Random geometric complexes
Abstract
We study the expected topological properties of Cech and VietorisRips 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.1649
 Bibcode:
 2009arXiv0910.1649K
 Keywords:

 Mathematics  Probability;
 Mathematics  Algebraic Topology;
 Mathematics  Combinatorics;
 Mathematics  Metric Geometry;
 60D05;
 55U10;
 05C80
 EPrint:
 26 pages, 3 figures, final revisions, to appear in Discrete &