Truncations of random unitary matrices drawn from Hua-Pickrell distribution

TL;DR

This paper studies the eigenvalues of truncated random unitary matrices from the Hua-Pickrell distribution, showing they form a determinantal point process on the unit disk with a universal limit independent of the distribution parameter.

Contribution

It establishes that the eigenvalues form a determinantal point process and proves the universal limiting process as matrix size grows, regardless of the Hua-Pickrell parameter.

Findings

Eigenvalues form a determinantal point process on the unit disk.

The limiting process is independent of the Hua-Pickrell parameter.

Explicit kernel for the limiting process is derived.

Abstract

Let $U$ be a random unitary matrix drawn from the Hua-Pickrell distribution $\mu_{\mathrm{U}(n+m)}^{(\delta)}$ on the unitary group $\mathrm{U}(n+m)$. We show that the eigenvalues of the truncated unitary matrix $[U_{i,j}]_{1\leq i,j\leq n}$ form a determinantal point process $\mathscr{X}_n^{(m,\delta)}$ on the unit disc $\mathbb{D}$ for any $\delta\in\mathbb{C}$ satisfying $\mathrm{Re}\,\delta>-1/2$. We also prove that the limiting point process taken by $n\to\infty$ of the determinantal point process $\mathscr{X}_n^{(m,\delta)}$ is always $\mathscr{X}^{[m]}$, independent of $\delta$. Here $\mathscr{X}^{[m]}$ is the determinantal point process on $\mathbb{D}$ with weighted Bergman kernel \begin{equation*} \begin{split} K^{[m]}(z,w)=\frac{1}{(1-z\overline w)^{m+1}} \end{split} \end{equation*} with respect to the reference measure $d\mu^{[m]}(z)=\frac{m}{\pi}(1-|z|)^{m-1}d\sigma(z)$, where $d\sigma(z)$ is the Lebesgue measure on $\mathbb{D}$.