Sinking timescales for a BH orbiting in a NFW DM halo. We plot the distance to the halo centre found through the shrinking sphere algorithm (Power et al. 2003) as a function of time. We plot the results from numerical simulation without any correction to the BH dynamics (top panel) and with the DF implementation introduced in Sect. 2.1 (bottom panel). The colourmap indicates the number of particles sampling the halo such that darker lines refer to higher numerical resolution. The dashed-dotted vertical lines are indicative sinking timescales and horizontal lines report the softening lengths, both are colour-coded with the simulation they refer to. The dotted lines are the analytical preditions for an infalling BH into a NFW halo, choosing the same BH and halo parameter of the numerical simulations, but varying the maximum impact paramter from b_max = 20 (red) and b_max = 80 (green).