Scaling in gravitational clustering, two- and three-dimensional dynamics

Perturbation theory (PT) applied to a cosmological density field with Gauss ian initial fluctuations suggests a specific hierarchy for the correlation functions when the variance is small. In particular quantitative prediction s have been made for the moments and the shape of the one-point probability distribution function (PDF) of the top-hat smoothed density. In this paper we perform a series of systematic checks of these predictions against N-bo dy computations in both 2D and 3D with a wide range of featureless power sp ectra. In agreement with previous studies, we found that the reconstructed PDFs work remarkably well down to very low probabilities, even when the var iance approaches unity. Our results for 2D reproduce the features for the 3 D dynamics. In particular we found that the PT predictions are more accurat e for spectra with less power on small scales. In the highly non-linear regime, on the other hand, different assumptions r egarding amplitudes of different tree topologies contributing to higher-ord er correlation functions lead to specific predictions regarding the scaling properties of the void probability function (VPF) and count probability di stribution function (CPDF). However most efforts to determine these amplitu des from dynamical theory of gravitational clustering have led to oversimpl ification of Born, Bogoliubov, Green, Kirkwood and Yvon (BBGKY) equations i n the highly non-linear regime. Generic predictions regarding VPF and CPDF were made assuming that these amplitudes can be constructed from the multip licative nature of vertex amplitudes appearing in the tree level approximat ion of correlation hierarchy. We test these predictions against simulations in 2D and 31D and determine the unknown parameters that appear as a result of the lack of complete knowledge of all hierarchal amplitudes. These stud ies have been done with unprecedented dynamical range, especially for the 2 D case, allowing in particular more robust determination of the asymptotic behaviour of the VPF, confirming scaling arguments. We have introduced a new method for correcting the finite-volume effect in determination of higher-order correlation functions that is based on scalin g properties and the use of factorial moments. Results of analysis using th is method are presented.