2002). The SMBH mass $m_{BH}$ grows through the accretion described above and by coalescence with other SMBHs during galaxy merging. As initial condition, we assume small seed BHs of mass $10^2\,M_{\odot}$ (Madau & Rees 2001) to be initially present in all galaxy progenitors; our results are insensitive to the specific value as long as it is smaller than $10^5\,M_{\odot}$.

In our Monte Carlo model, at each time step we assign to each galaxy the probability to interact after the rate given in eq. (1). According to such a probability, we assign an active BH accretion phase (with duration $\tau$) to the considered galaxy, and compute the accreted cold gas and associated QSO emission through equations (2) and (3).

The Feedback from AGNs
----------------------

To explore the dynamical effect of feedback occurring during the active AGN phase on the interstellar medium we adopt the model by Lapi, Cavaliere & Menci (2005). They compute the effect of an energy injection $\Delta E$ by AGNs on the surrounding gas by solving, in the ”shell approximation”, the equations for a blast-wave propagating outwards in the interstellas gas, including the effects of gravity and of the gas density gradient. The perturbed gas is confined to 