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Abstract: We study the expected topological properties of Cech and Vietoris-Ripscomplexes built on i.i.d. random points in R^d. We find higher dimensionalanalogues of known results for connectivity and component counts for randomgeometric graphs. However, higher homology H k is not monotone when k > 0. Inparticular for every k > 0 we exhibit two thresholds, one where homology passesfrom vanishing to nonvanishing, and another where it passes back to vanishing.We give asymptotic formulas for the expectation of the Betti numbers in thesparser regimes, and bounds in the denser regimes. The main technicalcontribution of the article is in the application of discrete Morse theory ingeometric probability.

Author: Matthew Kahle

Source: https://arxiv.org/

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