rigidity of eigenvalues of circular ensemble - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:30:06Z http://mathoverflow.net/feeds/question/84915 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84915/rigidity-of-eigenvalues-of-circular-ensemble rigidity of eigenvalues of circular ensemble John Jiang 2012-01-04T23:05:54Z 2012-01-04T23:05:54Z <p>Given a circular unitary ensemble, with the following joint density:</p> <p>$p(\theta_1,\ldots, \theta_n) = Z_n \prod_{j &lt; k} |e^{i \theta_j} - e^{i \theta_k}|^2$, </p> <p>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> <p>$P[\int_\alpha d(\{\theta_1, \ldots, \theta_n\}, \{0, 2\pi/n, \ldots, 2(n-1)\pi/n\} + \alpha (\text{mod }2\pi)) &lt; C] \to 1$ for some sufficiently large constant $C$?</p> <p>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. </p> <p>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.</p>