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Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group.

For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the unit circle.

Now, we are interested in the probability density function of $\Omega(U)$ as $U$ being distributed as Haar measure.

What is the probability that $\Omega(U)$ is smaller than some given constant $\epsilon$?

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This amounts to the question what is the probability distribution of the largest gap $\Delta=2\pi-\Omega$ between the eigenphases in the circular unitary ensemble. It was addressed in Extreme gaps between eigenvalues of random matrices (2010). Asymptotically for large $d$, the distribution $P(\Delta)$ is peaked around $\Delta_0=d^{-1}\sqrt{32\log d}$, with Poissonian fluctuations around this value.

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  • $\begingroup$ Thanks! Is there explicit probability density function? $\endgroup$
    – gondolf
    May 3, 2016 at 13:51
  • $\begingroup$ it's like all spacing distributions in random matrix theory, you only get explicit results asymptotically for large dimensionality. $\endgroup$
    – Carlo Beenakker
    May 3, 2016 at 14:35
  • $\begingroup$ Thank you! Just would like to make sure that the " Extreme gaps" is indeed "the length of the smallest arc containing all the eigenvalues”？ $\endgroup$
    – gondolf
    May 3, 2016 at 17:30
  • $\begingroup$ "extreme gap" is "length of largest arc containing no eigenvalues", so that's the $2\pi$ minus the length of the smallest arc containing all eigenvalues, isn't it? $\endgroup$
    – Carlo Beenakker
    May 3, 2016 at 19:37
  • $\begingroup$ Very promising! $\endgroup$
    – gondolf
    May 3, 2016 at 21:03

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