This paper gives a rigorous proof of a conjectured statistical self-similarity property of the eigenvalues random matrices from the Circular Unitary Ensemble. We consider on the one hand the eigenvalues of an $n \times n$ CUE matrix, and on the other hand those eigenvalues $e^{iϕ}$ of an $mn \times mn$ CUE matrix with $|ϕ| \le π/ m$, rescaled to fill the unit circle. We show that for a large range of mesoscopic scales, these collections of points are statistically indistinguishable for large $n$. The proof is based on a comparison theorem for determinantal point processes which may be of independent interest.

v4: published version reformatted with journal's style file