Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,862 answers , 20,637 comments
1,470 users with positive rep
502 active unimported users
More ...

Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

+ 10 like - 0 dislike
1903 views

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and $\bar{U}$ its complex conjugate, on the one hand, and the random orthogonal matrix $O$ with determinant $-1$, on the other hand. The random matrices $U$ and $O$ are uniformly distributed with respect to the Haar measure on the unitary group $U(N)$ and the orthogonal group $O(N+1)$.

One eigenvalue of $O$ is fixed at $-1$ (to ensure that det $O=-1$). The other $N$ eigenvalues have a distribution $p_O$ which was known since Girko (1985). We calculated the distribution $p_{U\bar{U}}$ of the eigenvalues of $U\bar{U}$ (which we did not find in the literature --- has anyone seen it before?). We discovered to our surprise that $p_{U\bar{U}}=p_O$. This holds for both $N$ even and odd (in the latter case both $U\bar{U}$ and $O$ have an eigenvalue fixed at $+1$).

Question: Is there a more direct route to arrive at this identity between the two eigenvalue distributions, without going through a separate calculation of each one? (You can find two such separate calculations in the Appendix of arXiv:1206.6687, but this seems a rather unsatisfactory way of understanding the correspondence.)


Some intuition for what is going on: for both $U\bar{U}$ and $O$ the eigenvalues different from $\pm 1$ come in complex conjugate pairs $e^{\pm i\theta}$. The matrix $O$ has an unpaired eigenvalue at $-1$, which repels $\theta$ from $\pi$. The matrix $U\bar{U}$ cannot have an unpaired eigenvalue at $-1$ by construction and somehow this leads to a repulsion of $\theta$ from $\pi$ which is mathematically equivalent to what happens for the matrix $O$.

For example, when $N=2$ the eigenvalue $e^{i\theta}$ has the same distribution $P(\theta)=(2\pi)^{-1}(1+\cos\theta)$ for both $U\bar{U}$ and $O$. For $N=3$ the distribution is $P(\theta)=\pi^{-1}(1-\cos^2\theta)$, in addition to an eigenvalue fixed at $+1$, again the same for $U\bar{U}$ and $O$. The correspondence continues for larger $N$, when factors $(\cos\theta_k-\cos\theta_l)^2$ appear in both distributions.

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Carlo Beenakker
asked Aug 26, 2012 in Mathematics by Carlo Beenakker (180 points) [ no revision ]
retagged Oct 21, 2014
Most voted comments show all comments
well, $U\bar{U}$ is not itself orthogonal (its inverse is not equal to its transpose); I can construct an orthogonal matrix $O$ with the same eigenvalues as $U\bar{U}$, but this orthogonal matrix is not uniformly distributed, so that does not seem to help much.

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Carlo Beenakker
argh...i made stupid calculation error!

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Suvrit
There is an identity for the eigenvalue distributions in this paper: web.williams.edu/go/math/sjmiller/public_html/ntandrmt/handouts/… (see page 6) Maybe this would help. It is also a known fact that the orthogonal group is similar to a standard normal distribution: www-stat.stanford.edu/~cgates/PERSI/papers/random_matrices.pdf (see page 56). Thus it is possible to solve equality of these two distributions to obtain the answer to your question.

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Jaivir Baweja
indeed, the eigenvalue distributions of the classical compact groups are related, notably $p_O$ for $N$ even is also the eigenvalue distribution of the compact symplectic group; the matrices of the form $U\bar{U}$ do not form a group, so this does not seem to help much; note also that the correspondence between $p_{U\bar{U}}$ and $p_O$ is not an asymptotic large-$N$ result (like the correspondence to a normal distribution): it holds exactly for any finite $N$.

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Carlo Beenakker
Is the observation that $U\bar{U}$ is a unitary symmetric matrix (and every symmetric unitary matrix $S$ can be written as such a product) of value to you here (I say that because such matrices seem to be used when studying orthogonal ensembles...)

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Suvrit
@Suvrit, $U\bar{U}$ is not a symmetric matrix, meaning it is not equal to its transpose; the probability distribution of unitary symmetric matrices is indeed well known, it is the circular orthogonal ensemble (COE) of random matrix theory; this is an unfortunate name, because it can create a confusion with the ensemble of random orthogonal matrices, but these are entirely different objects.

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Carlo Beenakker
@Carlo: thanks for the clarification. Time for me to go and read a book on this subject before further commenting :-) (I think I made all my $U\bar{U}$ related errors so far because of "wrongly" simulating a random unitary matrix. I was simulating random symmetric unitaries, not arbitrary ones). I still believe that there should be a slick solution to your problem!

This post imported from StackExchange MathOverflow at 2014-10-21 10:54 (UTC), posted by SE-user Suvrit

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverf$\varnothing$ow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...