论文标题
同源渗透：巨型K-Cycles的形成
Homological Percolation: The Formation of Giant k-Cycles
论文作者
论文摘要
在本文中，我们介绍并研究了连续渗透中巨型成分的更高维度类似物。使用代数拓扑的语言，我们定义了巨型k维循环的概念（具有连接的0个循环的组件）。考虑到平坦的D维圆环中的连续渗透模型，我们表明所有巨型K-cycles（K = 1，...，D-1）都出现在称为热力学极限的政权中。我们还证明，巨型K-Cycles出现的阈值在K中增加，并且与连续渗透中的临界值密切相关。最后，我们为出现巨型周期的概率的指数衰减提供了界限。
In this paper we introduce and study a higher-dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant k-dimensional cycles (with 0-cycles being connected components). Considering a continuum percolation model in the flat d-dimensional torus, we show that all the giant k-cycles (k=1,...,d-1) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant k-cycles are increasing in k and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.
