  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 function 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 re-derive the one and
two-point Green functions and show they are unchanged by choosing certain
ensembles with non-identically 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.
