  In order to have a better understanding of finite random matrices with
non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue
statistics in both the bulk and at the hard edge of the spectrum of random
matrices. This gives valuable information about the smallest singular value not
seen in universality laws. In particular, we show the dependence on the fourth
moment (or the kurtosis) of the entries. This work makes use of the so-called
complex Gaussian divisible ensembles for both Wigner and sample covariance
matrices.
