Comparison of the evolution of BH halo-centric distances during the BH sinking for different simulation setups. From the upper to the bottom row we show the results at increasing resolutions. Left column compares simulation using ϵ_BH = ϵ_DM or ϵ_BH = ϵ_∗ for simulation with and without the DF correction. The right panels display the results for a BH embedded either in the NFW halo alone (as described in Sect. 3.2) or with the addition of a central stellar bulge. The red curves show the analytical predictions computed with OTIS, with the lighter curve corresponding to the DM-only case and the darker curve including the central bulge. Dashed horizontal lines indicate the Plummer-equivalent softenings used in the simulations, with Z19 softenings marked in black and P03 softenings in grey.