Asymptotic properties of resolvents of large dilute Wigner random matrices

We study spectral properties of the dilute Wigner random real symmetric n × n matrices Hn,p such that the entries Hn,p(i, j) take the zero value with probability 1 – p/n. We prove that under rather general conditions on the probability distribution of Hn,p(i, j) the semicircle law is valid for the dilute Wigner ensemble in the limit n, p → ∞. In the second part of the paper we study the leading term of the correlation function of the resolvent Gn,p(z) = (Hn,p — zI)−1 with large enough |Imz| in the limit p,n → ∞, p = O(nα), 3/5 < α < 1. We show that this leading term, when considered on the local spectral scale, converges to the same limit as that of the resolvent correlation function of the Wigner ensemble of random matrices. This shows that the moderate dilution of the Wigner ensemble does not alter its universality class.