Separated determinantal point processes and generalized Fock spaces

Abstract

We study conditions so that the determinantal point process $\hat{I}{›}_{\ddot{I}}•$ associated to a generalized Fock space defined by a doubling subharmonic weight $\ddot{I}•$ is almost surely a separated sequence in $\mathbb{C}$. Under a natural assumption on $\ddot{I}•$, we provide a characterization of such processes. Additionally, we emphasize the role of intrinsic repulsion in determinantal processes by comparing $\hat{I}{›}_{\ddot{I}}•$ with the Poisson process of the same first intensity. As an application, we show that the determinantal process $\hat{I}{›}_{\hat{I}}\pm$ associated to the canonical weight $\ddot{I}{•}_{\hat{I}}\pm (z)=|z{|}^{\hat{I}}\pm$, $\hat{I}\pm >0$, is almost surely separated if and only if $\hat{I}\pm <4/3$. In contrast, the Poisson process $\hat{I}{›}_{\hat{I}}{\pm }^P$ having the same first intensity as $\hat{I}{›}_{\hat{I}}\pm$ is almost surely separated if and only if $\hat{I}\pm <1$.