THE SUPERDUSTER-VOID NETWORK .3. THE CORRELATION-FUNCTION AS A GEOMETRICAL STATISTIC

We investigate properties of the correlation function of clusters of g alaxies using geometrical models. We show that the correlation functio n contains useful information on the geometry of the distribution of c lusters. On small scales the correlation function depends on the shape and the size of superclusters. On large scales it describes the geome try of the distribution of superclusters. If superclusters are distrib uted randomly then the correlation function on large scales is feature less. If superclusters have a quasi-regular distribution then this reg ularity can be detected and measured by the correlation function. Supe rclusters of galaxies separated by large voids produce a correlation f unction with a minimum which corresponds to the mean separation betwee n centres of superclusters and voids, followed by a secondary maximum corresponding to the distance between superclusters across voids. If s uperclusters and voids have a tendency to form a regular lattice then the correlation function on large scales has quasi-regularly spaced ma xima and minima of decaying amplitude; i.e. it is oscillating. The per iod of oscillations is equal to the step size of the grid of the latti ce. We also calculate the power spectrum and the void diameter distrib ution for our models and compare the geometrical information of the co rrelation function with other statistics. We find that geometric prope rties (the regularity of the distribution of clusters on large scales) are better quantified by the correlation function. We also analyse er rors in the correlation function and the power spectrum by generating random realizations of models and finding the scatter of these realiza tions.