We consider determinantal point processes $\Lambda_\phi$ arising from generalized Fock spaces $\mathcal F_\phi$, defined by a doubling subharmonic weight $\phi$. We provide a complete characterization of when these processes are almost surely separated with respect to the euclidean metric on the plane. As a consequence, we also obtain a characterization of when these processes are almost surely interpolating for the classical Fock spaces. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\Lambda_\phi$ with the Poisson process with the same first intensity. Additionally, we demonstrated that, with probability one, the point process $\Lambda_\phi$ fails to be separated with respect to the distance induced by the reproducing kernel $K_\phi$.

This is a joint work with X. Massaneda.