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

TL;DR

This paper investigates the convergence rate of the spectral radius of a truncated circular unitary ensemble to the Gumbel distribution, providing precise asymptotic bounds for large matrix sizes.

Contribution

It derives explicit convergence rates for the spectral radius's distribution to the Gumbel law in the context of truncated Haar-invariant unitary matrices.

Findings

Convergence rate of spectral radius distribution to Gumbel law is quantified.

Asymptotic bounds involve logarithmic factors and are valid for large matrix sizes.

Results deepen understanding of spectral properties of truncated unitary matrices.

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 $A.$