Homological connectivity in random Čech complexes
Homological connectivity in random Čech complexes
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随机 Čech 复合体中的同源连接
DOI：
10.1007/s00440-022-01149-6
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发表时间：
2019
影响因子：
2
通讯作者：
O. Bobrowski
中科院分区：
文献类型：
--
作者：
O. Bobrowski
研究了由齐次Poisson过程生成的随机Čech复形的同调性。我们专注于“同调连通性”-随机复体足够密集的阶段，使其同调“稳定”并与底层拓扑空间同构。我们的结果形成了一个全面的高维模拟著名的现象有关的连通性在Erdens-Rényi图和随机几何图。我们首先证明，有一个序列的尖锐的相变描述同调连接在不同的维度。接下来，我们分析每个关键窗口内复合物的行为。我们表明，阻碍同调连接的周期有一个非常独特和简单的形状。此外，我们证明了过程计数的最后障碍收敛到一个泊松过程。我们大量使用了莫尔斯理论及其对距离函数的适应。特别是，我们的结果分类随机距离函数的临界点，根据其确切的影响同源性。
We study the homology of random Čech complexes generated by a homogeneous Poisson process. We focus on ‘homological connectivity’—the stage where the random complex is dense enough, so that its homology “stabilizes” and becomes isomorphic to that of the underlying topological space. Our results form a comprehensive high-dimensional analogue of well-known phenomena related to connectivity in the Erdős-Rényi graph and random geometric graphs. We first prove that there is a sequence of sharp phase transitions describing homological connectivity in different dimensions. Next, we analyze the behavior of the complex inside each of the critical windows. We show that the cycles obstructing homological connectivity have a very unique and simple shape. In addition, we prove that the process counting the last obstructions converges to a Poisson process. We make a heavy use of Morse theory, and its adaptation to distance functions. In particular, our results classify the critical points of random distance functions according to their exact effect on homology.
大型随机单纯复形，I
影响因子：
0.8
作者：
Costa A
通讯作者：
Costa A
热力学极限黎曼流形上随机几何复形的拓扑
影响因子：
1
作者：
Auffinger, Antonio;Lerario, Antonio;Lundberg, Erik
通讯作者：
Lundberg, Erik