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.



rigidity of eigenvalues of circular ensemble
If I understand correctly, what you are looking for is Lemma 10 (when m=1) in <http://arxiv.org/pdf/1210.2681v3.pdf> by Elisabeth and Mark Meckes. 


