Rotor-Router模型
The Rotor-Router Model
摘要：
在Diaconis和Fulton（1991）以及Lawler，Bramson和Griffeath（1992）的早期工作的基础上，Propp在2001年定义了内部扩散限制聚集的确定性模拟。这种增长模型是埃里克森（1996）研究的那种“收敛博弈”。在一维，我们表明，该模型是等价于一个简单的动力系统与三个整数值的参数，这个动力系统的不变量之间的联系，系统的行为和丢番图逼近的二次无理数。在两个维度上，我们给出了被占领区域的形状的限制，以及证明边界的渐近形状，随着时间的推移，适当地重新缩放到无穷大，是一个圆。虽然转子路由器模型显示了许多相同的有趣的功能，巴克唐魏森菲尔德阿贝尔沙堆模型，我们认为，新的模型可能更容易进行严格的分析。
英文摘要：
Building on earlier work of Diaconis and Fulton (1991) and Lawler, Bramson, and Griffeath (1992), Propp in 2001 defined a deterministic analogue of internal diffusion-limited aggregation. This growth model is a "convergent game" of the sort studied by Eriksson (1996). In one dimension, we show that the model is equivalent to a simple dynamical system with three integer-valued parameters; an invariant of this dynamical system yields links between the behavior of the system and diophantine approximation of quadratic irrationals. In two dimensions, we give constraints on the shape of the occupied region, as well as evidence for the proposition that the asymptotic shape of the boundary, suitably rescaled as time goes to infinity, is a circle. While the rotor-router model displays many of the same intriguing features as the Bak-Tang-Weisenfeld abelian sandpile model, we suggest that the new model is likely to be easier to analyze rigorously.
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