Perturbation Theory (PT) applied to a cosmological density field with
Gaussian initial fluctuations suggests a specific hierarchy for the correlation
functions when the variance is small. In particular quantitative predictions
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-body
computations both in 2D and 3D with a wide range of featureless power spectra.
In agreement with previous studies, we found that the reconstructed PDF-s work
remarkably well down to very low probabilities, even when the variance
approaches unity. Our results for 2D reproduce the features for the 3D
dynamics. In particular we found that the PT predictions are more accurate for
spectra with less power on small scales. The nonlinear regime has been explored
with various tools, PDF-s, moments and Void Probability Function (VPF). These
studies have been done with unprecedented dynamical range, especially for the
2D case, allowing in particular more robust determinations of the asymptotic
behaviour of the VPF. We have also introduced a new method to determine the
moments based on the factorial moments. Results using this method and taking
into account the finite volume effects are presented.Comment: 13 pages, Latex file, 9 Postscript Figure