We consider ensembles of N x N symmetric matrices whose entries are weakly
dependent random variables. We show that random dilution can change the lim
iting eigenvalue distribution of such matrices. We prove that under general
and natural conditions the normalised eigenvalue counting function coincid
es with the semicircle (Wigner) distribution in the limit N --> infinity.
This can be explained by the observation that dilution (or more generally,
random modulation) eliminates the weak dependence (or correlations) between
random matrix entries. It also supports our earlier conjecture that the Wi
gner distribution is stable to random dilution and modulation.