Inhomogeneous Poisson processes in the disk and interpolation

We investigate different geometrical properties of the inhomogeneous Poisson point process $\Lambda_{\mu}$ associated to a positive, locally finite, $\sigma$-finite measure $\mu$ on the unit disk. In particular, we characterize the processes $\Lambda_{\mu}$ such that almost surely: 1) $\Lambda_{\mu}$ is a Carleson-Newman sequence; 2) $\Lambda_{\mu}$ is the union of a given number M of separated sequences. We use these results to discuss the measures $\mu$ such that the associated process $\Lambda_{\mu}$ is almost surely an interpolating sequence for the Hardy, Bloch or weighted Dirichlet spaces.