On the Wigner law in dilute random matrices

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.