  We study, using numerical simulations, the dynamical evolution of
self-gravitating point particles in static euclidean space, starting from a
simple class of infinite ``shuffled lattice'' initial conditions. These are
obtained by applying independently to each particle on an infinite perfect
lattice a small random displacement, and are characterized by a power spectrum
(structure factor) of density fluctuations which is quadratic in the wave
number k, at small k. For a specified form of the probability distribution
function of the ``shuffling'' applied to each particle, and zero initial
velocities, these initial configurations are characterized by a single relevant
parameter: the variance $\delta^2$ of the ``shuffling'' normalized in units of
the lattice spacing $\ell$. The clustering, which develops in time starting
from scales around $\ell$, is qualitatively very similar to that seen in
cosmological simulations, which begin from lattices with applied correlated
displacements and incorporate an expanding spatial background. From very soon
after the formation of the first non-linear structures, a spatio-temporal
scaling relation describes well the evolution of the two-point correlations. At
larger times the dynamics of these correlations converges to what is termed
``self-similar'' evolution in cosmology, in which the time dependence in the
scaling relation is specified entirely by that of the linearized fluid theory.
Comparing simulations with different $\delta$, different resolution, but
identical large scale fluctuations, we are able to identify and study features
of the dynamics of the system in the transient phase leading to this behavior.
In this phase, the discrete nature of the system explicitly plays an essential
role.
