Random matrix theory addresses the asymptotic limit, $N \rightarrow \infty$. Are there results for spectral densities of small random matrices, $N \sim O(1)$? I am not interested in the results a-la Wigner that already ensembles of Hermitian matrices of size $N=3$ exhibit asymptotic universality (semicircle) but rather in the opposite - what are peculiarities of the spectral densities of small matrices? What make them different from the asymptotic densities?
It is evident that in the limit $N \sim O(1)$ everything becomes dependent on the particular pdf used to generate matrix elements. But is there a chance to derive something? F.e., consider $A = M - J$, where $M$ is non-negative matrix and $J$ diagonal non-negative matrix? (for the limit $N \rightarrow \infty$ free probability gives a nice answer).