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

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