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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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