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 \(\mathbb {D}\).