二维整数格上的欧拉行走者具有范围指数2/3
Eulerian walkers on $\mathbb{Z}^2$ have range exponent $2/3$
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中文总结 AI 辅助
研究证实二维整数格上的欧拉行走者模型的范围指数为2/3，验证了其探索区域半径量级为$t^{1/3}$的猜想，还证明该行走者无限次访问所有格点，其访问区域缩放后收敛为凸体。
中文摘要 AI 辅助
在欧拉行走者模型（亦称为转子行走）中，二维整数格$\boldsymbol{\text{Z}}^2$的每个格点初始时都有一个指向四个相邻格点之一的箭头。从原点出发的行走者会不断将当前格点的箭头顺时针旋转90度，并沿新方向移动。Priezzhev、Dhar、Dhar和Krishnamurthy于1996年将该模型作为自组织临界性的模型引入，并推测对于独立均匀的初始方向，行走者在前$t$步内探索的区域半径量级为$t^{1/3}$。我们证实了该猜想，还证明了该行走者会无限次访问每个格点，且在时间$t$时其已访问区域经$t^{1/3}$缩放后会收敛到一个凸体。
英文摘要
In the Eulerian walker model (also known as rotor walk), each site of the square lattice begins with an arrow pointing to one of its four neighbors. A walker that starts at the origin repeatedly turns the arrow at its current site clockwise by $90^\circ$ and steps in the new direction. Priezzhev, Dhar, Dhar, and Krishnamurthy (1996) introduced this as a model of self-organized criticality and conjectured that, for independent uniform initial directions, the region explored in the first $t$ steps has radius of order $t^{1/3}$. We establish this conjecture and further show that the walker visits every lattice site infinitely often, and that the region it has visited by time $t$, rescaled by $t^{1/3}$, converges to a convex body.
