Probability density distribution of the distances between sub-halos identified by SubFind and the closest BH particle within the R_HMS of each sub-halo of Cl 13, D9 regions, and Cosmobox. The rows show the results obtained at different redshifts: z = 3 (up), z = 1 (central), and z = 0 (bottom). The column report the results comparing DYNFRIC and REPOS on the left, and DYNFRIC and DYNMASS on the right side. We include a dashed-dot line in each plot indicating the softening length of the BH as a reference for the spatial resolution of the simulation.