Tropical curves in sandpiles

Abstract

We study a sandpile model on the set of the lattice points in a large lattice polygon. A small perturbation $\psi$ of the maximal stable state $\mu \equiv 3$ is obtained by adding extra grains at several points. It appears, that the result ${\psi }^{\circ }$ of the relaxation of $\psi$ coincides with $\mu$ almost everywhere; the set where ${\psi }^{\circ }\ne \mu$ 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. 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 $\mu \equiv 3$. Le résultat ${\psi }^{\circ }$ de la relaxation est presque partout égal à $\mu$. On appelle lieu de déviation l'ensemble des points où ${\psi }^{\circ }\ne \mu$. 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.

small corrections

References in corpus (5)

• Enumerative tropical algebraic geometry in R2
• Brief introduction to tropical geometry
• Multiple and inverse topplings in the Abelian Sandpile Model
• Conservation laws for strings in the Abelian Sandpile Model
• Pattern formation in fast-growing sandpiles

Cited by in corpus (8)

• Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry
• Harmonic dynamics of the Abelian sandpile
• Sandpile solitons via smoothing of superharmonic functions
• Sandpile solitons in higher dimensions
• The spectrum of the abelian sandpile model
• Some statistics about Tropical Sandpile Model
• Relaxation in one-dimensional tropical sandpile
• The number $\pi$ and summation by $SL(2,\mathbb{Z})$