  In this study, we present a detailed, statistical analysis of black hole
growth and the evolution of active galactic nuclei (AGN) using cosmological
hydrodynamic simulations run down to $z=0$. The simulations self-consistently
follow radiative cooling, star formation, metal enrichment, black hole growth
and associated feedback processes from both supernovae typeII/Ia and AGN. We
consider two simulation runs, one with a large co-moving volume of $(500\
\mathrm{Mpc})^3$ and one with a smaller volume of $(68\ \mathrm{Mpc})^3$ but
with a by a factor of almost 20 higher mass resolution. Consistently with
previous results, our simulations can widely match observed black hole
properties of the local Universe. Furthermore, our simulations can successfully
reproduce the evolution of the bolometric AGN luminosity function for both the
low-luminosity and the high-luminosity end up to $z=3.0$. In addition, the
smaller but higher resolution run is able to match the observational data of
the low bolometric luminosity end at higher redshifts $z=3-4$. We also perform
a direct comparison with the observed soft and hard X-ray luminosity functions
of AGN, including an empirical correction for a torus-level obscuration, and
find a similarly good agreement. These results nicely demonstrate that the
observed "anti-hierarchical" trend in the AGN number density evolution (i.e.
the number densities of luminous AGN peak at higher redshifts than those of
faint AGN) is self-consistently predicted by our simulations. Implications of
this downsizing behaviour on active black holes, their masses and
Eddington-ratios are discussed. Overall, the downsizing behaviour in the AGN
number density as a function of redshift can be mainly attributed to the
evolution of the gas density in the resolved vicinity of a (massive) black
hole. (shortened)
