Truncations of Random Unitary Matrices Drawn from Hua-Pickrell Distribution

Let U be a random unitary matrix drawn from the Hua-Pickrell distribution mu(U)(n+m)((delta)) on the unitary group U(n + m). We show that the eigenvalues of the truncated unitary matrix [U-i,U- j](1 <= i, j <= n) form a determinantal point process X-n((m,delta)) non the unit disc D for any delta is an element of C satisfying Re delta > -1/2. We also prove that the limiting point process taken by n -> infinity of the determinantal point process X(n)((m,delta) )is always X-[m], independent of delta. Here X[m] is the determinantal point process on 13 with weighted Bergman kernel K-[m](z, w) = 1/(1 - z <(w)over bar >)(m+1) with respect to the reference measure d mu([m])(z) = m/pi (1 - |z|)(m-1)d sigma (z), where d sigma (z) is the Lebesgue measure on D.