Given a (say real) random matrix $M=(M_{i,j})_{1\leq i, j \leq N}$, the moments method consists in computing (the limits in $N$ of) the quantities $$ \mathbb{E} \left(\mathrm{tr} M^k\right)^{1/k}, $$ where $\mathrm{tr}$ is the normalised trace and $k$ a natural number. Computing such limits in the case of Hermitian matrices yields the convergence of the empirical spectral measures to the semicircle law. There are several ways to computing these moments in the case of Hermitian matrices depending on the considered ensemble of matrices (e.g. using integer pairings or computing the number of graphs that satisfy certain conditions, etc.) I have two questions concerning these computations:
Computation of moments: while I am aware that computing the moments in the case of a random matrix $M$ with i.i.d. entries (not symmetric) does not yield useful information about the limiting spectral distribution (the circular law), are there any resources about computing such moments anyway (to simplify, we may assume that all the entries of $M$ are Gaussian)?
Genus expansion analysis: take $M$ to have standard Gaussian entries for example (put a condition on the symmetry of $M$ if necessary). What do we know about the terms in the expansion of $\mathbb{E} \mathrm{tr} M^k$ as a polynomial in $N$ (I believe this is known as the genus expansion) for $k$ fixed (e.g. do they form a unimodal sequence)? as a polynomial in $k$ for $N$ fixed?