Analysis of Large Scale Structure using Percolation, Genus and Shape Statistics

We probe gravitational clustering in N-body simulations using geometrical descriptors sensitive to `connectedness': the genus curve, percolation and shape statistics. We find that both genus and percolation curves provide complementary probes of large scale structure topology and could be used to discriminate between models of structure formation and the analysis of observational data such as galaxy catalogs and MBR maps. An analysis of `shapes' in N-body simulations has shown that filaments are more pronounced than pancakes. To probe shapes of clusters and superclusters more rigorously we propose a new shape statistic which does not fit isodensity surfaces by ellipsoids (as done earlier). Our shape statistic is derived from fundamental properties of a compact body: its Minkowski functionals. The new shape statistic gives sensible results for topologically simple surfaces such as the ellipsoid, and for more complicated surfaces such as the torus. (Invited talk, to appear in: Proceedings of the IAU Symposium No. 183, Kyoto, Japan Aug. 1997, ed. K. Sato, Kluwer Academic Publ.)