  The effects of particle discreteness in N-body simulations of Lambda Cold
Dark Matter (LambdaCDM) are still an intensively debated issue. In this paper
we explore such effects, taking into account the scatter caused by the
randomness of the initial conditions, and focusing on the statistical
properties of the cosmological density field. For this purpose, we run large
sets of LambdaCDM simulations and analyse them using a large variety of
diagnostics, including new and powerful wavelet statistics. Among other facts,
we point out (1) that dynamical evolution does not propagate discreteness noise
up from the small scales at which it is introduced, and (2) that one should aim
to satisfy the condition epsilon ~ 2d, where "epsilon" is the force resolution
and "d" is the interparticle distance. We clarify what such a condition means,
and how to implement it in modern cosmological codes.
