Separated determinantal point processes and generalized Fock spaces

Abstract: We study conditions so that the determinantal point process $\Lambda_\phi$ associated to a generalized Fock space defined by a doubling subharmonic weight $\phi$ is almost surely a separated sequence in $\mathbb C$. Under a natural assumption on $\phi$, we provide a characterization of such processes. Additionally, 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 to the canonical weight $\phi_\alpha(z)=|z|<sup>\alpha$, $\alpha&gt;0$, is almost surely separated if and only if $\alpha&lt;4/3$. In contrast, the Poisson process $\Lambda_\alpha<sup>P$ having the same first intensity as $\Lambda_\alpha$ is almost surely separated if and only if $\alpha&lt;1$.