Separation properties of a hybrid point process with determinantal radii and uniform arguments

We recently characterized the separated determinantal point processes $\Lambda_\phi$ associated with Fock spaces $\mathcal F_\phi$ in the plane with doubling weight $\phi$. We also showed that, as expected, a more restrictive condition is required to characterize the separated Poisson processes with the same first intensities as $\Lambda_\phi$. To gain further insight into this different behavior, we center our attention to radial weights $\phi(z)$ and introduce a hybrid process $\Lambda_\phi^M=\{r_k e^{i\theta_k}\}_{k=1}^\infty$, where the moduli $r_k$ are taken from $\Lambda_\phi$, while the arguments $\theta_k$ are chosen independently and uniformly in $[0,2\pi)$. Our main result is that $\Lambda_\phi^M$ is almost surely separated if and only if its first intensity satisfies the same condition as in the Poisson case.