Self-similarity and scaling behavior of scale-free gravitational clustering

We measure 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, with particular attention to the predictions of the stable clustering hypothesis. We concentrate our analysis on the ratios $S_Q(\mathrm{\ell })\equiv {\bar{\xi }}_Q\mathrm{/}{\bar{\xi }}_2^{Q-1}$, $Q\le 5$, where ${\bar \xi}_Q$istheaveraged$Q$-bodycorrelationfunctionoveracellofradius$\ell$. The behavior of the higher order correlations is studied through that of the void probability distribution function. As functions of ${\bar{\xi }}_2$, the quantities $S_Q$, $3\le Q\le 5$, exhibit two plateaus separated by a smooth transition around ${\bar \xi}_2 \sim 1$\mathrm{.}Intheweaklynonlinearregime,${\bar \xi}_2 \la 1$, the results are in reasonable agreement with the predictions of perturbation theory. In the nonlinear regime, ${\bar{\xi }}_2>1$, the function $S_Q({\bar{\xi }}_2)$ is larger than in the weakly nonlinear regime, and increasingly so with $-n$. It is well-fitted by the expression $S_Q= ({\bar \xi}_2/100)^{0.045(Q-2)}
{\widetilde S}_Q$forall$n$. This weak dependence on scale proves {\em 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.