The effects of particle discreteness in N -body simulations of Lambda Cold Dark Matter ( \Lambda CDM ) 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 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 \sim 2 d , 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 .