Inhomogeneous Poisson processes in the disk and interpolation

We investigate different geometrical properties of the inhomogeneous Poisson point process ${\mathrm{\Lambda }}_{\mu }$ associated to a positive, locally finite, $\sigma$-finite measure $\mu$ on the unit disk. In particular, we characterize the processes ${\mathrm{\Lambda }}_{\mu }$ such that almost surely: 1) ${\mathrm{\Lambda }}_{\mu }$ is a Carleson-Newman sequence; 2) ${\mathrm{\Lambda }}_{\mu }$ 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 ${\mathrm{\Lambda }}_{\mu }$ is almost surely an interpolating sequence for the Hardy, Bloch or weighted Dirichlet spaces.