Sinking path of a BH of mass M_BH infalling in a DM halo sampled with DM particles of mass M_DM such that M_BH/M_DM < 1. Each column refers to a different halo sampling, with increasing halo resolution from left to right. The grey curve corresponds to the simulation without the DF correction, whereas the blue curve refer to the simulation where the DF correction applied to the BH and computed according to Eq. (33) with b_max,c = 2ϵ_BH and b_max,c = ϵ_BH (lighter curve). We report in red the analytical estimation from OTIS. The Plummer-equivalent softening in each simulation is marked with the horizontal dashed line. For the details of each simulation, refer to Table 1.