On the eigenvalues' distribution of random unitary

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 $\epsilon$?

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.
• "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? – Carlo Beenakker May 3 '16 at 19:37