AGN luminosity functions at z = 1.0, 2.0, 4.0 for the model which includes super-Eddington events. The merger trees used correspond to the ones of Millennium. The error bars display the Poissionian error. The results are compared with the observations of Hopkins et al. (2007) (triangles) Aird et al. (2015) (squares) and Shen et al. (2020) (circles). The upper panels correspond to the predictions when varying ℛ^th and not imposing any ${\mathrm{M}}_{\mathrm{inflow}}^{\mathrm{th}}$ limit. The lower ones represent the LFs of the model when changing ${\mathrm{M}}_{\mathrm{inflow}}^{\mathrm{th}}$ without any ℛ^th threshold.