It is well-known that the semicircle law holds for a wide class of matrices with independent and identically distributed (mean zero) entries.

My question is: is there any study about the more general case, in which we drop the "identically distributed" condition? That is, the matrix still have independent entries, but their distributions may be different.

I understand that the completely general case might not permit very meaningful or strong results, but I am interested in settings even many restrictions and conditions are added. Maybe the limiting distribution of the eigenvalues is no longer semicircle?

Any pointer is appreciated. Thank you!