Introduction to tropical series and wave dynamic on them

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 curves in sandpiles

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.Comment: small correction

Tropical formulae for summation over a part of SL(2, Z)

Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$, let $(a,b,c,d)$ stand for $a,b,c,d\in\mathbb Z_{\geq 0}$ such that $ad-bc=1$. Define \begin{equation} \label{eq_main} F(s) = \sum_{(a,b,c,d)} f(a,b,c,d)^s. \end{equation} In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one. We prove that $F(s)$ converges when $s>1$ and diverges at $s=1/2$. (This papers differs from its published version: Fedor Petrov showed us how to easily prove that $F(s)$ converges for $s>2/3$ and diverges for $s\leq 2/3$, see below.) We also prove $$\sum\limits_{\substack{(a,b,c,d), 1\leq a\leq b, 1\leq c\leq d}} \frac{1}{(a+b)^2(c+d)^2(a+b+c+d)^2} = 1/24,$$ and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.Comment: typos corrected, a new proof adde

Quantum groups as hidden symmetries of quantum impurities

We present an approach to interacting quantum many-body systems based on the notion of quantum groups, also known as $q$-deformed Lie algebras. In particular, we show that if the symmetry of a free quantum particle corresponds to a Lie group $G$, in the presence of a many-body environment this particle can be described by a deformed group, $G_q$. Crucially, the single deformation parameter, $q$, contains all the information about the many-particle interactions in the system. We exemplify our approach by considering a quantum rotor interacting with a bath of bosons, and demonstrate that extracting the value of $q$ from closed-form solutions in the perturbative regime allows one to predict the behavior of the system for arbitrary values of the impurity-bath coupling strength, in good agreement with non-perturbative calculations. Furthermore, the value of the deformation parameter allows to predict at which coupling strengths rotor-bath interactions result in a formation of a stable quasiparticle. The approach based on quantum groups does not only allow for a drastic simplification of impurity problems, but also provides valuable insights into hidden symmetries of interacting many-particle systems.Comment: 5 pages, 2 figure

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset \mathbb{R}^2$ with colored directed edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we construct a monomorphism from the sandpile group $G_{\Gamma _1}=\mathbb{Z}^{\Gamma _1}/\Delta (\mathbb{Z}^{\Gamma _1})$ on $\Gamma _1=\hat{P}_1\cap \mathbb{Z}^2$ to the one on $\Gamma _2=\hat{P}_2\cap \mathbb{Z}^2$. We provide several examples of infinite series of such tilings converging to $\mathbb{R}^2$, and thus define the limit of the sandpile group on the plane

Sandpile monomorphisms and limits

We introduce a tiling problem between bounded open convex polyforms $\hat{P}\subset \mathbb{R}^2$ with colored directed edges. If there exists a tiling of the polyform $\hat{P}_2$ by $\hat{P}_1$, we construct a monomorphism from the sandpile group $G_{\Gamma _1}=\mathbb{Z}^{\Gamma _1}/\Delta (\mathbb{Z}^{\Gamma _1})$ on $\Gamma _1=\hat{P}_1\cap \mathbb{Z}^2$ to the one on $\Gamma _2=\hat{P}_2\cap \mathbb{Z}^2$. We provide several examples of infinite series of such tilings converging to $\mathbb{R}^2$, and thus define the limit of the sandpile group on the plane

Tropical curves, convex domains, sandpiles and amoebas

We show that heavy sandpiles on convex domains, in the scaling limit, are governed by tropical geometry