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Consider $A$ a $n \times n$ random matrix with centered Gaussian entries $A_{i,j}$ such that $$\mathbb{E}[A_{i,j}^2]=\sigma_j^2/n$$. The variances depend on the column only.

What do we know on the eigenvalues distribution ? In particular, if one assumes that $$\frac{1}{n}\sum_{j=1}^n \sigma_j^2 \to \bar{\sigma}^2$$ then is it true that :

$\rho(A):=max(|\lambda_k|) \to \bar{\sigma}^2$ when $n\to \infty$ ?

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Not really an answer, but you can check out and the references listed there. To my knowledge, nothing other than the IID case is known. – Stanley Yao Xiao Feb 24 '12 at 19:28

I'm not sure this answers your question exactly but there are some known bounds. For example, Latala provides the following high probability bound under the conditions of finite fourth moments for the matrix entries $$\|A\|_2 \leq C \left( \max_i \sqrt{\sum_j \mathbb{E}A_{ij}^2}+ \max_j \sqrt{\sum_i \mathbb{E}A_{ij}^2} + \sqrt[4]{\sum_{i,j}\mathbb{E}A_{ij}^4} \right).$$

It would also be worthwhile to check out some of the recent review papers. One is by Laszlo Erdos and another is by Terrence Tao.

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Your matrix model can be written as the product of a standard gaussian matrix with a deterministic diagonal matrix. As such, I believe that Theorem 4 of this paper of Bordenave: should give the asymptotic empirical spectral distribution under some reasonable hypotheses on the variances.

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@TerryTao Thank you for your answer. In Bordenave's paper I found another interesting reference which adress the case of a diagonal matrix with Cauchy distributed elements. Thanks. – user16215 Jun 4 '12 at 17:09

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