  We study the effect of nonperturbative corrections associated with the
behavior of particles after shell crossing on the matter power spectrum. We
compare their amplitude with the perturbative terms that can be obtained within
the fluid description of the system, to estimate the range of scales where such
perturbative approaches are relevant. We use the simple Zeldovich dynamics as a
benchmark, as it allows the exact computation of the full nonlinear power
spectrum and of perturbative terms at all orders. Then, we introduce a "sticky
model" that coincides with the Zeldovich dynamics before shell crossing but
shows a different behavior afterwards. Thus, their power spectra only differ in
their nonperturbative terms. We consider both the real-space and redshift-space
power spectra. We find that the potential of perturbative schemes is greater at
higher redshift for a $\Lambda$CDM cosmology. For the real-space power
spectrum, one can go up to order $66$ of perturbation theory at $z=3$, and to
order $9$ at $z=0$, before the nonperturbative correction surpasses the
perturbative correction of that order. This allows us to increase the upper
bound on $k$ where systematic theoretical predictions may be obtained by
perturbative schemes, beyond the linear regime, by a factor $\sim 26$ at $z=3$
and $\sim 6.5$ at $z=0$. This provides a strong motivation to study
perturbative resummation schemes, especially at high redshifts $z \geq 1$. We
find similar results for the redshift-space power spectrum, with characteristic
wavenumbers that are shifted to lower values as redshift-space distortions
amplify higher order terms of the perturbative expansions while decreasing the
resummed nonlinear power at high $k$.
