  We derive the growth of SMBHs relative to the stellar content of their host
galaxy predicted under the assumption of BH accretion triggered by galaxy
encounters occurring during their merging histories. We show that, within this
framework, the ratio Gamma=(M_BH/M_*)(z)/(M_BH/M_*)(z=0) between the Black Hole
mass and the galactic stellar mass (normalized to the local value) depends on
both BH mass and redshift. While the average value and the spread of Gamma(z)
increase with z, such an effect is larger for massive BHs, reaching values
Gamma=5 for massive Black Holes (M>10^9 M_{\odot}) at z>4, in agreement with
recent observations of high-redshift QSOs; this is due to the effectiveness of
interactions in triggering BH accretion in high-density environments at high
redshifts. To test such a model against observations, we worked out specific
predictions for sub-samples of the simulated galaxies corresponding to the
different observational samples for which measurements of Gamma have been
obtained. We found that for Broad Line AGNs at 1<z<2 values of Gamma=2 are
expected, with a mild trend toward larger value for increasing BH mass.
Instead, when we select from our Monte Carlo simulations only extremely gas
rich, rapidly star forming galaxies at 2<z<3, we find low values 0.3<Gamma<1.5,
consistent with recent observational findings on samples of sub-mm galaxies; in
the framework of our model, these objects end up at z=0 in low-to-intermediate
mass BHs (M<10^9 M_{\odot}), and they do not represent typical paths leading to
local massive galaxies. The latter have formed preferentially through paths
passing above the local M_*-M_BH relation. We discuss how the global picture
emerging from the model is consistent with a downsizing scenario, where massive
BHs accrete a larger fraction of their final mass at high redshifts z>4.
