  We present in this article an analysis of some of the properties of the
density field realized in numerical simulations for power-law initial
power-spectra in the case of a critical density universe. We compare our
numerical results in the non-linear regime with the predictions of a specific
scaling model, focusing on its much wider range of applicability, which is one
of its main advantages over the standard Press-Schechter approximation. We
first check that the two-point correlation functions agree with the
stable-clustering ansatz. Next we show that the statistics of the
counts-in-cells obey the scaling law predicted by our scaling model. Then, we
turn to mass functions of overdense and underdense regions. We first consider
the mass function of "just collapsed" objects defined by a density threshold
$\Delta~177$. We note that the usual Press-Schechter prescription agrees
reasonably well with the simulations (although there are some discrepancies)
while the numerical results are also consistent with the scaling model. Then,
we consider more general mass functions defined by different density thresholds
which can even be negative. This is out of reach of the Press-Schechter
approach while our scaling model can handle these mass functions and it shows a
reasonably good agreement with numerical results. Finally, we consider objects
defined by a constant radius condition. Thus, we find that the scaling model
allows one to study many different classes of objects and it clarifies the
links between various statistical tools.
