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 $\epsilon$?