Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution

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