Some statistics about tropical sandpile model

Nikita Kalinin, Yulieth Prieto

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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 Z2. Given a set P of points in a compact convex domain Ω ⊂ R2 this linearized model produces a tropical polynomial GP 0Ω. 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 = {p1, …, pn } ⊂ [0, 1]2 = Ω and the degree of the tropical polynomial GP 0Ω. We also study the distributions of the coefficients of GP 0Ω and the correlation between them. This paper’s main (experimental) result is that the tropical curve C(GP 0Ω) defined by GP 0Ω 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(GP 0Ω) are of directions (1, 0), (0, 1), (1, 1), (−1, 1). The main theoretical result is that C(GP 0Ω) \ (P ∩ ∂Ω), i.e. the tropical curve in Ω with marked points P removed, is a tree.

Original languageEnglish
Pages (from-to)9-19
Number of pages11
JournalCommunications in Mathematics
Issue number3
StatePublished - 2023


  • genus
  • power law
  • sandpile
  • tropical geometry


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