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Given a circular unitary ensemble, with the following joint density:

$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j < k} |e^{i \theta_j} - e^{i \theta_k}|^2$,

is the following statement true? With high probability the eigenvalues are within distance $\mathcal{O}(1)$ from the evenly spaced set of $n$ points $(0,2\pi/n, 4\pi/n, \ldots, 2(n-1)\pi/n)$, rotated by some angle $\theta$. More precisely, is it true that

$ P[\int_\alpha d(\{\theta_1, \ldots, \theta_n\}, \{0, 2\pi/n, \ldots, 2(n-1)\pi/n\} + \alpha (\text{mod }2\pi)) < C] \to 1$ for some sufficiently large constant $C$?

Here the distance is the induced Riemannian distance on $\mathbb{T}^n/ S_n$, where the action of $S_n$ on $\mathbb{T}^n$ is permutation of the coordinates.

I know Erdos Schlein Yau have proved a rigidity theorem for Wigner ensembles, but their result is slightly weaker than what I need. It seems natural to investigate this question for the exactly solvable case.

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up vote 2 down vote accepted

If I understand correctly, what you are looking for is Lemma 10 (when m=1) in by Elisabeth and Mark Meckes.

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Thanks a lot Adrien! If you are interested I hope to talk to you more about related problems. – John Jiang Dec 1 '14 at 21:06
@JohnJiang You're the welcome. I don't know what you have in mind, but I think this type of questions is interesting and I'd be glad to discuss more about it; please write to me at ahardy(you know what) – Adrien Hardy Dec 2 '14 at 20:02

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