Self-similarity in the circular unitary ensemble (Q2826227)

The authors give a rigorous proof of a conjectured statistical self-similarity property of the eigenvalues random matrices from the circular unitary ensemble (CUE) (cf. [\textit{M. Coram} and \textit{P. Diaconis}, J. Phys. A, Math. Gen. 36, No. 12, 2883--2906 (2003; Zbl 1074.11046)]). They consider on the one hand the eigenvalues of an \(n\times n\) CUE matrix, and on the other hand those eigenvalues \(e^{i\phi}\) of an \(mn\times mn\) CUE matrix with \(|\phi|\leq\pi /m\), rescaled to fill the unit circle. They 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.