Timeline for An integral with respect to the Haar measure on a unitary group
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2020 at 9:58 | answer | added | Peter Forrester | timeline score: 6 | |
| Feb 3, 2015 at 7:44 | comment | added | Peter | @AbdelmalekAbdesselam : Thank you. The problem is that using the formula for Harish-Chandra-Itzykson-Zuber integral, we can come up with $\int (\mathrm{tr}(A^{-1}HDH^{\dagger}))^p dH$. But we need the answer for $\int \mathrm{tr}((A^{-1}HDH^{\dagger})^p) dH$. In addition, the formula for Harish-Chandra-Itzykson-Zuber integral is valid for nonzero $t$, while I need to take $p$ consecutive derivatives and set $t=0$. Am I right? | |
| Jan 28, 2015 at 23:04 | comment | added | Abdelmalek Abdesselam | @Peter: Just a follow up on Carlo's suggestion. I don't think you want to compute the integral of exp sum of traces as in your last comment. $\int\exp(t\ {\rm tr}(A^{-1}HDH^{\dagger}))dH$ is enough, then take derivatives in $t$. That's the famous Harish-Chandra-Itzykson-Zuber integral. See e.g. terrytao.wordpress.com/2013/02/08/… | |
| Jan 28, 2015 at 22:45 | history | edited | Vít Tuček |
more appropriate tags
|
|
| Jan 28, 2015 at 21:10 | comment | added | Peter | Thanks a lot. In my Problem $A>D$, and the first few terms in the expansion may give a good approximate. I still do not know how to evaluate $\int \exp (\sum_{p}tr(A^{-1}HDH^{\dagger})^p) dH$. Can you please help me with that? | |
| Jan 28, 2015 at 20:04 | comment | added | Carlo Beenakker | there is no hope for an exact answer in closed form; if $D$ is small or $A$ is large, you can evaluate a few terms in the power series, of the form $\int {\rm tr}\,(A^{-1}HDH^\dagger)^p\,dH$, which can be done in closed form for small $p$ (but not for any $p$). | |
| Jan 28, 2015 at 18:55 | comment | added | Peter | Thanks, this is going to help for large $n$. But I need the exact answer. | |
| Jan 28, 2015 at 18:17 | comment | added | Yemon Choi | Do you need an exact answer or would large $n$ asymptotics suffice? In the second situation, in the case where $A$ and $D$ are diagonal with free entries, one might be able to use an approximation of the spectral distribution of $A-HDH^*$ ($H$ chosen uniformly at random) by the free convolution of $\mu_A$ with $\mu_D$, see e.g. the first few pages of arxiv.org/abs/math/9809193 | |
| Jan 28, 2015 at 17:36 | history | edited | Peter | CC BY-SA 3.0 |
deleted 55 characters in body
|
| Jan 28, 2015 at 17:28 | comment | added | Johannes Hahn | Wlog one can assume that $A$ is diagonal because the Haar measure is invariant under left- and rightmultiplication. | |
| Jan 28, 2015 at 17:23 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
added 12 characters in body
|
| Jan 28, 2015 at 17:22 | comment | added | Peter | Oops, Sorry for these typos! I edited them. Does it make sense now? Thanks. | |
| Jan 28, 2015 at 17:17 | history | edited | Peter | CC BY-SA 3.0 |
edited body
|
| Jan 28, 2015 at 12:58 | review | Close votes | |||
| Jan 28, 2015 at 19:37 | |||||
| Jan 28, 2015 at 12:08 | comment | added | Vít Tuček | First of all, the matrices should probably be in $\mathbb{C}^{n^2}$ not in $\mathbb{C}^n$. Second, what role does $D$ play? I don't see it in the integral, perhaps $D = L$? Third, what does $H'$ mean? Is it the transpose of $H$? Fourth, are the words "deterministic" really relevant here? | |
| S Jan 28, 2015 at 9:08 | history | edited | David Roberts♦ | CC BY-SA 3.0 |
removed shouting (and re-capitalised Haar)
|
| S Jan 28, 2015 at 9:08 | history | suggested | bummi | CC BY-SA 3.0 |
removed shouting
|
| Jan 28, 2015 at 9:06 | review | Suggested edits | |||
| S Jan 28, 2015 at 9:08 | |||||
| Jan 28, 2015 at 9:03 | history | asked | Peter | CC BY-SA 3.0 |