How fast does spectral radius of truncated circular unitary ensemble converge?

Abstract

Let $z_1, \cdots, z_p$ be the eigenvalues of $A,$ which is the left-top $p\times p$ submatrix of an $n\times n$ Haar-invariant unitary matrix. Suppose there exist two constants $0<h_1<h_2<1$ such that $h_1<\frac pn<h_2.$ Then, $$\sup_{x\in \mathbb{R}}|\mathbb{P}(X_n\le x)-e^{-e^{-x}}|=\frac{(\log \log n)^{2}}{2e\log n}(1+o(1))$$ and further $$ W_{1}\left(\mathcal{L}(X_n),\Lambda\right)=\frac{(\log\log n)^2}{2\log n}(1+o(1))$$ for $n$ large enough. Here, $\Lambda$ is the Gumbel distribution and $\mathcal{L}(X_n)$ is the distribution of $X_n$ with $X_n$ being some rescaled version of $\max_{1\le i\le p}|z_i|,$ the spectral radius of $A.$