Random geometric complexes

Details Random geometric complexes Random geometric complexes

2009-10-09

arxiv.org/abs/0910.1649v3

Discrete Comput Geom (2011) 45: 553-573

Mathematics

Probability

26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometry

Scientific paper

10.1007/s00454-010-9319-3

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.

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