Universality conjecture for the edge spectrum of truncated unitary matrices

Let ${\mathcal{Q}}(\bm \phi)$ be the ensemble of matrices considered in the paper, with eigenvalues $\lambda_i(\bm \phi)$, and let $\rho(x)$ denote their local eigenvalue density. Let $\rho_{\scriptscriptstyle\mathrm{TrUE}}(x)$ be the eigenvalue density for the corresponding ensemble of truncated unitary matrices with invariant measure. For $n=t\sqrt{N}$, let

$$\mathcal{K}(t):=2^n\sqrt{N}K(t\sqrt{N},N),\qquad \mathcal{K}_{\scriptscriptstyle\mathrm{TrUE}}(t):=2^n\sqrt{N}K_{\scriptscriptstyle\mathrm{TrUE}}(t\sqrt{N},N)$$

be the rescaled spectral form factors. Universality conjecture. The local density is universal on the scale $1/\sqrt{N}$ around $1/2$: for fixed $s$,

$$\Delta\rho(s):=\rho\left(\frac{1}{2}-\frac{s}{\sqrt{N}}\right)-\rho_{\scriptscriptstyle\mathrm{TrUE}}\left(\frac{1}{2}-\frac{s}{\sqrt{N}}\right)=O\left(N^{-\frac{1}{2}}\right).$$

Moreover, for fixed $t$ and $N\to\infty$,

$$\Delta\mathcal{K}(t):=\mathcal{K}(t)-\mathcal{K}_{\scriptscriptstyle\mathrm{TrUE}}(t)=O\left(N^{-\frac{1}{2}}\right).$$

The factor $2^n$ corresponds to multiplying each eigenvalue by $\sqrt{2}$ and hence moving the spectral edge to $1$. The conjecture asserts universality of both the eigenvalue density and spectral correlations near the edge between the flat ensemble of ${\mathcal{Q}}(\bm \phi)$ and the truncated CUE ensemble; the stated estimates are based on numerical simulations, and no resolution is supplied here.