Some statistics about Tropical Sandpile Model

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 $Ω\subset \mathbb R^2$ 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.

Figures (6)

• Figure 1: $\Omega$-tropical curve $C(f)$ corresponding to $f(x,y)=\min(1/3,x,y,1-x,1-y)$.
• Figure 2: The operator $G_p$ shrinks the face $\Phi$ where $p$ belongs to. Note that combinatorics of the new curve can change when shrinking.
• Figure 3: A typical example of $G_P{\bf 0}_{[0,1]^2}$.
• Figure 4: Minimum and mean degree of $5000$ experiments for each $s \in S$ (different curves) and $n \in N$ ($x$-axe).
• Figure 5: Graph of $\overline{c_{00}}(s,n)+\overline{c_{11}}(s,n)-\overline{c_{10}}(s,n)-\overline{c_{01}}(s,n)$.

Theorems & Definitions (1)

• proof