We determine the scaling properties of the probability distribution of the smoothed density field in N-body simulations of expanding universes with scale-free initial power-spectra, < ∣ δ_k ∣~2 > ∝ k~n, with particular attention to the predictions of the stable clustering hypothesis. We concentrate our analysis on the ratios S_Q(l) ≡ ξ_Q/ ξ_2~(Q-1) where ξ_Q is the average Q-body correlation function over a cell of radius l. According to the stable clustering hypothesis, S_Q should not depend on scale. We performed measurements for Q ≤ 5. The behavior of the higher order correlations is studied through that of the void probability distribution, P_0(l), which is the probability of finding an empty cell of radius l. If the stable clustering hypothesis applies, the function P_0 should also exhibit remarkable scaling properties. In our analysis, we take carefully into account various misleading effects, such as initial grid contamination, loss of dynamics due to the short-range softening of the forces, and finite volume size of our simulations. Only after correcting for the latter do we find agreement of the measured S_Q with the expected self-similar behavior. Otherwise, P_0 is only weakly sensitive to such effects and closely follows the expected self-similar behavior. The ratios S_Q exhibit two plateaus separated by a smooth transition when ξ_2(l) ～1. In the weakly nonlinear regime, ξ_2 approx < 1, the results are in reasonable agreement with the predictions of perturbation theory. In the strongly nonlinear regime, ξ_2 > 1, the S_Q values are larger than in the weakly nonlinear regime, and increasingly so with — n. They are well fitted by the expression S_Q = (ξ_2/100)~(0.045(Q-2))S_Q for all n and 3 ≤ Q ≤ 5. This weak dependence on scale proves a small but significant departure from the stable clustering predictions, at least for n = 0 and n = + 1. -The analysis of P_0 confirms that the expected scale invariance of the functions S_Q is not exactly attained in the part of the nonlinear regime we probe, except possibly for n = -2 and marginally for n= -1. In these two cases, our measurements are not accurate enough to be discriminant. On the other hand, we could demonstrate that the observed power-law behavior of S_Q cannot be generalized as such to arbitrary order in Q. Indeed, we show that this would induce scaling properties of P_0 that are incompatible with those measured.
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