Separated Determinantal Point Processes and Generalized Fock Spaces

Abstract Let $\Lambda _\phi $ denote the determinantal point processes associated with the generalized Fock space defined by a doubling subharmonic weight $\phi $. We provide a characterization of the weights $\phi $ such that the process $\Lambda _\phi $ forms a separated sequence in $\mathbb{C}$ almost surely. Furthermore, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\Lambda _\phi $ with the Poisson process of the same first intensity. As an application, we show that the determinantal process $\Lambda _\alpha $ associated with the canonical weight $\phi _\alpha (z)=|z|^\alpha $, $\alpha>0$, is almost surely separated if and only if $\alpha <4/3$. In contrast, the Poisson process $\Lambda _\alpha ^{P}$ having the same first intensity as $\Lambda _\alpha $ is almost surely separated if and only if $\alpha <1$.