关于Tropical Sandpile Model的一些统计性质
Some statistics about Tropical Sandpile Model
AI总结：
本文研究了Tropical sandpile模型的统计特征，通过实验发现其生成的tropical曲线是标准方格线的微小扰动，并从理论上证明了移除标记点后的曲线是一棵树。
摘要：
Tropical sandpile model（或线性化沙堆模型）是目前已知唯一表现出自组织临界性的连续几何模型。该模型描述了在$\mathbb Z^2$的一个大子集上，最大稳定沙堆态的微小扰动下的标度极限行为。给定紧凸域$Ω\subset \mathbb R^2$中的一个点集$P$，此线性化模型生成一个tropical多项式$G_P{\bf 0}_Ω$。本文提出了该模型的一些定量统计特征及一些推测性解释。具体而言，我们研究了在$[0,1]^2=Ω$中随机投放的点数$n$与tropical多项式$G_{P}{\bf 0}_Ω$的次数之间的依赖关系。我们还研究了$G_{P}{\bf 0}_Ω$系数的分布及其相关性。本文的主要（实验）结果是，由$G_{P}{\bf 0}_Ω$定义的tropical曲线$C(G_{P}{\bf 0}_Ω)$是标准方格线的一个微小扰动。这解释了先前已知的一个事实：tropical曲线$C(G_{P}{\bf 0}_Ω)$的大部分边具有$(1,0),(0,1),(1,1),(-1,1)$的方向。主要理论结果是，$C(G_{P}{\bf 0}_Ω)\setminus (P\cap \partialΩ)$，即在$Ω^\circ$中移除标记点$P$后的tropical曲线，是一棵树。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
