Separated determinantal point processes and generalized Fock spaces

We study conditions so that the determinantal point process ${\mathrm{\Lambda }}_{\phi }$ 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 ${\mathrm{\Lambda }}_{\phi }$ with the Poisson process of the same first intensity. As an application, we show that the determinantal process ${\mathrm{\Lambda }}_{\alpha }$ associated to the canonical weight ${\phi }_{\alpha }(z)=\mathrm{\mid }z{\mathrm{\mid }}^{\alpha }$, $\alpha >0$, is almost surely separated if and only if $\alpha <4\mathrm{/}3$. In contrast, the Poisson process ${\mathrm{\Lambda }}_{\alpha }^P$ having the same first intensity as ${\mathrm{\Lambda }}_{\alpha }$ is almost surely separated if and only if $\alpha <1$.