Separation properties of a hybrid point process with determinantal radii and uniform arguments
Giuseppe Lamberti, Xavier Massaneda
TL;DR
This work studies a hybrid point process Λ_φ^M on the plane, combining radii from a radial determinantal process Λ_φ with independently uniform angular components. The authors establish a sharp 0-1 law: Λ_φ^M is almost surely separated exactly when the first intensity satisfies the Poisson-type condition ∫_{ℂ} dm(z)/ρ^4(z) < ∞, mirroring the Poisson case rather than the determinantal one. The analysis reduces to a discretized partition of the plane into annuli and angular sectors, and uses probabilistic tools (Borel-Cantelli, Chernoff bounds, Le Cam’s Poisson approximation) to connect separation to the divergence or convergence of ∑_{n} μ_n^2/n, where μ_n ≍ ∫_{I_n} dm(z)/ρ^2(z). The results illuminate the differing roles of radial and angular components in separation and provide precise criteria for the hybrid model that interpolate between determinantal and Poisson behavior. Applications include understanding interpolation and sampling phenomena for associated Fock spaces with radial weights.
Abstract
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.