Timeline for Riemann-Hilbert approach to Selberg integral
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jan 13, 2022 at 15:51 | history | bounty ended | Marcel | ||
| Jan 10, 2022 at 18:35 | vote | accept | Marcel | ||
| Jan 7, 2022 at 21:15 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jan 7, 2022 at 21:13 | comment | added | Carlo Beenakker | concerning the $V=-a\log(1-X)$ potential: to be able to use the RH approach in the large-$N$ limit the coefficient $a=\alpha N$ must increase linearly with $N$; the calculation then proceeds along the same lines as for the quadratic potential, it's just a more complicated RH equation that one has to solve, in particular the support of the solution will be an $\alpha$-dependent sub-interval $(0,c)$ of $(0,1)$, with a $1/\sqrt{c-x}$ singularity at the upper limit. | |
| Jan 7, 2022 at 17:08 | comment | added | Carlo Beenakker | I fixed the error in the coefficient, the asymptotic result now agrees with the Selberg expression in the large-$N$ limit. | |
| Jan 7, 2022 at 17:07 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
remaining error fixed
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| Jan 7, 2022 at 16:08 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jan 7, 2022 at 15:39 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jan 7, 2022 at 13:34 | comment | added | Marcel | Ok, I still wonder if the case $V=\log(1-X)$ can be done, but thank you for your answer. | |
| Jan 7, 2022 at 13:26 | comment | added | Carlo Beenakker | I have added an explicit comparison between the RH asymptotic result and the large-N limit of the Selberg integral for the case of a quadratic $V$ (where I know how to solve the RH equation). There is probably still an error in my calculation, and I'm unable to carry out the large-N limit of the product of Gamma functions, so I have compared them numerically. | |
| Jan 7, 2022 at 13:24 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jan 7, 2022 at 11:49 | comment | added | Marcel | If you consider $0<X<1$ from the start, you can take $V$ to be the well defined function $a\log(1-X)$. Although there is no direct connection between the two techniques, the question is whether R-H is capable of reproducing the Selberg result, at least to leading orders. | |
| Jan 7, 2022 at 11:27 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Jan 7, 2022 at 11:20 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |