Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution

Abstract

Let U be a random unitary matrix drawn from the Hua-Pickrell distribution $\mu _{\textrm{U}(n+m)}^{(\delta )}$ on the unitary group $\textrm{U}(n+m)$. We show that the eigenvalues of the truncated unitary matrix $[U_{i,j}]_{1\le i,j\le n}$ form a determinantal point process $\mathscr {X}_n^{(m,\delta )}$ on the unit disc $\mathbb {D}$ for any $\delta \in \mathbb {C}$ satisfying $\textrm{Re}\,\delta >-1/2$. We also prove that the limiting point process taken by $n\rightarrow \infty$ 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

$$\begin{aligned} \begin{aligned} K^{[m]}(z,w)=\frac{1}{(1-z{\overline{w}})^{m+1}} \end{aligned} \end{aligned}$$

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 $\mathbb {D}$.