# Some statistics about Tropical Sandpile Model

## TL;DR

This paper investigates the statistical properties of the tropical sandpile model, revealing that its tropical curves are small perturbations of grid lines and form trees when points are removed, with implications for understanding self-organized criticality.

## Contribution

It provides the first quantitative statistical analysis of the tropical sandpile model and establishes that the tropical curve forms a tree after removing boundary points.

## Key findings

- Tropical curves are small perturbations of grid lines.
- Most edges have directions (1,0), (0,1), (1,1), (-1,1).
- The curve minus boundary points is a tree.

## Abstract

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number $n$ of randomly dropped points $P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropical polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap \partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points $P$ removed, is a tree.