Timeline for Generating function of $SO(N)$ random matrix
Current License: CC BY-SA 4.0
34 events
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| Jun 13, 2018 at 7:17 | vote | accept | Adam | ||
| Jun 13, 2018 at 7:16 | comment | added | Adam | As I show at the other question, the Laplace transform can be written explicitly in terms of the invariants ${\rm Tr}(J J^T)$${\rm Tr}((J J^T)^2)$ and $\det J$. This is already a great result ! | |
| Jun 13, 2018 at 6:29 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 12, 2018 at 21:06 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 12, 2018 at 21:01 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 11, 2018 at 14:49 | comment | added | Adam | Using this representation, one can also compute the case $\lambda_3=0$ and $\lambda_1=\lambda_2=\lambda$, which reads $I_0(2\lambda)+\frac\pi2 L_1(2\lambda) I_0(2\lambda)-\frac\pi2 L_0(2\lambda) I_1(2\lambda)$ with $L_\nu(x)$ the Struve L function. | |
| Jun 9, 2018 at 8:00 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 21:39 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 16:11 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 16:01 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 15:34 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 8:35 | comment | added | Adam | Side notes: One can check that $Z_3$ as given above is indeed an eigenfunction of the laplacian operator described in the question (e.g. apply the laplacian, the expand in the $\lambda$'s and perform the integrals). Unfortunately this representation of $Z_3$ breaks the explicit invariance under permutation of the $\lambda$'s. Also, il $\lambda_i=\lambda$, the $Z_3=e^{\lambda}\left(I_0(2\lambda)-I_1(2\lambda)\right)$. | |
| Jun 8, 2018 at 8:28 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 8:19 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 8:01 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 7:48 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 7:40 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 7:27 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 8, 2018 at 7:09 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 21:42 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 21:35 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
[Edit removed during grace period]
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| Jun 7, 2018 at 21:19 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 21:07 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 21:02 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 20:47 | comment | added | Carlo Beenakker | @NeilStrickland --- I corrected a typo (the integrals over $\alpha$ and $\alpha'$ should run from $0$ to $2\pi$, not from $0$ to $\pi$), and now it is symmetric; thanks for catching this. | |
| Jun 7, 2018 at 20:46 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 16:30 | comment | added | Neil Strickland | Your integral for $Z_3$ seems to give $Z_3(0.3,0.4,0.5)\neq Z_3(0.4,0.3,0.5)$, but it seems clear that the true $Z_3$ should be symmetric in $\lambda$. | |
| Jun 7, 2018 at 14:55 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 14:34 | comment | added | Adam | Sure, it is not hard to find an integral form of $Z_3$, but I'm looking for a more explicit form. One of the main issue with this representation is that the symmetry under permutation of $\lambda_i$ is not explicit (this would be needed for further calculation), see also math.stackexchange.com/questions/2808873/… | |
| Jun 7, 2018 at 14:33 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 14:27 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 14:08 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jun 7, 2018 at 14:07 | comment | added | Adam | There is at least a closed form for $N=2$, see the edit part of my question, and my hope is that it can be compute for some specific $N$ (say $N=3$). Thanks for the large $N$ limit, though. | |
| Jun 7, 2018 at 14:05 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |