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- μ_infinity_ (eq. [3]), characterizing the size of the largest cluster, and μ^2^ (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 P_c_ = 0.023 +/- 0.005 in the C + HDM and P_c_ = 0.044 +/- 0.005 in CDM models compared to P_c_ = 0.16 for a Gaussian random field. For galaxy samples with a few thousand galaxies the thresholds are P_c_,C+HDM = 0.06 +/- 0.02 and P_c_,CDM = 0.10 +/- 0.02 compared to P_c_ = 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.

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