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.
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