Timeline for Spectral density of $D + XX^T$
Current License: CC BY-SA 4.0
14 events
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| Jan 1, 2019 at 0:19 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Dec 13, 2018 at 21:30 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Dec 13, 2018 at 20:52 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Dec 13, 2018 at 20:19 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Dec 13, 2018 at 19:31 | comment | added | Carlo Beenakker | I would recommend Forrester's press.princeton.edu/titles/9237.html | |
| Dec 13, 2018 at 18:03 | comment | added | valle | As a physicist, do you recommend any texts on random matrix theory? | |
| Dec 13, 2018 at 17:51 | comment | added | valle | Now the equation for $g$ is a cubic. There is only one real solution (when $z$ is real). But the solution is considerably more complicated. | |
| Dec 13, 2018 at 17:32 | comment | added | valle | Yes, that means that in the large $N$ limit, a fraction $w$ of the diagonal entries of $D$ are 1, the remaining are zero. | |
| Dec 13, 2018 at 17:31 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Dec 13, 2018 at 17:30 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Dec 13, 2018 at 16:10 | comment | added | valle | In this case the integral equation is a simple quadratic in $g(z)$. Which root should I take? | |
| Dec 13, 2018 at 16:01 | comment | added | valle | $\rho(t) = w \delta(t - 1) + w \delta(t)$ for some $0 < w < 1$. | |
| Dec 13, 2018 at 14:34 | comment | added | valle | Thanks. So in general (Hermitian $D$), $g(z)$ is the solution of this integral equation, and then the spectral density of $D+XX^T$ is the inverse Stieltjes transform of $g(z)$? In my case $D$ is deterministic and diagonal, so $\rho(t)$ (in your notation) is a sum of Dirac delta's on the diagonal elements. But I'm not sure if this helps me. In the end I need to compute an expectation over the density of $D+XX^T$. Specifically, $(1/m)\log \det(D+XX^T) = \langle\log(\lambda)\rangle$, where $\lambda$ are the eigenvalues of $D+XX^T$ and the expectation is over its spectral density. | |
| Dec 13, 2018 at 8:37 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |