Timeline for Determinant of the random matrix $X^2+Y^2$
Current License: CC BY-SA 4.0
13 events
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| Oct 25, 2023 at 7:37 | comment | added | Wolfgang | What about the further generalization $\mathbb{E}[{\rm det}(\sum_i X_i^2+k\sum_{i\ne j} X_i X_j)]$ for a given $k\in \mathbb{R}$? (and would it be different from $\mathbb{E}[{\rm det}(\sum_i X_i^2+2k\sum_{i<j} X_i X_j)]$?) | |
| Oct 19, 2023 at 16:02 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| Oct 19, 2023 at 15:45 | comment | added | Abdelmalek Abdesselam | @TheSimpliFire: The beginning and the end of the apply. The computation of the contribution of $\omega$ is also the sign of that permutation. But one would also get isolated/disconnected loops contributing a factor of $n$ each. The part which would likely be too complicated is counting the analogues of the permutations $\sigma_i$ which would become much more complicated combinatorial objects. Follwing up on Richard's comment, one could start by looking up the article by Beck arxiv.org/abs/2207.09311 | |
| Oct 19, 2023 at 13:21 | comment | added | Richard Stanley | There is a nice generating function for $\mathbb{E}\det X^4$ so maybe this can be extended to $\mathbb{E}\det \sum X_i^4$. For $\mathbb{E}\det X^6$ see arXiv:2206.11356. It looks like $\mathbb{E} \det X^8$ will be very complicated. | |
| Oct 19, 2023 at 10:40 | comment | added | TheSimpliFire | Can this method be used to compute $\Bbb E\det\sum X_i^k$ for integer $k>2$? | |
| Oct 18, 2023 at 23:40 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| Oct 18, 2023 at 23:25 | comment | added | Abdelmalek Abdesselam | Most relevant in Cvitanovic's book are equations (6.38) and (6.46). | |
| Oct 18, 2023 at 23:21 | comment | added | Abdelmalek Abdesselam | See, e.g., the book by Cvitanovic birdtracks.eu/version9.0/GroupTheory.pdf If you have never seen this, you have to practice a bit. | |
| Oct 18, 2023 at 22:10 | comment | added | mathworker21 | Can someone explain to me what on Earth is going on in this answer? Any references appreciated. | |
| Oct 18, 2023 at 21:21 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| Oct 18, 2023 at 20:45 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| Oct 18, 2023 at 20:34 | history | edited | Abdelmalek Abdesselam | CC BY-SA 4.0 |
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| Oct 18, 2023 at 20:29 | history | answered | Abdelmalek Abdesselam | CC BY-SA 4.0 |