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 $h_1<\frac{p}{n}<h_2.$ Then, $$\underset{x\in \mathbb{R}}{\sup }|\mathbb{P}(X_n\le x)-e^{-e^{-x}}|=\frac{(\log \log n{)}^2}{2e\log n}(1+o(1))$$ 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 $A.$