Correlations between eigenvalues of large random matrices with independent entries

We derive the connected correlation functions for eigenvalues of large Hermitian random matrices with independently distributed elements using both a diagrammatic and a renormalization-group (RG) inspired approach. With the diagrammatic method we obtain a general form for the one-, two-, and three-point connected Green functions for this class of ensembles when matrix elements are identically distributed, and then discuss the derivation of higher-order functions by the same approach. Using the RG approach we rederive the one- and two-point Green functions and show they are unchanged by choosing certain ensembles with nonidentically distributed elements. Throughout, we compare the Green functions we obtain to those from the class of ensembles with unitary invariant distributions and discuss universality in both ensemble classes.