Top panel: illustration of the behaviour of the function for an active fraction ξ = 0.6, using the same conventions as in Fig. 5. As expected, for an increased active fraction, the secular evolution of the system is fastened leading to a faster increase of the distance . Middle panel: behaviour of the function , for an active fraction ξ = 0.6, using the same conventions as in Fig. 6. Its linear fit takes the form . One recovers the expected scaling of the Poisson shot noise sampling derived in Eq. (59). Bottom panel: behaviour of the function , for an active fraction ξ = 0.6, using the same conventions as in Fig. 6. Its linear fit takes the form . One recovers the expected collisional scaling with N obtained in Eq. (64).