余维1中同调生成元的渗流
Percolation on Homology Generators in Codimension One
摘要：
本文介绍了一种由高分子材料激发的新的渗流模型。该数学模型定义在$d$维空间$\mathbb{R}^d$中的随机立方体集上，并着重于作为高维拓扑对象的$(d-1)$维孔的生成和分解。这里，随机立方集是由在$d-1$维中以概率$p$随机和独立出现的单位面的并集构造的，并且孔由同调生成元表示。在此模型下，本文给出了空穴渗流临界概率$p_c^{\rm hole}$的上、下估计，暗示了相变的存在。证明了无限空穴簇的唯一性。该结果表明，当$p > p_c^{\rm hole}$时，对偶晶格$(\mathbb{Z}^d)^*$中的两个点属于同一空穴簇的概率$P_p(x^*\overset{\rm hole}{\longleftrightarrow} y^*)$均匀地大于0。
英文摘要：
This paper introduces a new percolation model motivated from polymer materials. The mathematical model is defined over a random cubical set in the $d$-dimensional space $\mathbb{R}^d$ and focuses on generations and percolations of $(d-1)$-dimensional holes as higher dimensional topological objects. Here, the random cubical set is constructed by the union of unit faces in dimension $d-1$ which appear randomly and independently with probability $p$, and holes are formulated by the homology generators. Under this model, the upper and lower estimates of the critical probability $p_c^{\rm hole}$ of the hole percolation are shown in this paper, implying the existence of the phase transition. The uniqueness of infinite hole cluster is also proven. This result shows that, when $p > p_c^{\rm hole}$, the probability $P_p(x^*\overset{\rm hole}{\longleftrightarrow} y^*)$ that two points in the dual lattice $(\mathbb{Z}^d)^*$ belong to the same hole cluster is uniformly greater than 0.
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