Random Geometric Complexes

We study the expected topological properties of ech and Vietoris-Rips complexes built on random points in a"e (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.