I am considering a real random matrix whose off-diagonal entries are i.i.d. and normally distributed (with non-zero mean and non-unit variance), and whose diagonal entries are uniform and deterministic. When I calculate its eigenvalues, I get they are distributed as a circle (see figure), plus an isolated eigenvalue which is due to the Perron–Frobenius theorem. I know about the Girko's circular law, but this is usually proved when all the entries (not only the off-diagonal ones) are i.i.d. with zero mean and unit variance. Do you know about any proof or article of the circular law in the case of the matrix I have just described?
Many thanks in advance!
EDIT
This is the only explanation I have found (from the book Food Webs and Biodiversity: Foundations, Models, Data), even if it doesn't sound very rigorous:
where (probably) $Var\alpha_{12}$ and $E\alpha_{12}$ are the variance and the mean of the off-diagonal entries.