随机Čech复形的同调连通性
Homological Connectivity in Random Čech Complexes
摘要：
研究了由齐次泊松过程生成的随机Čech复形的同调性。我们专注于“同调连通性”--随机复体足够稠密的阶段，这样它的同调“稳定”并与底层拓扑空间的同调同构。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。我们首先证明了有一个尖锐的相变描述同调连接。接下来，我们分析复合物在临界窗口中的行为。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，我们还证明了计数最后一个障碍物的过程收敛于一个泊松过程。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，我们的结果分类的随机距离函数的临界点，根据其确切的效果同调。
英文摘要：
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 sharp phase transition describing homological connectivity. Next, we analyze the behavior of the complex in the critical window. 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.
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