First panel: Redshift evolution of the fraction of galaxies whose central MBH underwent at least one (solid line) or five (pale lines) super-Eddington phases (f_SupEdd). Second panel: Redshift evolution of the epoch at which MBHs within a given stellar mass bin experienced their most recent super-Eddington accretion phase ($z_{\mathrm{SupEdd}}^{\mathrm{Last}}$). Third panel: Redshift evolution of the epoch at which MBHs within a given stellar mass bin experienced their first super-Eddington accretion phase ($z_{\mathrm{SupEdd}}^{\mathrm{First}}$). Fourth panel: Median value of the largest mass inflow (Ṁ_inflow) that triggered the super-Eddington phase in the MBH. The colour coding and mass dependence are the same as in Fig. 4.