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

Abstract

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