On dilute unitary random matrices

We study random dilution of random matrices H-N = UNFNUN dagger uniformly distributed over the group of N x N unitary matrices and F-N are non-random Hermitian matrices. We show that the eigenvalue distribution function of dilute random matrices [H-N](d) converges to the semicircle (Wigner) distribution in the limit N --> infinity, p --> infinity, where p is the dilution parameter. This convergence can be explained by the observation that the dilution eliminates statistical dependence between the entries of [H-N](d). The same statement is valid for the entries of [U-N](d) Our results support the conjecture that the Wigner law is valid for wide classes of dilute Hermitian random matrices.