Point processes and interpolation

The Kotelnikov theorem recovers a Paley-Wiener function from its restriction onto an arithmetic progression. A Paley-Wiener function can also be recovered from its restriction onto a realization of the sine-process with one particle removed. If no particles are removed, then the possibility of such interpolation for the sine-process is due to Ghosh, for general determinantal point processes governed by orthogonal projections, to Qiu, Shamov and the speaker. If two particles are removed, then there exists a nonzero Paley-Wiener function vanishing at all the remaining particles.
How explicitly to interpolate a function belonging to Hilbert space that admits a reproducing kernel, given the restriction of our function onto a realization of the determinantal pont process governed by the kernel? In the case of the zero set of the Gaussian analytic function, or, in other words, the determinantal point process governed by the Bergman kernel, in joint work with Qiu, the Patterson-Sullivan construction is used for uniform interpolation in dense subspaces of the Bergman space. The invariance of our point process under Lobachevskian isometries plays a key role.
For the sine-process, the Ginibre process, the determinantal point process with the Bessel kernel and the determinantal point process with the Airy kernel, A.A. Borichev, A.V. Klimenko and the speaker proved that if the function decays as a sufficiently high negative power of the distance to the origin, then the answer is given by the Lagrange interpolation formula.