Rotor-Router聚集和可分沙堆的强球渐近性
Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
摘要：
转子路由器模型是随机游动的确定性模拟。它可以用来定义一个类似于内部DLA的确定性增长模型。我们证明了这个模型的渐近形状是一个欧几里得球，在某种意义上，这是强于我们以前的工作。对于由$n=ω_d r^d$位点组成的形状，其中$ω_d$是$\R^d$中单位球的体积，我们证明了对于任何$α> 1-1/d$，占据位点的集合的内径至少为$r-O(\log r)$，而外径至多为$r+O(r^α)$。对于一个相关的模型，可分的沙堆，我们表明，该域的占领网站是一个欧几里得球的半径误差一个常数独立的总质量。对于经典的二维阿贝尔沙堆模型，当粒子为$n=πr^2$时，我们证明了内半径至少为$r/\sqrt{3}$，外半径至多为$(r+o(r))/\sqrt{2}$ .这改进了Le Borgne和Rossin的边界。类似的界限适用于更高的维度。
英文摘要：
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work. For the shape consisting of $n=ω_d r^d$ sites, where $ω_d$ is the volume of the unit ball in $\R^d$, we show that the inradius of the set of occupied sites is at least $r-O(\log r)$, while the outradius is at most $r+O(r^α)$ for any $α> 1-1/d$. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with $n=πr^2$ particles, we show that the inradius is at least $r/\sqrt{3}$, and the outradius is at most $(r+o(r))/\sqrt{2}$. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions.
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。
