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.

We give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link [after this work was accomplished, the authors learned about a paper by Nakabo (J. Knot Theory Ramif 11(4): [565][566][567][568][569][570][571][572][573][574] 2002) where a similar result was proved. However, Nakabo's formula is different from ours, and his proof is longer and less clear].

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

Tropical geometry, an established field in pure mathematics, is a place where string theory, mirror symmetry, computational algebra, auction theory, and so forth meet and influence one another. In this paper, we report on our discovery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a certain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and proportional growth phenomena and discuss the dichotomy between continuous and discrete models in several contexts. Our aim in this context is to present an idealized tropical toy model (cf. Turing reaction-diffusion model), requiring further investigation.