articles/10.1016/j.crma.2015.11.003/
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☆ 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.
