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What positive semi-definite random matrices have (roughly) $n^{-\alpha}$ for $n^{th}$ singular value? The power law decay need not be exact.

I want to find random matrix ensembles that naturally produce power law in their eigenvalues. The Laplacian matrix of a scale-free network is one example. As in the scale-free example, it would be great if the random matrix describes some stochastic system.

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    $\begingroup$ Well, you can just fix a matrix with the required singular values and say that you are equal to this matrix with probability $1$, but that's probably not what you want... Add more details please. $\endgroup$
    – Aleksei Kulikov
    Commented May 10 at 5:19
  • $\begingroup$ Thank you for the comment. I edited my question. $\endgroup$
    – CWC
    Commented May 10 at 13:16
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    $\begingroup$ The inverse of a GUE matrix (or more generally a Wigner matrix under some mild assumptions) will have roughly this behavior for $\alpha=1$, thanks to a lot of work in recent decades on least singular values of Wigner matrices as well as local semicircular laws. One can then make such matrices positive definite by taking the matrix absolute value, and raise to various powers to get other values of $\alpha$. $\endgroup$
    – Terry Tao
    Commented May 11 at 0:13
  • $\begingroup$ @TerryTao Would you know some good references to the last part? $\endgroup$
    – ABIM
    Commented Aug 9 at 4:14
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    $\begingroup$ I'm just referring here to the simple observation that if a Hermitian matrix $A$ has eigenvalues $\lambda_i$, then a power $|A|^\theta$ of its absolute value will have eigenvalues $|\lambda_i|^\theta$. In particular, if the eigenvalues of $A$ enjoyed a power law with exponent $\alpha$, then the eigenvalues of $|A|^\theta$ will enjoy a power law with exponent $\alpha \theta$. $\endgroup$
    – Terry Tao
    Commented Aug 9 at 20:47

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