  The unitary group with the Haar probability measure is called Circular
Unitary Ensemble. All the eigenvalues lie on the unit circle in the complex
plane and they can be regarded as a determinantal point process on
$\mathbb{S}^1$. It is also known that the scaled point processes converge
weakly to the determinantal point process associated with the so-called sine
kernel as the size of matrices tends to $\infty$. We extend this result to the
case of high-dimensional spheres and show that the scaling limit processes are
determinantal point processes associated with the kernels expressed by the
Bessel functions of the first kind.
