Tropical curves in sandpiles

Abstract

Abstract. 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 ≡ 3 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 ψ • = 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.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 ≡ 3. Le résultat ψ • de la relaxation est presque partoutégalà µ. On appelle lieu de déviation l'ensemble des points où ψ • = 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.

Stability of Patterns in the Abelian Sandpile

“…This phenomenon has been known experimentally since at least the works of Ostojic [11] and Dhar-Sadhu-Chandra [4]. The recent work of Kalinin-Shkolnikov [5] identifies the defects, in a more restricted context, as tropical curves.…”

Harmonic dynamics of the abelian sandpile

“…While tropical geometry is still a relatively young mathematical field, it was already applied in the study of the BTW sandpile model leading to the identification and mathematical description of the thin "curves" or "strings" visible in the sandpile identity ( Figure 1B) as being tropical curves [18,19,20,21]. In tropical algebra (see e.g.…”

“…At the time of writing, however, only rigorous proofs for some specific structural aspects of the sandpile identity are available [17]. For example, the thin (usually one pixel wide) curves or "strings" visible in the identity ( Figure 1B) were recently identified as tropical curves [18,19,20,21], structures from tropical geometry arising e.g. in some areas of string theory and statistical physics.…”

Self-organized criticality and pattern emergence through the lens of tropical geometry

“…The state ϕ • and the evolution of the relaxation can be described by means of tropical geometry (the final picture (D) of Figure 4 is a tropical curve). This was discovered by Caracciolo et al (37) while a rigorous mathematical theory to prove this fact has been given by Kalinin et al (38), which we review presently. It is a remarkable fact that, in this case, the toppling function H(i, j) is piecewise linear (after passing to the scaling limit).…”