Mixing
Real version of Harish-Chandra-Itzykson-Zuber integral
$begingroup$
I'm interested in an integral of the form
$$
int_{O(d)} expleft(-frac{1}{2}mathrm{trace}(CUAU^T)right)dU
$$
where the integration is with respect to the Haar measure on the orthogonal group, i.e., uniform measure on real $UU^T=U^TU=I$, where $A$ is a diagonal matrix, and let's say for simplicity $C$ is symmetric (and hence might as well be diagonal too, WLOG).
Following a suggestion from a previous question about this, I found this note on arxiv which derives an approximation, assuming the the parameters are rational.
Do you know of any other references for this? I get that there are no nice closed form solutions unlike the unitary case, but something exact?
Thanks.
integration lie-groups lie-algebras random-matrices
edited Apr 13 '17 at 12:20
Community♦
1
asked Jun 19 '15 at 1:24
martinmartin
263
$endgroup$
add a comment |
$begingroup$
I'm interested in an integral of the form
$$
int_{O(d)} expleft(-frac{1}{2}mathrm{trace}(CUAU^T)right)dU
$$
where the integration is with respect to the Haar measure on the orthogonal group, i.e., uniform measure on real $UU^T=U^TU=I$, where $A$ is a diagonal matrix, and let's say for simplicity $C$ is symmetric (and hence might as well be diagonal too, WLOG).
Following a suggestion from a previous question about this, I found this note on arxiv which derives an approximation, assuming the the parameters are rational.
Do you know of any other references for this? I get that there are no nice closed form solutions unlike the unitary case, but something exact?
Thanks.
integration lie-groups lie-algebras random-matrices
edited Apr 13 '17 at 12:20
Community♦
1
asked Jun 19 '15 at 1:24
martinmartin
263
$endgroup$
add a comment |
$begingroup$
I'm interested in an integral of the form
$$
int_{O(d)} expleft(-frac{1}{2}mathrm{trace}(CUAU^T)right)dU
$$
where the integration is with respect to the Haar measure on the orthogonal group, i.e., uniform measure on real $UU^T=U^TU=I$, where $A$ is a diagonal matrix, and let's say for simplicity $C$ is symmetric (and hence might as well be diagonal too, WLOG).
Following a suggestion from a previous question about this, I found this note on arxiv which derives an approximation, assuming the the parameters are rational.
Do you know of any other references for this? I get that there are no nice closed form solutions unlike the unitary case, but something exact?
Thanks.
integration lie-groups lie-algebras random-matrices
edited Apr 13 '17 at 12:20
Community♦
1
asked Jun 19 '15 at 1:24
martinmartin
263
$endgroup$
I'm interested in an integral of the form
$$
int_{O(d)} expleft(-frac{1}{2}mathrm{trace}(CUAU^T)right)dU
$$
where the integration is with respect to the Haar measure on the orthogonal group, i.e., uniform measure on real $UU^T=U^TU=I$, where $A$ is a diagonal matrix, and let's say for simplicity $C$ is symmetric (and hence might as well be diagonal too, WLOG).
Following a suggestion from a previous question about this, I found this note on arxiv which derives an approximation, assuming the the parameters are rational.
Do you know of any other references for this? I get that there are no nice closed form solutions unlike the unitary case, but something exact?
Thanks.
integration lie-groups lie-algebras random-matrices
integration lie-groups lie-algebras random-matrices
edited Apr 13 '17 at 12:20
Community♦
1
asked Jun 19 '15 at 1:24
martinmartin
263
edited Apr 13 '17 at 12:20
Community♦
1
asked Jun 19 '15 at 1:24
martinmartin
263
edited Apr 13 '17 at 12:20
Community♦
1
edited Apr 13 '17 at 12:20
Community♦
1
edited Apr 13 '17 at 12:20
Community♦
1
1
asked Jun 19 '15 at 1:24
martinmartin
263
asked Jun 19 '15 at 1:24
martinmartin
263
asked Jun 19 '15 at 1:24
martinmartin
263
263
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.
answered Jan 23 at 11:25
erachangerachang
1109
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1330951%2freal-version-of-harish-chandra-itzykson-zuber-integral%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.
answered Jan 23 at 11:25
erachangerachang
1109
$endgroup$
add a comment |
$begingroup$
It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.
answered Jan 23 at 11:25
erachangerachang
1109
$endgroup$
add a comment |
$begingroup$
It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.
answered Jan 23 at 11:25
erachangerachang
1109
$endgroup$
It seems that the $O(n)$ case does not give a determinantal form. However the fact that the Lie algebra of $O(n)$ is the set of $n×n$ anti-symmetric matrices gives similar integral formulas for anti-symmetric matrices. See (1.3) and (1.4) in this paper.
answered Jan 23 at 11:25
erachangerachang
1109
answered Jan 23 at 11:25
erachangerachang
1109
answered Jan 23 at 11:25
erachangerachang
1109
answered Jan 23 at 11:25
erachangerachang
1109
1109
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1330951%2freal-version-of-harish-chandra-itzykson-zuber-integral%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
