Separated determinantal point processes and generalized Fock spaces
Giuseppe Lamberti, Xavier Massaneda
TL;DR
The paper establishes a sharp criterion for when the determinantal point process Λ_φ, arising from a generalized Fock space 𝔉_φ with a doubling subharmonic weight φ, is almost surely separated, linking separation to the integrability of 1/ρ^6 over ℂ where ρ(z) encodes Δφ. The authors develop a cell-partition analysis and a key proposition showing that the probability of two points landing in the same cell scales as ρ(z)^{-6}, enabling a Borel–Cantelli dichotomy that yields a precise separation condition and zero upper density when satisfied. They also compare intrinsic repulsion in determinantal processes to a Poisson process with the same first intensity, showing that separation for the Poisson model requires the weaker ρ^{-4} integrability; canonical weights φ_α yield separation iff α<4/3 for Λ_α and α<1 for Λ_α^P, respectively, with implications for interpolating sequences in classical Fock spaces. The results illuminate the balance between repulsion and density and have consequences for interpolation problems in Fock-type spaces and related analytic frameworks.
Abstract
We study conditions so that the determinantal point process $Λ_φ$ associated to a generalized Fock space defined by a doubling subharmonic weight $φ$ is almost surely a separated sequence in $\mathbb C$. Under a natural assumption on $φ$, we provide a characterization of such processes. Additionally, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $Λ_φ$ with the Poisson process of the same first intensity. As an application, we show that the determinantal process $Λ_α$ associated to the canonical weight $φ_α(z)=|z|^α$, $α>0$, is almost surely separated if and only if $α<4/3$. In contrast, the Poisson process $Λ_α^P$ having the same first intensity as $Λ_α$ is almost surely separated if and only if $α<1$.