Self-similarity in the circular unitary ensemble

Abstract

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 $e^{i\varphi }$ of an $mn\times mn$ CUE matrix with $|\varphi |\le \pi /m$, 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.

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Cited by in corpus (1)

• Operator limit of the circular beta ensemble