Separated determinantal point processes and generalized Fock spaces

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 $α<1$.