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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!

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Yes, there is a central limit theorem that guarantees convergence in probability to the Wigner semicircle law, see for example Central limit theorem for linear eigenvalue statistics of random matrices with independent entries, or Chapter 2 of Tao's Topics in random matrix theory.

The convergence to the semicircle law when the matrix dimension $n\rightarrow\infty$ requires that the variances $\sigma_{ij}^2$ of the off-diagonal matrix elements are comparable, meaning that $$0<c_{\rm inf}\leq n\sigma_{ij}^2\leq c_{\rm sup},\;\;1\leq i<j\leq n,$$ for $n$-independent constants $c_{\rm inf}$ and $c_{\rm sup}$, see Universality of Wigner random matrices: a Survey of Recent Results.

One important class of random matrices where the universality breaks down is the class of banded random matrices, where the nonzero matrix elements are those near the diagonal, see for example Limiting eigenvalue distribution for band random matrices.

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  • $\begingroup$ Thank you!! The first literature seems even more helpful. Do you happen to know about any literature that relaxes the condition that "all off-diagonal entries must have the same variance"? It seemed that the literature I could find (with your help) cannot relax "identically distributed" beyond this line. (Or maybe the convergence to semicircle fails without this constraint...) $\endgroup$
    – yuanz07
    Commented Dec 14, 2015 at 13:29
  • $\begingroup$ I added a comment on the condition of equal variance. Note that you can also remove any component of the matrix of rank $m$ that grows slower than $n$ (which is why the diagonal elements don't matter and can be set to zero). $\endgroup$
    – Carlo Beenakker
    Commented Dec 14, 2015 at 14:26

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