In the past decades we have assisted to an exponential growth of the galaxy redshift surveys data which have revealed that galaxies are organized in a large scale network of filaments and voids. Statistical analyses that we have performed of these surveys have shown that the galaxy distribution is characterized by power-law correlations in the range of scales [0.1-20] Mpc/h with a correlation exponent of abou 1 corresponding to a fractal dimension D=2.
Further we found that the density depends, for 20<r<80 Mpc/h, only weakly on the system size, i.e. D = 2.7 but density fluctuations are not self-averaging. Correspondingly, we have found that density fluctuations follow the Gumbel distribution of extreme-value statistics, different from a Gaussian distribution which would arise for a homogeneous spatial galaxy configuration.
Whether or not on scales r > 80 Mpc/h correlations decay and the distribution crossovers to uniformity, is still matter of considerable debate. This debate was originated by the use of different statistical methods to measure two-point correlations, to estimate statistical and systematic errors and to control the selection effects that maybe present in the data. In particular, the critical points concern the a priori assumptions which are usually used, without being directly tested, in the statistical analysis of the data and the a posteriori hypotheses that are invoked to interpret the results. Among the former, there are the assumptions of spatial homogeneity and of translational and rotational invariance (i.e. statistical homogeneity) which are built in the definition of the standard estimators of galaxy correlations.
Ongoing galaxy surveys, such as the Dark Energy Survey, will create in the next few years the largest three-dimensional map of galaxies to date that, covering a contiguous large spatial volume and controlling luminosity selection effects, will allow to study galaxy correlations on scales larger than 100 Mpc/h. Analyses of such sample represent one key-objective for our activities. %in the next years.