  Using a non-perturbative method developed in a previous article (paper II) we
investigate the tails of the probability distribution $P(\rho_R)$ of the
overdensity within spherical cells. We show that our results for the
low-density tail of the pdf agree with perturbative results when the latter are
finite (up to the first subleading term), that is for power-spectra with
$-3<n<-1$. Over the range $-1<n<1$ some shell-crossing occurs (which leads to
the break-up of perturbative approaches) but this does not invalidate our
approach. In particular, we explain that we can still obtain an approximation
for the low-density tail of the pdf. This feature also clearly shows that
perturbative results should be viewed with caution (even when they are finite).
We point out that our results can be recovered by a simple spherical model but
they cannot be derived from the stable-clustering ansatz in the regime $\sigma
\gg 1$ since they involve underdense regions which are still expanding. Second,
turning to high-density regions we explain that a naive study of the radial
spherical dynamics fails. Indeed, a violent radial-orbit instability leads to a
fast relaxation of collapsed halos (over one dynamical time) towards a roughly
isotropic equilibrium velocity distribution. Then, the transverse velocity
dispersion stabilizes the density profile so that almost spherical halos obey
the stable-clustering ansatz for $-3<n<1$. We again find that our results for
the high-density tail of the pdf agree with a simple spherical model (which
takes into account virialization). Moreover, they are consistent with the
stable-clustering ansatz in the non-linear regime. Besides, our approach
justifies the large-mass cutoff of the Press-Schechter mass function (although
the various normalization parameters should be modified).
