The effects of particle discreteness in N-body Lambda CDM simulations 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 Lambda CDM simulations and analyze them using a wide 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 image 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.