Self-similarity in the circular unitary ensemble

TL;DR

This paper rigorously proves a conjectured self-similarity property of eigenvalues in the Circular Unitary Ensemble, showing that eigenvalues of different matrix sizes are statistically indistinguishable at certain scales.

Contribution

It provides the first rigorous proof of the self-similarity conjecture for CUE eigenvalues using a novel comparison theorem for determinantal point processes.

Findings

Eigenvalues of large CUE matrices exhibit self-similarity at mesoscopic scales.

Eigenvalues of smaller and larger CUE matrices are statistically indistinguishable after appropriate scaling.

The proof introduces a new comparison theorem for determinantal point processes.

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