Bobrowski Omer, Skraba Primoz
Viterbi Faculty of Electrical Engineering Technion, Israel Institute of Technology, Haifa 32000, Israel.
School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom.
Phys Rev E. 2020 Mar;101(3-1):032304. doi: 10.1103/PhysRevE.101.032304.
In this paper we study the connection between the zeros of the expected Euler characteristic curve and the phenomenon which we refer to as homological percolation-the formation of "giant" cycles in persistent homology, which is intimately related to classical notions of percolation. We perform an experimental study that covers four different models: site percolation on the cubical and permutahedral lattices, the Poisson-Boolean model, and Gaussian random fields. All the models are generated on the flat torus T^{d} for d=2,3,4. The simulation results strongly indicate that the zeros of the expected Euler characteristic curve approximate the critical values for homological percolation. Our results also provide some insight about the approximation error. Further study of this connection could have powerful implications both in the study of percolation theory and in the field of topological data analysis.
在本文中，我们研究了期望欧拉特征曲线的零点与我们称为同调渗流现象之间的联系——持久同调中“巨大”循环的形成，这与经典渗流概念密切相关。我们进行了一项实验研究，涵盖四种不同模型：立方格和排列多面体格上的位点渗流、泊松 - 布尔模型以及高斯随机场。所有模型均在二维、三维、四维的平坦环面(T^{d})上生成。模拟结果有力地表明，期望欧拉特征曲线的零点近似于同调渗流的临界值。我们的结果还提供了关于近似误差的一些见解。对这种联系的进一步研究可能在渗流理论研究和拓扑数据分析领域都产生重大影响。
