## Abstract

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model repre-sents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of Z^{2}. Given a set P of points in a compact convex domain Ω ⊂ R^{2} this linearized model produces a tropical polynomial G_{P} 0_{Ω}. 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}, …, p_{n} } ⊂ [0, 1]^{2} = Ω and the degree of the tropical polynomial G_{P} 0_{Ω}. We also study the distributions of the coefficients of G_{P} 0_{Ω} and the correlation between them. This paper’s main (experimental) result is that the tropical curve C(G_{P} 0_{Ω}) defined by G_{P} 0_{Ω} 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} 0_{Ω}) are of directions (1, 0), (0, 1), (1, 1), (−1, 1). The main theoretical result is that C(G_{P} 0_{Ω}) \ (P ∩ ∂Ω), i.e. the tropical curve in Ω^{◦} with marked points P removed, is a tree.

Original language | English |
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Pages (from-to) | 9-19 |

Number of pages | 11 |

Journal | Communications in Mathematics |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - 2023 |

## Keywords

- genus
- power law
- sandpile
- tropical geometry