Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution

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