Negative slope β of the mass function of BH seeds as they merge with the central SMBH, ψ ∝ m^−β. It is derived using Eq. (42) assuming an original BH mass function ϕ ∝ m⁻² everywhere in the disk, a disk of BHs with a power-law surface density profile Σ ∝ r^−γ, and a total circular velocity V_c ∝ r^μ (which may include the additional contribution of a halo and a post-compaction central mass). At time t and mass m, these two power laws are relevant for the radial region from which BHs of mass m arrive to the center at time t, as derived for a hot disk using t_df​ = ​0.5V_c/a_df and a_df from Eq. (19). Self-gravitating disks with the shown μ are marked by red circles. We read, for example, that for a flat rotation curve (μ​ = ​0) and for the BHs that arrive from the disk range where Σ ∝ r⁻¹ (a self-gravitating Mestel disk) the mass function at the center is ψ ∝ m^−1.5. These values of β refer to the mass function used in the recoil simulations summarized in Fig. 13.