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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/1906.02802 |
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| _version_ | 1866909103443410944 |
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| author | Kalinin, Nikita Prieto, Yulieth |
| author_facet | Kalinin, Nikita Prieto, Yulieth |
| contents | 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 $Ω\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_Ω$.
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 $P=\{p_1,\dots,p_n\}\subset[0,1]^2=Ω$ and the degree of the tropical polynomial $G_{P}{\bf 0}_Ω$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_Ω$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_Ω)$ defined by $G_{P}{\bf 0}_Ω$ 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 $C(G_{P}{\bf 0}_Ω)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$.
The main theoretical result is that $C(G_{P}{\bf 0}_Ω)\setminus (P\cap \partialΩ)$, i.e. the tropical curve in $Ω^\circ$ with marked points $P$ removed, is a tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_02802 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Some statistics about Tropical Sandpile Model Kalinin, Nikita Prieto, Yulieth Combinatorics Dynamical Systems 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 $Ω\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_Ω$. 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 $P=\{p_1,\dots,p_n\}\subset[0,1]^2=Ω$ and the degree of the tropical polynomial $G_{P}{\bf 0}_Ω$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_Ω$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_Ω)$ defined by $G_{P}{\bf 0}_Ω$ 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 $C(G_{P}{\bf 0}_Ω)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_Ω)\setminus (P\cap \partialΩ)$, i.e. the tropical curve in $Ω^\circ$ with marked points $P$ removed, is a tree. |
| title | Some statistics about Tropical Sandpile Model |
| topic | Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/1906.02802 |