Truncations of random unitary matrices drawn from Hua-Pickrell distribution

Let $U$ be a random unitary matrix drawn from the Hua-Pickrell distribution $μ_{\mathrm{U}(n+m)}^{(δ)}$ 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 $\mathscr{X}_n^{(m,δ)}$ on the unit disc $\mathbb{D}$ for any $δ\in\mathbb{C}$ satisfying $\mathrm{Re}\,δ>-1/2$. We also prove that the limiting point process taken by $n\to\infty$ of the determinantal point process $\mathscr{X}_n^{(m,δ)}$ is always $\mathscr{X}^{[m]}$, independent of $δ$. 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 $dμ^{[m]}(z)=\frac{m}π(1-|z|)^{m-1}dσ(z)$, where $dσ(z)$ is the Lebesgue measure on $\mathbb{D}$.

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