Random geometric complexes in the thermodynamic regime

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• 作者：D. Yogeshwaran ; Eliran Subag ; Robert J. Adler
• 关键词：Point processes ; Boolean model ; Random geometric complexes ; Limit theorems
• 刊名：Probability Theory and Related Fields
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：167
• 期：1-2
• 页码：107-142
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathem
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-2064
• 卷排序：167

文摘

We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality.