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Is there a method using random matrix theory and NOT using free probability to determine the asymptotic eigenvalue distribution of the random matrix $\mathbf{M}=\mathbf{X}_1+ \mathbf{X}_2$? where:

$\mathbf{X}_i=\mathbf{H}_i\mathbf{H}_i^{*}$ for $i\in\{1,2\}$ where $\mathbf{H}_i$ are both square $N\times N$ random matrices whose entries are i.i.d and follow a normalised Gaussian distribution $\sim {\mathcal{N}}(\mu=0 ,\sigma^{2}=\frac{1}{N})$, so that the asymptotic eigenvalue distributions of $\mathbf{X}_i$ are given by the Marchenko-Pastur law with $\beta=1$.

More generally, is there such a method for finding the AED of the linear combination $\mathbf{M}_p=\alpha\mathbf{X}_1+ \beta\mathbf{X}_2$ for $\alpha, \beta \in \mathbb{R}$?

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    $\begingroup$ Why not using free probability? Solving problems with prescribed methods rather belongs to philosophy, and that's far away from RMT, which is applied math. $\endgroup$
    – Richard
    Commented Apr 9, 2019 at 15:20
  • $\begingroup$ I know how to solve it using FPT, I was interested to know if there were a simpler method. $\endgroup$
    – user40130
    Commented Feb 23, 2020 at 23:29

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Wouldn't it be helpful, if $\alpha$ and $\beta$ are positive to use the fact that $\mathbf{M}$ has the same law than $\mathbf{H}\Sigma\mathbf{H}^{*}$ with $\mathbf{H}$ the $N\times 2N$ matrix filled with independant entries of law ${\mathcal{N}}(\mu=0 ,\sigma^{2}=\frac{1}{N})$ and $\Sigma$ the $2N$ diagonal matrix $Diag(\alpha,\dots,\alpha,\beta,\dots,\beta)$ ?

Then I think that most classical method to get the MP law can be used on $\mathbf{M}$.

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  • $\begingroup$ Thank you for this. Yes my thoughts were that it could be solved using this decomposition, however I am unclear on the details (I am not familiar with generalizing the MP law). I can see how to find the AED for $\mathbf{H}\mathbf{H}^{*}$ in your solution, but I don't know how that extends to $\Sigma\mathbf{H}\mathbf{H}^{*}$. Presumably you just weight using $\alpha$ and $\beta$ and normalize using $N$? Do you have a reference or exercise that uses the classical method you mention? $\endgroup$
    – user40130
    Commented Feb 23, 2020 at 23:40

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