Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

Question

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.)

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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

Comments

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

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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

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@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

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@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

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@Z254R: as far as I know, the Weingarten formula gives averages of polynomials in the matrix elements of $U$ or $O$; that does not seem a viable route to find the distribution of the eigenvalues of $U\bar{U}$, nor to show that the eigenvalues of $U\bar{U}$ and $O$ have the same distribution.

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

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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

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argh...i made stupid calculation error!

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