Truncations of random unitary matrices drawn from Hua-Pickrell distribution

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 $[U_{i,j}{]}_{1\le i,j\le n}$ form a determinantal point process ${\mathcal{X}}_n^{(m,\delta )}$ 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 ${\mathcal{X}}_n^{(m,\delta )}$ is always ${\mathcal{X}}^{[m]}$, independent of $\delta$. Here ${\mathcal{X}}^{[m]}$ 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 $\mathbb{D}$.

20 pages