Separated determinantal point processes and generalized Fock spaces

TL;DR

This paper characterizes when determinantal point processes from generalized Fock spaces are almost surely separated, highlighting the role of intrinsic repulsion and providing specific conditions for weights like lpha.

Contribution

It offers a new characterization of separated determinantal processes associated with generalized Fock spaces under certain conditions.

Findings

Determinantal process lpha is separated iff lpha<4/3.

Poisson process with same intensity is separated iff lpha<1.

Intrinsic repulsion influences separation properties of these processes.

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 $\phi_\alpha(z)=|z|^\alpha$, $\alpha>0$, 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 $\alpha<1$.