Tropical curves in sandpiles

TL;DR

This paper investigates the scaling limit of deviation loci in a sandpile model on lattice polygons, revealing that they converge to a special tropical curve passing through perturbation points.

Contribution

It establishes a connection between sandpile deviation loci and tropical geometry, showing the limit shape is a distinguished tropical curve.

Findings

Deviation loci converge to a tropical curve in the scaling limit

The tropical curve passes through the perturbation points

The model links sandpile dynamics with tropical geometry

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 consid\'erons le mod\`ele 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 r\'esultat $\psi^\circ$ de la relaxation est presque partout \'egal \`a $\mu$. On appelle lieu de d\'eviation l'ensemble des points o\`u $\psi^\circ\ne \mu$. La limite au sens de la distance de Hausdorff du lieu de d\'eviation est une courbe tropicale sp\'eciale, qui passe par les points de perturbation.