Distributions (normalized to one) of the semimajor axis (top), the eccentricity (middle) and the inclination (bottom) for 2.5 × 10⁴ realizations of an embedded BH, evaluated after 20 initial orbital periods. The initial conditions are identical to those in Fig. 3. In the top panel, the blue line marks the final semimajor axis from the nonturbulent simulation, while the green curve shows a Gaussian distribution with standard deviation given by Eq. (44). In the middle and bottom panels, the dot-dashed green line is the ratio between the turbulent velocity dispersion σ_turb and the circular velocity v_circ, while the green curves indicate Rayleigh distributions with mean value σ_turb/v_circ. The insets show the same distributions in logarithm scale, highlighting the presence of fatter tails with respect to the Rayleigh prediction.