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

TL;DR

This paper investigates the separation properties of a hybrid point process combining determinantal radii with uniform arguments, revealing conditions under which it is almost surely separated, thus bridging determinantal and Poisson processes.

Contribution

It introduces a new hybrid point process with determinantal radii and uniform arguments, analyzing its separation properties and linking them to Poisson process conditions.

Findings

Hybrid process is almost surely separated under Poisson-like intensity conditions.

Separation properties depend on the first intensity of the process.

Provides insight into the behavior of determinantal versus Poisson processes.

Abstract

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.