网格上沙堆暂态的近似紧致界限
Nearly Tight Bounds for Sandpile Transience on the Grid
AI总结：
本文利用电网理论技术，通过对电势和随机游走进行精细全局分析，给出了二维及d维网格上Abelian沙堆模型暂态类的近似紧致上下界。
摘要：
我们使用来自电网理论的技术，针对二维网格上的 Abelian sandpile model（阿贝尔沙堆模型）的暂态类，给出了在多对数因子范围内近似紧致的界限。Abelian sandpile model 是图上的一个离散过程，与自组织临界现象密切相关。在此过程中，顶点接收沙粒，一旦沙粒数量超过其度数，它们就会通过将沙粒发送给邻居而崩塌。模型的暂态类是指系统在必然达到其稳态行为或等价的循环态之前，可以添加到系统中的最大沙粒数量。通过对电势和随机游走进行更精细和全局的分析，我们为 $n \times n$ 网格的暂态类给出了 $O(n^4\log^4{n})$ 的上界和 $Ω(n^4)$ 的下界。我们的方法自然地扩展到 $n^d$ 大小的 $d$ 维网格，给出了 $O(n^{3d - 2}\log^{d+2}{n})$ 的上界和 $Ω(n^{3d -2})$ 的下界。
英文摘要：
We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $Ω(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $Ω(n^{3d -2})$ lower bounds.
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