Evolution of the BH’s distance from the halo centre in low-resolution simulations including both a DM halo and a stellar bulge. The figure compares different setups: no DF correction (lighter and darker lines for different softening choices), DF correction with b_max,c = ϵ_BH and ϵ_BH = ϵ_DM or ϵ_BH = ϵ_∗(lighter and darker purple lines), and DF correction with an extended impact parameter b_max,c = 2ϵ_BH, 4ϵ_BH (lighter and darker orange lines). The brown curve represents the analytical sinking timescale from OTIS.