Percolation technique for galaxy clustering

We study percolation in mass and galaxy distributions obtained in three-dimensional simulations of the CDM, C + HDM, and the power law (n = - 1) models in the OMEGA = 1 universe. Percolation statistics is used here as a quantitative measure of the degree to which a mass or galaxy distribution is of a filamentary or cellular type. We have developed a very fast code (based on the 1985 algorithm described by Stauffer) which calculates the statistics of clusters along with the direct detection of percolation. We found that two parameters- μᵢnfinity_ (eq. 3), characterizing the size of the largest cluster, and μ²^ (eq. 4), characterizing the weighted mean size of all clusters excluding the largest one-are extremely useful for evaluating the percolation threshold. An advantage of using these parameters is their low sensitivity to boundary effects. We show that both the CDM and the C + HDM models are extremely filamentary both in mass and galaxy distribution. The percolation thresholds for the mass distributions are Pc_ = 0. 023 +/- 0. 005 in the C + HDM and Pc_ = 0. 044 +/- 0. 005 in CDM models compared to Pc_ = 0. 16 for a Gaussian random field. For galaxy samples with a few thousand galaxies the thresholds are Pc_, C+HDM = 0. 06 +/- 0. 02 and Pc_, CDM = 0. 10 +/- 0. 02 compared to Pc_ = 0. 31 for a Poisson distribution. Percolation in regions having the shape of a parallelepiped is discussed in the context of the applications of percolation statistics to real galaxy catalogs.