Identifying large-scale structure in terms of statistics

We propose a method to analyze the clustering property of samples of discrete sources. With that method one can determine the probability of forming a cluster with a given number of galaxy members and a certain number density by chance. A relative number density quantity, called the neighborhood function, is introduced so that the crowded nature of the neighborhood of individual sources could be described. To avoid those effects arising from the distance of objects, we compare the real number density concerned with the mean number density measured in the corresponding local area rather than the mean of the whole sample. The scale used to sort out clustering sources is determined by that mean local number density, and thus it is a redshift-dependent quantity. The method is applied to a sample drawn from the 2dF survey, analyzed in redshift space. We find from the analysis that the probability of forming the resulting large-scale structures by chance is very small; the phenomenon of clustering is dominant in the local universe; within the $3\sigma $ confidence level, a coherent cluster with its scale as large as $357h^{-1}Mpc$ and another with its number of galaxy members as large as 12966 are identified from the sample; there exist some galaxies which are not affected by the gravitation of clusters and hence are suspected to rest on the co-moving frame of the universe; voids are likely the volumes within which no very crowded sources are present and they are likely formed in embryo by fluctuation in the very early epoch of the universe; large-scale structures are coral-like and they are likely made up of smaller ones; sources with large values of the neighborhood function are mainly distributed within the structure of prominent clusters and it is them who form the frame of the large-scale structure