Coram–Diaconis self-similarity conjecture for CUE eigenvalues

Let $U$ be an $N\times N$ random matrix from the circular unitary ensemble with eigenvalues $\{e^{i\theta_j}\}_{1\le j\le N}$, where $0\le \theta_1\le\dots\le\theta_N<2\pi$. Choose an eigenvalue $e^{i\theta_K}$ uniformly, and let $T$ be the length of the counter-clockwise circular arc from $\theta_K$ to $\theta_{K+k}$, with indices interpreted modulo $N$. Let $\phi\in[0,2\pi)$ be a uniformly chosen random angle independent of $U$. Coram–Diaconis's conjecture. If $k$ and $N$ are both large, then the random set

$$\left\{e^{i\left(\phi+\frac{2\pi\theta_j}{T}\right)}\right\}_{K\le j<K+k}$$

is statistically indistinguishable from the eigenvalues of a $k\times k$ random matrix from the circular unitary ensemble. This folklore conjecture motivates the paper's rigorous results on statistical self-similarity, while the precise asymptotic formulation and meaning of “statistically indistinguishable” require quantitative interpretation.